Non-equilibrium work distributions from fluctuating lattice-Boltzmann model
NASA Astrophysics Data System (ADS)
Nasarayya Chari, S. Siva; Murthy, K. P. N.
2012-06-01
We switch a system from an equilibrium to a non-equilibrium state, by changing the value of a macroscopic control variable as per a specified protocol. The distribution of work performed during the process is obtained for various switching times. The free energy difference (ΔF) is determined from the work fluctuation relation. Some of the work values in the ensemble shall be less than ΔF. We term these as the second law violating switching. We employ fluctuating lattice-Boltzmann model to simulate a switching experiment on an ideal gas system. Our results show that, the probability of violation of second law increases as the switching time increases and in the reversible limit goes to one-half. We explain this result by invoking Callen-Welton theorem.
Boltzmann equation solver adapted to emergent chemical non-equilibrium
Birrell, Jeremiah; Wilkening, Jon; Rafelski, Johann
2015-01-15
We present a novel method to solve the spatially homogeneous and isotropic relativistic Boltzmann equation. We employ a basis set of orthogonal polynomials dynamically adapted to allow for emergence of chemical non-equilibrium. Two time dependent parameters characterize the set of orthogonal polynomials, the effective temperature T(t) and phase space occupation factor ϒ(t). In this first paper we address (effectively) massless fermions and derive dynamical equations for T(t) and ϒ(t) such that the zeroth order term of the basis alone captures the particle number density and energy density of each particle distribution. We validate our method and illustrate the reduced computational cost and the ability to easily represent final state chemical non-equilibrium by studying a model problem that is motivated by the physics of the neutrino freeze-out processes in the early Universe, where the essential physical characteristics include reheating from another disappearing particle component (e{sup ±}-annihilation)
Revised lattice Boltzmann model for traffic flow with equilibrium traffic pressure
NASA Astrophysics Data System (ADS)
Shi, Wei; Lu, Wei-Zhen; Xue, Yu; He, Hong-Di
2016-02-01
A revised lattice Boltzmann model concerning the equilibrium traffic pressure is proposed in this study to tackle the phase transition phenomena of traffic flow system. The traditional lattice Boltzmann model has limitation to investigate the complex traffic phase transitions due to its difficulty for modeling the equilibrium velocity distribution. Concerning this drawback, the equilibrium traffic pressure is taken into account to derive the equilibrium velocity distribution in the revised lattice Boltzmann model. In the proposed model, a three-dimensional velocity-space is assumed to determine the equilibrium velocity distribution functions and an alternative, new derivative approach is introduced to deduct the macroscopic equations with the first-order accuracy level from the lattice Boltzmann model. Based on the linear stability theory, the stability conditions of the corresponding macroscopic equations can be obtained. The outputs indicate that the stability curve is divided into three regions, i.e., the stable region, the neutral stability region, and the unstable region. In the stable region, small disturbance appears in the initial uniform flow and will vanish after long term evolution, while in the unstable region, the disturbance will be enlarged and finally leads to the traffic system entering the congested state. In the neutral stability region, small disturbance does not vanish with time and maintains its amplitude in the traffic system. Conclusively, the stability of traffic system is found to be enhanced as the equilibrium traffic pressure increases. Finally, the numerical outputs of the proposed model are found to be consistent with the recognized, theoretical results.
Exponential trend to equilibrium for the inelastic Boltzmann equation driven by a particle bath
NASA Astrophysics Data System (ADS)
Cañizo, José A.; Lods, Bertrand
2016-05-01
We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres (with constant restitution coefficient α \\in (0,1) ) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove that the solution to the associated initial-value problem converges exponentially fast towards the unique equilibrium solution. The proof combines a careful spectral analysis of the linearised semigroup as well as entropy estimates. The trend towards equilibrium holds in the weakly inelastic regime in which α is close to 1, and the rate of convergence is explicit and depends solely on the spectral gap of the elastic linear collision operator.
Stable Equilibrium Based on Lévy Statistics:A Linear Boltzmann Equation Approach
NASA Astrophysics Data System (ADS)
Barkai, Eli
2004-06-01
To obtain further insight on possible power law generalizations of Boltzmann equilibrium concepts, we consider stochastic collision models. The models are a generalization of the Rayleigh collision model, for a heavy one dimensional particle M interacting with ideal gas particles with a mass m<< M. Similar to previous approaches we assume elastic, uncorrelated, and impulsive collisions. We let the bath particle velocity distribution function to be of general form, namely we do not postulate a specific form of power-law equilibrium. We show, under certain conditions, that the velocity distribution function of the heavy particle is Lévy stable, the Maxwellian distribution being a special case. We demonstrate our results with numerical examples. The relation of the power law equilibrium obtained here to thermodynamics is discussed. In particular we compare between two models: a thermodynamic and an energy scaling approaches. These models yield insight into questions like the meaning of temperature for power law equilibrium, and into the issue of the universality of the equilibrium (i.e., is the width of the generalized Maxwellian distribution functions obtained here, independent of coupling constant to the bath).
NASA Astrophysics Data System (ADS)
Asinari, P.
2011-03-01
Boltzmann equation is one the most powerful paradigms for explaining transport phenomena in fluids. Since early fifties, it received a lot of attention due to aerodynamic requirements for high altitude vehicles, vacuum technology requirements and nowadays, micro-electro-mechanical systems (MEMs). Because of the intrinsic mathematical complexity of the problem, Boltzmann himself started his work by considering first the case when the distribution function does not depend on space (homogeneous case), but only on time and the magnitude of the molecular velocity (isotropic collisional integral). The interest with regards to the homogeneous isotropic Boltzmann equation goes beyond simple dilute gases. In the so-called econophysics, a Boltzmann type model is sometimes introduced for studying the distribution of wealth in a simple market. Another recent application of the homogeneous isotropic Boltzmann equation is given by opinion formation modeling in quantitative sociology, also called socio-dynamics or sociophysics. The present work [1] aims to improve the deterministic method for solving homogenous isotropic Boltzmann equation proposed by Aristov [2] by two ideas: (a) the homogeneous isotropic problem is reformulated first in terms of particle kinetic energy (this allows one to ensure exact particle number and energy conservation during microscopic collisions) and (b) a DVM-like correction (where DVM stands for Discrete Velocity Model) is adopted for improving the relaxation rates (this allows one to satisfy exactly the conservation laws at macroscopic level, which is particularly important for describing the late dynamics in the relaxation towards the equilibrium).
Distributional Monte Carlo Methods for the Boltzmann Equation
2013-03-01
Examples of such violations arise in rarefied gas dynamics, hypersonic flows , and micro-scale flows . Additionally, there is an “equilibrium hypothesis...are rarefied flows and flows containing non-equilibrium phenomena. Applications of rarefied gas dynamics typically involve high-altitude flight and...1 1.1 Kinetic Theory and Rarefied Gas Dynamics . . . . . . . . . . . . . . . . . 3 1.2 Computational Methods for the Boltzmann equation
Liñán-García, Ernesto; Sánchez-Hernández, Juan Paulo; González-Barbosa, J. Javier; González-Flores, Carlos
2016-01-01
A new hybrid Multiphase Simulated Annealing Algorithm using Boltzmann and Bose-Einstein distributions (MPSABBE) is proposed. MPSABBE was designed for solving the Protein Folding Problem (PFP) instances. This new approach has four phases: (i) Multiquenching Phase (MQP), (ii) Boltzmann Annealing Phase (BAP), (iii) Bose-Einstein Annealing Phase (BEAP), and (iv) Dynamical Equilibrium Phase (DEP). BAP and BEAP are simulated annealing searching procedures based on Boltzmann and Bose-Einstein distributions, respectively. DEP is also a simulated annealing search procedure, which is applied at the final temperature of the fourth phase, which can be seen as a second Bose-Einstein phase. MQP is a search process that ranges from extremely high to high temperatures, applying a very fast cooling process, and is not very restrictive to accept new solutions. However, BAP and BEAP range from high to low and from low to very low temperatures, respectively. They are more restrictive for accepting new solutions. DEP uses a particular heuristic to detect the stochastic equilibrium by applying a least squares method during its execution. MPSABBE parameters are tuned with an analytical method, which considers the maximal and minimal deterioration of problem instances. MPSABBE was tested with several instances of PFP, showing that the use of both distributions is better than using only the Boltzmann distribution on the classical SA. PMID:27413369
Frausto-Solis, Juan; Liñán-García, Ernesto; Sánchez-Hernández, Juan Paulo; González-Barbosa, J Javier; González-Flores, Carlos; Castilla-Valdez, Guadalupe
2016-01-01
A new hybrid Multiphase Simulated Annealing Algorithm using Boltzmann and Bose-Einstein distributions (MPSABBE) is proposed. MPSABBE was designed for solving the Protein Folding Problem (PFP) instances. This new approach has four phases: (i) Multiquenching Phase (MQP), (ii) Boltzmann Annealing Phase (BAP), (iii) Bose-Einstein Annealing Phase (BEAP), and (iv) Dynamical Equilibrium Phase (DEP). BAP and BEAP are simulated annealing searching procedures based on Boltzmann and Bose-Einstein distributions, respectively. DEP is also a simulated annealing search procedure, which is applied at the final temperature of the fourth phase, which can be seen as a second Bose-Einstein phase. MQP is a search process that ranges from extremely high to high temperatures, applying a very fast cooling process, and is not very restrictive to accept new solutions. However, BAP and BEAP range from high to low and from low to very low temperatures, respectively. They are more restrictive for accepting new solutions. DEP uses a particular heuristic to detect the stochastic equilibrium by applying a least squares method during its execution. MPSABBE parameters are tuned with an analytical method, which considers the maximal and minimal deterioration of problem instances. MPSABBE was tested with several instances of PFP, showing that the use of both distributions is better than using only the Boltzmann distribution on the classical SA.
NASA Astrophysics Data System (ADS)
He, Ping
2012-01-01
The long-standing puzzle surrounding the statistical mechanics of self-gravitating systems has not yet been solved successfully. We formulate a systematic theoretical framework of entropy-based statistical mechanics for spherically symmetric collisionless self-gravitating systems. We use an approach that is very different from that of the conventional statistical mechanics of short-range interaction systems. We demonstrate that the equilibrium states of self-gravitating systems consist of both mechanical and statistical equilibria, with the former characterized by a series of velocity-moment equations and the latter by statistical equilibrium equations, which should be derived from the entropy principle. The velocity-moment equations of all orders are derived from the steady-state collisionless Boltzmann equation. We point out that the ergodicity is invalid for the whole self-gravitating system, but it can be re-established locally. Based on the local ergodicity, using Fermi-Dirac-like statistics, with the non-degenerate condition and the spatial independence of the local microstates, we rederive the Boltzmann-Gibbs entropy. This is consistent with the validity of the collisionless Boltzmann equation, and should be the correct entropy form for collisionless self-gravitating systems. Apart from the usual constraints of mass and energy conservation, we demonstrate that the series of moment or virialization equations must be included as additional constraints on the entropy functional when performing the variational calculus; this is an extension to the original prescription by White & Narayan. Any possible velocity distribution can be produced by the statistical-mechanical approach that we have developed with the extended Boltzmann-Gibbs/White-Narayan statistics. Finally, we discuss the questions of negative specific heat and ensemble inequivalence for self-gravitating systems.
Operational derivation of Boltzmann distribution with Maxwell’s demon model
NASA Astrophysics Data System (ADS)
Hosoya, Akio; Maruyama, Koji; Shikano, Yutaka
2015-11-01
The resolution of the Maxwell’s demon paradox linked thermodynamics with information theory through information erasure principle. By considering a demon endowed with a Turing-machine consisting of a memory tape and a processor, we attempt to explore the link towards the foundations of statistical mechanics and to derive results therein in an operational manner. Here, we present a derivation of the Boltzmann distribution in equilibrium as an example, without hypothesizing the principle of maximum entropy. Further, since the model can be applied to non-equilibrium processes, in principle, we demonstrate the dissipation-fluctuation relation to show the possibility in this direction.
Operational derivation of Boltzmann distribution with Maxwell’s demon model
Hosoya, Akio; Maruyama, Koji; Shikano, Yutaka
2015-01-01
The resolution of the Maxwell’s demon paradox linked thermodynamics with information theory through information erasure principle. By considering a demon endowed with a Turing-machine consisting of a memory tape and a processor, we attempt to explore the link towards the foundations of statistical mechanics and to derive results therein in an operational manner. Here, we present a derivation of the Boltzmann distribution in equilibrium as an example, without hypothesizing the principle of maximum entropy. Further, since the model can be applied to non-equilibrium processes, in principle, we demonstrate the dissipation-fluctuation relation to show the possibility in this direction. PMID:26598363
The Boltzmann Equation for a Multi-species Mixture Close to Global Equilibrium
NASA Astrophysics Data System (ADS)
Briant, Marc; Daus, Esther S.
2016-12-01
We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the three dimensional torus. The ultimate aim of this work is to obtain the existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in {L^1_vL^∞_x(m)}, where {m˜ (1+ |v|^k)} is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann operator. Then we develop an {L^2-L^∞} theory à la Guo for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann operator which prevents the direct adaptation of standard mono-species methods (for example Carleman representation, Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the {L^1_vL^∞_x} framework is dealt with for any {k > k_0}, recovering the optimal physical threshold of finite energy {k_0=2} in the particular case of a multi-species hard spheres mixture with the same masses.
Non-Boltzmann population distributions during single-bubble sonoluminescence.
Flannigan, David J; Suslick, Kenneth S
2013-12-12
Single-bubble sonoluminescence (SBSL) spectra from aqueous sulfuric acid solutions containing dissolved neon show widely varying emission despite being similar in chemical composition. From a 65 wt % solution, emission from hydroxyl radicals is observed, with the rovibronic progression being well-described by a single temperature of 7600 K. From an 80 wt % solution, however, emission spectra reveal vibrationally hot sulfur monoxide (SO; Tv = 2400 K) that is also rotationally cold (Tr = 280 K). Further, the SO vibrational population distribution is best-described by a non-Boltzmann distribution. Excited neon atom emission observed from the 80 wt % solution gives an estimated temperature of only 3400 K, indicative of emission from a cool outer shell at the interfacial region. The neon atom excited-state population is also best-described by a non-Boltzmann distribution. These observations are consistent with SBSL emission having both a spatial and temporal component, and the implications for these effects are discussed.
Non-Boltzmann stationary distributions and nonequilibrium relations in active baths
NASA Astrophysics Data System (ADS)
Argun, Aykut; Moradi, Ali-Reza; Pinçe, ErçaÇ§; Bagci, Gokhan Baris; Imparato, Alberto; Volpe, Giovanni
2016-12-01
Most natural and engineered processes, such as biomolecular reactions, protein folding, and population dynamics, occur far from equilibrium and therefore cannot be treated within the framework of classical equilibrium thermodynamics. Here we experimentally study how some fundamental thermodynamic quantities and relations are affected by the presence of the nonequilibrium fluctuations associated with an active bath. We show in particular that, as the confinement of the particle increases, the stationary probability distribution of a Brownian particle confined within a harmonic potential becomes non-Boltzmann, featuring a transition from a Gaussian distribution to a heavy-tailed distribution. Because of this, nonequilibrium relations (e.g., the Jarzynski equality and Crooks fluctuation theorem) cannot be applied. We show that these relations can be restored by using the effective potential associated with the stationary probability distribution. We corroborate our experimental findings with theoretical arguments.
Distributional monte carlo methods for the boltzmann equation
NASA Astrophysics Data System (ADS)
Schrock, Christopher R.
Stochastic particle methods (SPMs) for the Boltzmann equation, such as the Direct Simulation Monte Carlo (DSMC) technique, have gained popularity for the prediction of flows in which the assumptions behind the continuum equations of fluid mechanics break down; however, there are still a number of issues that make SPMs computationally challenging for practical use. In traditional SPMs, simulated particles may possess only a single velocity vector, even though they may represent an extremely large collection of actual particles. This limits the method to converge only in law to the Boltzmann solution. This document details the development of new SPMs that allow the velocity of each simulated particle to be distributed. This approach has been termed Distributional Monte Carlo (DMC). A technique is described which applies kernel density estimation to Nanbu's DSMC algorithm. It is then proven that the method converges not just in law, but also in solution for Linfinity(R 3) solutions of the space homogeneous Boltzmann equation. This provides for direct evaluation of the velocity density function. The derivation of a general Distributional Monte Carlo method is given which treats collision interactions between simulated particles as a relaxation problem. The framework is proven to converge in law to the solution of the space homogeneous Boltzmann equation, as well as in solution for Linfinity(R3) solutions. An approach based on the BGK simplification is presented which computes collision outcomes deterministically. Each technique is applied to the well-studied Bobylev-Krook-Wu solution as a numerical test case. Accuracy and variance of the solutions are examined as functions of various simulation parameters. Significantly improved accuracy and reduced variance are observed in the normalized moments for the Distributional Monte Carlo technique employing discrete BGK collision modeling.
NASA Astrophysics Data System (ADS)
Hora, Heinrich; Miley, George H.; Osman, Frederick
2005-07-01
As laser plasma interactions access ever-increasing ranges of plasma temperatures and densities, it is interesting to consider whether they will some day shed light on questions concerning nuclear synthesis. One such open question is the process of endothermic nuclear synthesis for elements with A > 60, thought to have taken place at a point in time during the big bang, or currently in supernovae. We present an explanation based on a Boltzmann equilibrium condition, in combination with the change of the Fermi-statistics from the relativistic branch for hadrons from higher than nuclear densities to the lower density subrelativistic branch. The Debye length confinement of nuclei breaks down at the relativistic change, thus leading to the impossibility of nucleation of the quark-gluon state at higher than nuclear densities. Taking the increment for the proton number Z as Z‧ = 10 of the measured standard abundance distribution (SAD) of the elements for a Boltzmann probability for heavy element synthesis, a sequence 3 n was found with the exponent n for the sequence of the magic numbers. The jump between the magic numbers 20 and 28 does not need then the usual spin-orbit explanation.
Solutions of Boltzmann Equation for Simulation of Particle Distributions in Plasmas
NASA Astrophysics Data System (ADS)
Hammond, Jason
2014-10-01
We are investigating the time evolution of the electron and excited state distribution functions. To accomplish this, we solve the time dependent Boltzmann equation to overcome some typical limitations of modeling high pressure plasmas using Monte Carlo methods. Here we focus on the numerical approach to solving the time dependent Boltzmann equation using a multi-term approximation of the electron distribution function. We also compare Boltzmann results for electron distribution evolution against multiple plasma simulations using experimental collisional cross-section data.
Derivation of the Second Law of Thermodynamics from Boltzmann's Distribution Law.
ERIC Educational Resources Information Center
Nelson, P. G.
1988-01-01
Shows how the thermodynamic condition for equilibrium in an isolated system can be derived by the application of Boltzmann's law to a simple physical system. States that this derivation could be included in an introductory course on chemical equilibrium to help prepare students for a statistical mechanical treatment presented in the curriculum.…
Equilibrium Tail Distribution Due to Touschek Scattering
Nash,B.; Krinsky, S.
2009-05-04
Single large angle Coulomb scattering is referred to as Touschek scattering. In addition to causing particle loss when the scattered particles are outside the momentum aperture, the process also results in a non-Gaussian tail, which is an equilibrium between the Touschek scattering and radiation damping. Here we present an analytical calculation for this equilibrium distribution.
NASA Astrophysics Data System (ADS)
Qin, Feng; Zhao, Hua; Cai, Wei; Zhang, Zhiguo; Cao, Wenwu
2016-06-01
Noncontact monitoring temperature is very important in modern medicine, science, and technologies. The fluorescence intensity ratio (FIR) technique based on the Boltzmann distribution law exhibits excellent application potential, but the observed FIR deviates from the Boltzmann distribution law in the low temperature range. We propose a fluorescence intensity ratio relation FIR* = ηFIR by introducing a quantity η representing thermal population degree, which can be obtained from measured fluorescence decay curves of the upper emitting level. Using Eu3+ as an example, the method is confirmed that the deviated FIR is able to be corrected and return to follow the Boltzmann law.
Spatial distribution of thermal energy in equilibrium.
Bar-Sinai, Yohai; Bouchbinder, Eran
2015-06-01
The equipartition theorem states that in equilibrium, thermal energy is equally distributed among uncoupled degrees of freedom that appear quadratically in the system's Hamiltonian. However, for spatially coupled degrees of freedom, such as interacting particles, one may speculate that the spatial distribution of thermal energy may differ from the value predicted by equipartition, possibly quite substantially in strongly inhomogeneous or disordered systems. Here we show that for systems undergoing simple Gaussian fluctuations around an equilibrium state, the spatial distribution is universally bounded from above by 1/2k(B)T. We further show that in one-dimensional systems with short-range interactions, the thermal energy is equally partitioned even for coupled degrees of freedom in the thermodynamic limit and that in higher dimensions nontrivial spatial distributions emerge. Some implications are discussed.
A modified double distribution lattice Boltzmann model for axisymmetric thermal flow
NASA Astrophysics Data System (ADS)
Wang, Zuo; Liu, Yan; Wang, Heng; Zhang, Jiazhong
2017-04-01
In this paper, a double distribution lattice Boltzmann model for axisymmetric thermal flow is proposed. In the model, the flow field is solved by a multi-relaxation-time lattice Boltzmann scheme while the temperature field by a newly proposed lattice-kinetic-based Boltzmann scheme. Chapman-Enskog analysis demonstrates that the axisymmetric energy equation in the cylindrical coordinate system can be recovered by the present lattice-kinetic-based Boltzmann scheme for temperature field. Numerical tests, including the thermal Hagen-Poiseuille flow and natural convection in a vertical annulus, have been carried out, and the results predicted by the present model agree well with the existing numerical data. Furthermore, the present model shows better numerical stability than the existing model.
Stationary equilibrium singularity distributions in the plane
NASA Astrophysics Data System (ADS)
Newton, P. K.; Ostrovskyi, V.
2012-02-01
We characterize all stationary equilibrium point singularity distributions in the plane of logarithmic type, allowing for real, imaginary or complex singularity strengths. The dynamical system follows from the assumption that each of the N singularities moves according to the flow field generated by all the others at that point. For strength vector \\vec{\\Gamma} \\in {\\Bbb R}^N , the dynamical system is the classical point vortex system obtained from a singular discrete representation of the vorticity field from ideal, incompressible fluid flow. When \\vec{\\Gamma} \\in \\Im , it corresponds to a system of sources and sinks, whereas when \\vec{\\Gamma} \\in {\\Bbb C}^N the system consists of spiral sources and sinks discussed in Kochin et al (1964 Theoretical Hydromechanics 1 (London: Interscience)). We formulate the equilibrium problem as one in linear algebra, A \\vec{\\Gamma} = 0 , A \\in {\\Bbb C}^{N \\times N} , \\vec{\\Gamma} \\in {\\Bbb C}^N , where A is a N × N complex skew-symmetric configuration matrix which encodes the geometry of the system of interacting singularities. For an equilibrium to exist, A must have a kernel and \\vec{\\Gamma} must be an element of the nullspace of A. We prove that when N is odd, A always has a kernel, hence there is a choice of \\vec{\\Gamma} for which the system is a stationary equilibrium. When N is even, there may or may not be a non-trivial nullspace of A, depending on the relative position of the points in the plane. We provide examples of evenly and randomly distributed points on curves such as circles, figure eights, flower-petal configurations and spirals. We then show how to classify the stationary equilibria in terms of the singular spectrum of A.
Wealth distribution and collective knowledge: a Boltzmann approach.
Pareschi, L; Toscani, G
2014-11-13
We introduce and discuss a nonlinear kinetic equation of Boltzmann type that describes the influence of knowledge in the evolution of wealth in a system of agents that interact through the binary trades, an equation first introduced by Cordier et al. (2005 J. Stat. Phys. 120, 253-277 (doi:10.1007/S10955-005-5456-0)). The trades, which include both saving propensity and the risks of the market, are here modified in the risk and saving parameters, which now are assumed to depend on the personal degree of knowledge. The numerical simulations show that the presence of knowledge has the potential to produce a class of wealthy agents and to account for a larger proportion of wealth inequality.
NASA Astrophysics Data System (ADS)
Lucia, Umberto
2016-02-01
The balance of forces and processes between the system and the environment and the processes inside the system are the result of the flows of the quanta. Moreover, the transition between two thermodynamic states is the consequence of absorption or emission of quanta, but, during the transition, the entropy variation due to the irreversibility occurs and it breaks any symmetry of time. Consequently, the irreversibility is the result of a transition, a process, an interaction between the system and its environment. This interaction results completely time-irreversible for any real process because of irreversibility. As a consequence, a proof of the third law is obtained proving that the zero temperature state can be achieved only for an infinite work lost for dissipation or in an infinite time. The fundamental role of time both in equilibrium and in non equilibrium analysis is pointed out. Moreover, the non equilibrium temperature is related to the entropy generation and its fluctuation rate; indeed, non-stationary temperature means that the system has not yet attained free energy minimum state, i.e., the maximum entropy state; the consequence is that the zero temperature state can be achieved only for an infinite work lost for dissipation or in an infinite time. In engineering thermodynamics the efficiency is always obtained without any consideration on time, while, here, just the time is introduced as a fundamental quantity of the analysis of non equilibrium states.
Consistent Application of the Boltzmann Distribution to Residual Entropy in Crystals
ERIC Educational Resources Information Center
Kozliak, Evguenii I.
2007-01-01
Four different approaches to residual entropy (the entropy remaining in crystals comprised of nonsymmetric molecules like CO, N[subscript 2]O, FClO[subscript 3], and H[subscript 2]O as temperatures approach 0 K) are analyzed and a new method of its calculation is developed based on application of the Boltzmann distribution. The inherent connection…
ERIC Educational Resources Information Center
Castle, Karen J.
2007-01-01
In this undergraduate physical chemistry laboratory experiment, students acquire a high-resolution infrared absorption spectrum of carbon dioxide and use their data to show that the rotational-vibrational state populations follow a Boltzmann distribution. Data are acquired with a mid-infrared laser source and infrared detector. Appropriate…
Lattice models, packing density, and Boltzmann-like distribution of cavities in proteins.
Rashin, Alexander A; Rashin, Abraham H L
2005-02-15
A model reproducing the experimental Boltzmann-like distribution of empty cavity sizes in proteins is introduced. Proteins are represented by lattices of different dimensionalities, corresponding to different numbers of nearest neighbor contacts. Small cavities emerge and join into larger ones in a random process that can be related to random mutations. Simulations of cavity creation are performed under the constraint of a limiting total packing density. Cavities sufficiently large (20 A(3) or more), that they might accommodate at least one additional methyl group produced by a mutation, are counted and compared to the distribution of cavities according to their sizes from protein statistics. The distributions calculated with this very simple model within a realistic range of packing densities are in good agreement with the empirical cavity distribution. The results suggest that the Boltzmann-like distribution of cavities in proteins might be affected by a mechanism controlled by limiting packing density and maximum allowed protein destabilization. This supports an earlier suggestion that the agreement between the free energies of cavity formation from the mutational experiments and from the statistics of the empty cavity distribution in X-ray protein structures is nonfortuitous. A possible relation of the suggested model to the Boltzmann hypothesis is discussed.
Energy Distributions in Small Populations: Pascal versus Boltzmann
ERIC Educational Resources Information Center
Kugel, Roger W.; Weiner, Paul A.
2010-01-01
The theoretical distributions of a limited amount of energy among small numbers of particles with discrete, evenly-spaced quantum levels are examined systematically. The average populations of energy states reveal the pattern of Pascal's triangle. An exact formula for the probability that a particle will be in any given energy state is derived.…
NASA Astrophysics Data System (ADS)
Bergeron, H.; Curado, E. M. F.; Gazeau, J. P.; Rodrigues, Ligia M. C. S.
2016-02-01
Asymptotic behavior (with respect to the number of trials) of symmetric generalizations of binomial distributions and their related entropies is studied through three examples. The first one has the q-exponential as the generating function, the second one involves the modified Abel polynomials, and the third one has Hermite polynomials. We prove analytically that the Rényi entropy is extensive for these three cases, i.e., it is proportional (asymptotically) to the number n of events and that q-exponential and Hermite cases have also extensive Boltzmann-Gibbs. The Abel case is exceptional in the sense that its Boltzmann-Gibbs entropy is not extensive and behaves asymptotically as the square root of n. This result is obtained numerically and also confirmed analytically, under reasonable assumptions, by using a regularization of the beta function and its derivative. Probabilistic urn and genetic models are presented for illustrating this remarkable case.
Equilibrium of Global Amphibian Species Distributions with Climate
Munguía, Mariana; Rahbek, Carsten; Rangel, Thiago F.; Diniz-Filho, Jose Alexandre F.; Araújo, Miguel B.
2012-01-01
A common assumption in bioclimatic envelope modeling is that species distributions are in equilibrium with contemporary climate. A number of studies have measured departures from equilibrium in species distributions in particular regions, but such investigations were never carried out for a complete lineage across its entire distribution. We measure departures of equilibrium with contemporary climate for the distributions of the world amphibian species. Specifically, we fitted bioclimatic envelopes for 5544 species using three presence-only models. We then measured the proportion of the modeled envelope that is currently occupied by the species, as a metric of equilibrium of species distributions with climate. The assumption was that the greater the difference between modeled bioclimatic envelope and the occupied distribution, the greater the likelihood that species distribution would not be at equilibrium with contemporary climate. On average, amphibians occupied 30% to 57% of their potential distributions. Although patterns differed across regions, there were no significant differences among lineages. Species in the Neotropic, Afrotropics, Indo-Malay, and Palaearctic occupied a smaller proportion of their potential distributions than species in the Nearctic, Madagascar, and Australasia. We acknowledge that our models underestimate non equilibrium, and discuss potential reasons for the observed patterns. From a modeling perspective our results support the view that at global scale bioclimatic envelope models might perform similarly across lineages but differently across regions. PMID:22511938
NASA Astrophysics Data System (ADS)
Capitelli, M.; Colonna, G.; D’Ammando, G.; Laricchiuta, A.; Pietanza, L. D.
2017-03-01
Non-equilibrium vibrational distributions (vdf) and non-equilibrium electron energy distribution functions (eedf) in a nitrogen plasma at low pressure (mtorr) have been calculated by using a time-dependent plasma physics model coupled to the Boltzmann equation and heavy particle kinetics. Different case studies have been selected showing the non-equilibrium character of both vdf and eedf under discharge and post-discharge conditions in the presence of large concentrations of electrons. Particular attention is devoted to the electron-molecule resonant vibrational excitation cross sections acting in the whole vibrational ladder. The results in the post-discharge conditions show the interplay of superelastic vibrational and electronic collisions in forming structures in the eedf. The link between the present results in the mtorr afterglow regime with the existing eedf in the torr and atmospheric regimes is discussed.
The equilibrium size distribution of rouleaux.
Perelson, A S; Wiegel, F W
1982-01-01
Rouleaux are formed by the aggregation of red blood cells in the presence of macromolecules that bridge the membranes of adherent erythrocytes. We compute the size and degree of branching of rouleaux for macroscopic systems in thermal equilibrium in the absence of fluid flow. Using techniques from statistical mechanics, analytical expressions are derived for (a) the average number of rouleaux consisting of n cells and having m branch points; (b) the average number of cells per rouleau; (c) the average number of branch points per rouleau; and (d) the number of rouleaux with n cells, n = 1, 2, ..., in a system containing a total of N cells. We also present the results of numerical evaluations to establish the validity of asymptotic expressions that simplify our formal analytic results. Images FIGURE 1 PMID:7059653
NASA Astrophysics Data System (ADS)
Chiloyan, Vazrik; Zeng, Lingping; Huberman, Samuel; Maznev, Alexei A.; Nelson, Keith A.; Chen, Gang
2016-04-01
The phonon Boltzmann transport equation (BTE) is a powerful tool for studying nondiffusive thermal transport. Here, we develop a new universal variational approach to solving the BTE that enables extraction of phonon mean free path (MFP) distributions from experiments exploring nondiffusive transport. By utilizing the known Fourier heat conduction solution as a trial function, we present a direct approach to calculating the effective thermal conductivity from the BTE. We demonstrate this technique on the transient thermal grating experiment, which is a useful tool for studying nondiffusive thermal transport and probing the MFP distribution of materials. We obtain a closed form expression for a suppression function that is materials dependent, successfully addressing the nonuniversality of the suppression function used in the past, while providing a general approach to studying thermal properties in the nondiffusive regime.
Boltzmann-Machine Learning of Prior Distributions of Binarized Natural Images
NASA Astrophysics Data System (ADS)
Obuchi, Tomoyuki; Koma, Hirokazu; Yasuda, Muneki
2016-11-01
Prior distributions of binarized natural images are learned by using a Boltzmann machine. According the results of this study, there emerges a structure with two sublattices in the interactions, and the nearest-neighbor and next-nearest-neighbor interactions correspondingly take two discriminative values, which reflects the individual characteristics of the three sets of pictures that we process. Meanwhile, in a longer spatial scale, a longer-range, although still rapidly decaying, ferromagnetic interaction commonly appears in all cases. The characteristic length scale of the interactions is universally up to approximately four lattice spacings ξ ≈ 4. These results are derived by using the mean-field method, which effectively reduces the computational time required in a Boltzmann machine. An improved mean-field method called the Bethe approximation also gives the same results, as well as the Monte Carlo method does for small size images. These reinforce the validity of our analysis and findings. Relations to criticality, frustration, and simple-cell receptive fields are also discussed.
NASA Astrophysics Data System (ADS)
Gyenis, Balázs
2017-02-01
We investigate Maxwell's attempt to justify the mathematical assumptions behind his 1860 Proposition IV according to which the velocity components of colliding particles follow the normal distribution. Contrary to the commonly held view we find that his molecular collision model plays a crucial role in reaching this conclusion, and that his model assumptions also permit inference to equalization of mean kinetic energies (temperatures), which is what he intended to prove in his discredited and widely ignored Proposition VI. If we take a charitable reading of his own proof of Proposition VI then it was Maxwell, and not Boltzmann, who gave the first proof of a tendency towards equilibrium, a sort of H-theorem. We also call attention to a potential conflation of notions of probabilistic and value independence in relevant prior works of his contemporaries and of his own, and argue that this conflation might have impacted his adoption of the suspect independence assumption of Proposition IV.
Technology Transfer Automated Retrieval System (TEKTRAN)
The distribution coefficient (KD) for the human drug carbamazepine was measured using a non-equilibrium technique. Repacked soil columns were prepared using an Airport silt loam (Typic Natrustalf) with an average organic matter content of 2.45%. Carbamazepine solutions were then leached through th...
Rigorous Proof of the Boltzmann-Gibbs Distribution of Money on Connected Graphs
NASA Astrophysics Data System (ADS)
Lanchier, Nicolas
2017-02-01
Models in econophysics, i.e., the emerging field of statistical physics that applies the main concepts of traditional physics to economics, typically consist of large systems of economic agents who are characterized by the amount of money they have. In the simplest model, at each time step, one agent gives one dollar to another agent, with both agents being chosen independently and uniformly at random from the system. Numerical simulations of this model suggest that, at least when the number of agents and the average amount of money per agent are large, the distribution of money converges to an exponential distribution reminiscent of the Boltzmann-Gibbs distribution of energy in physics. The main objective of this paper is to give a rigorous proof of this result and show that the convergence to the exponential distribution holds more generally when the economic agents are located on the vertices of a connected graph and interact locally with their neighbors rather than globally with all the other agents. We also study a closely related model where, at each time step, agents buy with a probability proportional to the amount of money they have, and prove that in this case the limiting distribution of money is Poissonian.
Rigorous Proof of the Boltzmann-Gibbs Distribution of Money on Connected Graphs
NASA Astrophysics Data System (ADS)
Lanchier, Nicolas
2017-04-01
Models in econophysics, i.e., the emerging field of statistical physics that applies the main concepts of traditional physics to economics, typically consist of large systems of economic agents who are characterized by the amount of money they have. In the simplest model, at each time step, one agent gives one dollar to another agent, with both agents being chosen independently and uniformly at random from the system. Numerical simulations of this model suggest that, at least when the number of agents and the average amount of money per agent are large, the distribution of money converges to an exponential distribution reminiscent of the Boltzmann-Gibbs distribution of energy in physics. The main objective of this paper is to give a rigorous proof of this result and show that the convergence to the exponential distribution holds more generally when the economic agents are located on the vertices of a connected graph and interact locally with their neighbors rather than globally with all the other agents. We also study a closely related model where, at each time step, agents buy with a probability proportional to the amount of money they have, and prove that in this case the limiting distribution of money is Poissonian.
ERIC Educational Resources Information Center
Angeli, Celestino; Cimiraglia, Renzo; Dallo, Federico; Guareschi, Riccardo; Tenti, Lorenzo
2013-01-01
The dependence on the temperature of the population of the "i"th state, "P"[subscript "i"], in the Boltzmann distribution is analyzed by studying its derivative with respect to the temperature, "T." A simple expression is found, involving "P"[subscript "i"], the energy of the state,…
Development of equilibrium raindrop size distribution in natural rain.
NASA Astrophysics Data System (ADS)
Pio D'Adderio, Leo; Porcu, Federico; Tokay, Ali
2015-04-01
The NASA's Global Precipitation Measurement (GPM) mission dual-frequency precipitation radar retrieval has adopted a three-parameter gamma distribution to retrieve the raindrop size distribution (DSD) from dual-frequency precipitation radar (DPR) measurements. Recent analysis from disdrometric measurements collected during GPM ground validation (GV) field experiments shows that the three-parameter gamma distribution does not well fit the observed spectra in the presence of collisional break-up, i.e. when the DSD reaches the equilibrium stage. An automatic algorithm is used to select equilibrium DSD in six datasets for a total number of more than 12,000 minutes with rain rate higher than 5 mmh-1 collected from 2-DVD disdrometers. The algorithm is based on the analysis of the DSD slope in the interval 1.0-2.5 mm diameter. The 1-minute time series are studied in order to assess the conditions more favorable for equilibrium DSD to take place, showing the transition between the one-peak DSD to the 2-peak DSD, for selected case studies, over a wide range of rainrate values. The results are discussed in terms of precipitation type and intensity, showing a very rapid onset and dissipation of equilibrium DSD conditions. The temporal evolution of some DSD parameters is also analyzed, and, for two of the six datasets (MC3E and Wallops), was also possible to evaluate the small-scale spatial structure of equilibrium DSD.
NASA Astrophysics Data System (ADS)
Norouzi, Ali; Esfahani, Javad Abolfazli
2016-02-01
In this study, gaseous flow through a micro/nano-channel is investigated via a novel two relaxation time lattice Boltzmann method. In this method, the slip velocity at the fluid-solid interface is realized by defining the free relaxation parameter. Furthermore, in order to capture the non-linear phenomena associated with the Knudsen layer, the wall function correction is employed. To this respect, different available wall functions are implemented. The objective of the study is to provide a comparative study on the accuracy, range of applicability and computational efficiency of these wall functions in a wide range of Knudsen numbers. The results of the present study are compared against direct simulation Mont Carlo and information preservation data. It is found that only a few of the implemented wall functions are capable of predicting the flow behavior with reasonable accuracy, particularly when the Knudsen number lies in the transition flow regime.
Thermodynamic Derivation of the Equilibrium Distribution Functions of Statistical Mechanics.
ERIC Educational Resources Information Center
Stoeckly, Beth
1979-01-01
Presents a simplified derivation of the equilibrium distribution functions. The derivation proceeds from the change in the Helmholtz free energy when a particle is added to a system of fixed temperature, volume, and chemical potential. The derivations show the relationship between statistical mechanics and macroscopic thermodynamics. (Author/GA)
Equilibrium Distribution of Mutators in the Single Fitness Peak Model
NASA Astrophysics Data System (ADS)
Tannenbaum, Emmanuel; Deeds, Eric J.; Shakhnovich, Eugene I.
2003-09-01
This Letter develops an analytically tractable model for determining the equilibrium distribution of mismatch repair deficient strains in unicellular populations. The approach is based on the single fitness peak model, which has been used in Eigen’s quasispecies equations in order to understand various aspects of evolutionary dynamics. As with the quasispecies model, our model for mutator-nonmutator equilibrium undergoes a phase transition in the limit of infinite sequence length. This “repair catastrophe” occurs at a critical repair error probability of ɛr=Lvia/L, where Lvia denotes the length of the genome controlling viability, while L denotes the overall length of the genome. The repair catastrophe therefore occurs when the repair error probability exceeds the fraction of deleterious mutations. Our model also gives a quantitative estimate for the equilibrium fraction of mutators in Escherichia coli.
Cervantes-Sanchez, Fernando; Hernandez-Aguirre, Arturo; Solorio-Meza, Sergio; Ornelas-Rodriguez, Manuel; Torres-Cisneros, Miguel
2016-01-01
This paper presents a novel method for improving the training step of the single-scale Gabor filters by using the Boltzmann univariate marginal distribution algorithm (BUMDA) in X-ray angiograms. Since the single-scale Gabor filters (SSG) are governed by three parameters, the optimal selection of the SSG parameters is highly desirable in order to maximize the detection performance of coronary arteries while reducing the computational time. To obtain the best set of parameters for the SSG, the area (Az) under the receiver operating characteristic curve is used as fitness function. Moreover, to classify vessel and nonvessel pixels from the Gabor filter response, the interclass variance thresholding method has been adopted. The experimental results using the proposed method obtained the highest detection rate with Az = 0.9502 over a training set of 40 images and Az = 0.9583 with a test set of 40 images. In addition, the experimental results of vessel segmentation provided an accuracy of 0.944 with the test set of angiograms. PMID:27738422
Cervantes-Sanchez, Fernando; Cruz-Aceves, Ivan; Hernandez-Aguirre, Arturo; Aviña-Cervantes, Juan Gabriel; Solorio-Meza, Sergio; Ornelas-Rodriguez, Manuel; Torres-Cisneros, Miguel
2016-01-01
This paper presents a novel method for improving the training step of the single-scale Gabor filters by using the Boltzmann univariate marginal distribution algorithm (BUMDA) in X-ray angiograms. Since the single-scale Gabor filters (SSG) are governed by three parameters, the optimal selection of the SSG parameters is highly desirable in order to maximize the detection performance of coronary arteries while reducing the computational time. To obtain the best set of parameters for the SSG, the area (Az ) under the receiver operating characteristic curve is used as fitness function. Moreover, to classify vessel and nonvessel pixels from the Gabor filter response, the interclass variance thresholding method has been adopted. The experimental results using the proposed method obtained the highest detection rate with Az = 0.9502 over a training set of 40 images and Az = 0.9583 with a test set of 40 images. In addition, the experimental results of vessel segmentation provided an accuracy of 0.944 with the test set of angiograms.
Distributions of Hardy-Weinberg equilibrium test statistics.
Rohlfs, R V; Weir, B S
2008-11-01
It is well established that test statistics and P-values derived from discrete data, such as genetic markers, are also discrete. In most genetic applications, the null distribution for a discrete test statistic is approximated with a continuous distribution, but this approximation may not be reasonable. In some cases using the continuous approximation for the expected null distribution may cause truly null test statistics to appear nonnull. We explore the implications of using continuous distributions to approximate the discrete distributions of Hardy-Weinberg equilibrium test statistics and P-values. We derive exact P-value distributions under the null and alternative hypotheses, enabling a more accurate analysis than is possible with continuous approximations. We apply these methods to biological data and find that using continuous distribution theory with exact tests may underestimate the extent of Hardy-Weinberg disequilibrium in a sample. The implications may be most important for the widespread use of whole-genome case-control association studies and Hardy-Weinberg equilibrium (HWE) testing for data quality control.
NASA Astrophysics Data System (ADS)
Dechant, Andreas; Shafier, Shalom Tzvi; Kessler, David A.; Barkai, Eli
2016-08-01
The Boltzmann-Gibbs density, a central result of equilibrium statistical mechanics, relates the energy of a system in contact with a thermal bath to its equilibrium statistics. This relation is lost for nonthermal systems such as cold atoms in optical lattices, where the heat bath is replaced with the laser beams of the lattice. We investigate in detail the stationary phase-space probability for Sisyphus cooling under harmonic confinement. In particular, we elucidate whether the total energy of the system still describes its stationary state statistics. We find that this is true for the center part of the phase-space density for deep lattices, where the Boltzmann-Gibbs density provides an approximate description. The relation between energy and statistics also persists for strong confinement and in the limit of high energies, where the system becomes underdamped. However, the phase-space density now exhibits heavy power-law tails. In all three cases we find expressions for the leading-order phase-space density and corrections which break the equivalence of probability and energy and violate energy equipartition. The nonequilibrium nature of the steady state is corroborated by explicit violations of detailed balance. We complement these analytical results with numerical simulations to map out the intricate structure of the phase-space density.
Equilibrium models of mass distribution and collisional lifetimes of asteroids
NASA Technical Reports Server (NTRS)
Williams, David R.; Wetherill, George
1993-01-01
An understanding of the steady state distribution expected in the present day asteroid belt is important to our understanding of the collisional evolution of the asteroids and their physical properties. We have extended earlier work to show that, in the absence of gravity, a simple power law distribution as a function of mass with constant exponent will give an equilibrium distribution of asteroids for all bodies much smaller than the largest asteroids. This result holds for realistic fragmentation mechanisms and is independent of the physical properties of the asteroids. Inclusion of the effects of gravity on disruption and fragmentation of asteroids precludes an analytic solution to this problem, and rules out a simple power law distribution. We are currently calculating numerical solutions in order to determine the expected steady state mass distribution in the asteroid belt.
Equilibrium Distributions and the Nanostructure Diagram for Epitaxial Quantum Dots
Rudd, R E; Briggs, G D; Sutton, A P; Medeiros-Ribeiro, G; Williams, R S
2006-05-01
We present in detail a thermodynamic equilibrium model for the growth of nanostructures on semiconductor substrates in heteroepitaxy and its application to germanium deposition on silicon. Some results of this model have been published previously, but the details of the formulation of the model are given here for the first time. The model allows the computation of the shape and size distributions of the surface nanostructures, as well as other properties of the system. We discuss the results of the model, and their incorporation into a nanostructure diagram that summarizes the relative stability of domes and pyramids in the bimodal size distributions.
Equilibrium distribution from distributed computing (simulations of protein folding).
Scalco, Riccardo; Caflisch, Amedeo
2011-05-19
Multiple independent molecular dynamics (MD) simulations are often carried out starting from a single protein structure or a set of conformations that do not correspond to a thermodynamic ensemble. Therefore, a significant statistical bias is usually present in the Markov state model generated by simply combining the whole MD sampling into a network whose nodes and links are clusters of snapshots and transitions between them, respectively. Here, we introduce a depth-first search algorithm to extract from the whole conformation space network the largest ergodic component, i.e., the subset of nodes of the network whose transition matrix corresponds to an ergodic Markov chain. For multiple short MD simulations of a globular protein (as in distributed computing), the steady state, i.e., stationary distribution determined using the largest ergodic component, yields more accurate free energy profiles and mean first passage times than the original network or the ergodic network obtained by imposing detailed balance by means of symmetrization of the transition counts.
NASA Technical Reports Server (NTRS)
Lanyi, Gabor E.
2003-01-01
This viewgraph presentation reviews the 1901 work in Planck's constant and blackbody radiation law and the 1916 Einstein rederivation of the blackbody radiation law. It also reviews Wien's law. It also presents equations that demonstrate the thermal balance between radiation and matter.
NASA Astrophysics Data System (ADS)
Rabhi, Raja; Amami, Bayssain; Dhahri, Hacen; Mhimid, Abdallah
2016-12-01
The present paper centered on a numerical investigation of irreversibility within a porous micro duct subjected to an external oriented magnetic field. At the wall, slip velocity and temperature jump are used as types of boundary conditions. The flow is described by Darcy-Brinkman-Forchheimer model. The Local Thermal Non Equilibrium (LTNE) is adopted including viscous dissipation effects into the energy equation of fluid phase. The study has been carried out for slip-flow regime for wide range of Knudsen numbers, 10-3 ≤ Kn ≤ 10-1 . The obtained governing system equations are solved using the modified Lattice Boltzmann Method (LBM). Efforts are focused on identifying the influence of magnetic field on the entropy generation and Bejan number with a change of various parameters such as Knudsen, Eckert, Biot, Darcy numbers and thermal conductivity ratio. The obtained results show that the irreversibility and the contribution of heat transfer irreversibility and fluid flow irreversibility are strongly affected by the presence of magnetic field.
Williams, C F; Watson, J E; Nelson, S D
2014-01-01
The distribution coefficient (KD) for the human drug carbamazepine was measured using a non-equilibrium technique. Repacked soil columns were prepared using an Airport silt loam (Typic Natrustalf) with an average organic matter content of 2.45%. Carbamazepine solutions were then leached through the columns at 0.5, 1.0 and 1.5 mL min(-1) representing average linear velocities of 1.8, 3.5 and 5.3 cm h(-1) respectively. Each flow rate was replicated three times and three carbamazepine pulses were applied to each column resulting in a total of 9 columns with 27 total carbamazepine pulses. Breakthrough curves were used to determine KD using the parameter fitting software CXTFIT. Results indicate that as flow rate decreased from 5.3 to 1.8 cm h(-1), KD increased an average of 21%. Additionally, KD determined by column leaching (14.7-22.7 L kg(-1)) was greater than KD determined by a 2h batch equilibrium adsorption (12.6 L kg(-1)). Based on these KD's carbamazepine would be generally characterized as non-mobile in the soil investigated. However, repeated carbamazepine applications resulted in an average 22% decrease in KD between the first and third applications. Decreasing KD is attributed to differences in sorption site kinetics and carbamazepine residence time in contact with the soil. This would indicate that the repeated use of reclaimed wastewater at high application rates for long-term irrigation or groundwater recharge has the potential to lead to greater transport of carbamazepine than KD determined by batch equilibrium would predict.
Graph-distance distribution of the Boltzmann ensemble of RNA secondary structures
2014-01-01
Background Large RNA molecules are often composed of multiple functional domains whose spatial arrangement strongly influences their function. Pre-mRNA splicing, for instance, relies on the spatial proximity of the splice junctions that can be separated by very long introns. Similar effects appear in the processing of RNA virus genomes. Albeit a crude measure, the distribution of spatial distances in thermodynamic equilibrium harbors useful information on the shape of the molecule that in turn can give insights into the interplay of its functional domains. Result Spatial distance can be approximated by the graph-distance in RNA secondary structure. We show here that the equilibrium distribution of graph-distances between a fixed pair of nucleotides can be computed in polynomial time by means of dynamic programming. While a naïve implementation would yield recursions with a very high time complexity of O(n6D5) for sequence length n and D distinct distance values, it is possible to reduce this to O(n4) for practical applications in which predominantly small distances are of of interest. Further reductions, however, seem to be difficult. Therefore, we introduced sampling approaches that are much easier to implement. They are also theoretically favorable for several real-life applications, in particular since these primarily concern long-range interactions in very large RNA molecules. Conclusions The graph-distance distribution can be computed using a dynamic programming approach. Although a crude approximation of reality, our initial results indicate that the graph-distance can be related to the smFRET data. The additional file and the software of our paper are available from http://www.rna.uni-jena.de/RNAgraphdist.html. PMID:25285153
Extended Tonks-Langmuir-type model with non-Boltzmann-distributed electrons and cold ion sources
NASA Astrophysics Data System (ADS)
Kamran, M.; Kuhn, S.; Tskhakaya, D. D.; Khan, M.; Khan
2013-04-01
kinetic Tonks-Langmuir model. Phys. Plasmas 13, 063508) or bi-Maxwellian (Godyak, V. A. et al. 1995 Tonks-Langmuir problem for a bi-Maxwellian plasma. IEEE Trans. Plasma Sci. 23, 728) electron velocity distribution functions (VDFs), which satisfy the zero-CSS-term (Vlasov) kinetic equation and imply zero electron currents, we here propose a more general class of electron VDFs allowing, in an approximate manner, for non-zero CSS terms and finite electron currents inside the plasma region. The sheath-edge and floating-wall potentials are calculated by balancing the ion and electron current densities at sheath-edge singularities. In a first detailed application, the type-t and type-p electron VDFs are assumed to be `inner' and `outer' cut-off Maxwellians respectively, with different amplitudes and `formal' temperatures, implying the perfectly CSS-free limit. For the special case of equal type-t and type-p electron VDF amplitudes and formal temperatures, the classical Boltzmann distribution for electrons is formally retrieved. Special cases with other amplitude and formal-temperature ratios show significant deviations from the classical case.
NASA Astrophysics Data System (ADS)
Nauenberg, Michael
2005-03-01
In 1916 Einstein published a remarkable paper entitled ``On the Quantum Theory of Radiation''ootnotetextA. Einstein ``On the Quantum theory of Radiation,'' Phys. Zeitschrift 18 (1917) 121. First printed in Mitteilungender Physikalischen Gesellschaft Zurich. No 18, 1916. Translated into English in Van der Waerden ``Sources of Quantum Mechanics'' (North Holland 1967) pp. 63-77. in which he obtained Planck's formula for black-body radiation by introducing a new statistical hypothesis for the emmision and absorption of electromagneic radiation based on discrete bundles of energy and momentum which are now called photons. Einstein radiation theory replaced Maxwell's classical theory by a stochastic process which, when properly interpreted, also gives well known statistics of massless particles with even spin.^2 This quantum distribution, however, was not discovered by Einstein but was communicated to him by Bose in 1924. Like Boltzmann's classical counterpart, Einstein's statistical theory leads to an irreversible approach to thermal equilibrium, but because this violates time reversal, Einstein theory can not be regarded as a fundamental theory of physical process.ootnotetextM. Nauenberg ``The evolution of radiation towards thermal equilibrium: A soluble model which illustrates the foundations of statistical mechanics,'' American Journal of Physics 72 (2004) 313 Apparently Einstein and his contemporaries were unaware of this problem, and even today this problem is ignored in contemporary discussions of Einstein's treatment of the black-body spectrum.
Lahonian, Mansour; Golneshan, Ali Akbar
2011-12-01
This work applies a three-dimensional lattice Boltzmann method (LBM), to solve the Pennes bio-heat equation (BHE), in order to predict the temperature distribution in a spherical tissue, with blood perfusion, metabolism and magnetic nanoparticles (MNPs) heat sources, during magnetic fluid hyperthermia (MFH). So, heat dissipation of MNPs under an alternating magnetic field has been studied and effect of different factors such as induction and frequency of magnetic field and volume fraction of MNPs has been investigated. Then, effect of MNPs dispersion on temperature distribution inside tumor and its surrounding healthy tissue has been shown. Also, effect of blood perfusion, thermal conductivity of tumor, frequency and amplitude of magnetic field on temperature distribution has been explained. Results show that the LBM has a good accuracy to solve the bio-heat transfer problems.
Sels, Dries; Brosens, Fons
2013-10-01
The equation of motion for the reduced Wigner function of a system coupled to an external quantum system is presented for the specific case when the external quantum system can be modeled as a set of harmonic oscillators. The result is derived from the Wigner function formulation of the Feynman-Vernon influence functional theory. It is shown how the true self-energy for the equation of motion is connected with the influence functional for the path integral. Explicit expressions are derived in terms of the bare Wigner propagator. Finally, we show under which approximations the resulting equation of motion reduces to the Wigner-Boltzmann equation.
Learning thermodynamics with Boltzmann machines
NASA Astrophysics Data System (ADS)
Torlai, Giacomo; Melko, Roger G.
2016-10-01
A Boltzmann machine is a stochastic neural network that has been extensively used in the layers of deep architectures for modern machine learning applications. In this paper, we develop a Boltzmann machine that is capable of modeling thermodynamic observables for physical systems in thermal equilibrium. Through unsupervised learning, we train the Boltzmann machine on data sets constructed with spin configurations importance sampled from the partition function of an Ising Hamiltonian at different temperatures using Monte Carlo (MC) methods. The trained Boltzmann machine is then used to generate spin states, for which we compare thermodynamic observables to those computed by direct MC sampling. We demonstrate that the Boltzmann machine can faithfully reproduce the observables of the physical system. Further, we observe that the number of neurons required to obtain accurate results increases as the system is brought close to criticality.
NASA Astrophysics Data System (ADS)
Coelho, Rodrigo C. V.; Ilha, Anderson S.; Doria, Mauro M.
2016-10-01
A lattice Boltzmann method is proposed based on the expansion of the equilibrium distribution function in powers of a new set of generalized orthonormal polynomials which are here presented. The new polynomials are orthonormal under the weight defined by the equilibrium distribution function itself. The D-dimensional Hermite polynomials is a sub-case of the present ones, associated to the particular weight of a Gaussian function. The proposed lattice Boltzmann method allows for the treatment of semi-classical fluids, such as electrons in metals under the Drude-Sommerfeld model, which is a particular case that we develop and validate by the Riemann problem.
Aquilanti, Vincenzo; Coutinho, Nayara Dantas; Carvalho-Silva, Valter Henrique
2017-04-28
This article surveys the empirical information which originated both by laboratory experiments and by computational simulations, and expands previous understanding of the rates of chemical processes in the low-temperature range, where deviations from linearity of Arrhenius plots were revealed. The phenomenological two-parameter Arrhenius equation requires improvement for applications where interpolation or extrapolations are demanded in various areas of modern science. Based on Tolman's theorem, the dependence of the reciprocal of the apparent activation energy as a function of reciprocal absolute temperature permits the introduction of a deviation parameter d covering uniformly a variety of rate processes, from those where quantum mechanical tunnelling is significant and d < 0, to those where d > 0, corresponding to the Pareto-Tsallis statistical weights: these generalize the Boltzmann-Gibbs weight, which is recovered for d = 0. It is shown here how the weights arise, relaxing the thermodynamic equilibrium limit, either for a binomial distribution if d > 0 or for a negative binomial distribution if d < 0, formally corresponding to Fermion-like or Boson-like statistics, respectively. The current status of the phenomenology is illustrated emphasizing case studies; specifically (i) the super-Arrhenius kinetics, where transport phenomena accelerate processes as the temperature increases; (ii) the sub-Arrhenius kinetics, where quantum mechanical tunnelling propitiates low-temperature reactivity; (iii) the anti-Arrhenius kinetics, where processes with no energetic obstacles are rate-limited by molecular reorientation requirements. Particular attention is given for case (i) to the treatment of diffusion and viscosity, for case (ii) to formulation of a transition rate theory for chemical kinetics including quantum mechanical tunnelling, and for case (iii) to the stereodirectional specificity of the dynamics of reactions strongly hindered by the
Lattice Boltzmann model for the complex Ginzburg-Landau equation.
Zhang, Jianying; Yan, Guangwu
2010-06-01
A lattice Boltzmann model with complex distribution function for the complex Ginzburg-Landau equation (CGLE) is proposed. By using multiscale technique and the Chapman-Enskog expansion on complex variables, we obtain a series of complex partial differential equations. Then, complex equilibrium distribution function and its complex moments are obtained. Based on this model, the rotation and oscillation properties of stable spiral waves and the breaking-up behavior of unstable spiral waves in CGLE are investigated in detail.
Equilibrium distribution of ions in a muscle fiber.
Maughan, D W; Godt, R E
1989-01-01
We have developed a mathematical description of the equilibrium (Donnan) distribution of mobile ions between two phases containing fixed charges. This differs from the classical Donnan derivation by including mobile polyvalent ions such as those present in intact muscle fibers and in solutions used with skinned muscle fibers. Given the average concentrations of ionic species present in intact frog muscle, we calculate that the myofibrillar fixed charge density (-42 meq/liter cytoplasmic fluid) is in close agreement with estimates from amino acid analysis of myofibrillar proteins. As expected, with negative fixed charges in the myofibril, anions are excluded from the myofibrillar space while cations are concentrated in this space; the ratio between the average intra- and extramyofibrillar concentrations for an ion of valence n is (1.11)n. This model allowed us to design a bathing solution for skinned muscle fibers in which the intramyofibrillar ion concentrations closely approximate those found in intact frog muscle cells. Our model, applied to the A- and I-bands of the sarcomere, suggests that likely differences in fixed charge densities in these regions accounts for only a small fraction of the extreme concentration of phosphocreatine observed in the I-bands of intact frog muscle. PMID:2819235
On the full Boltzmann equations for leptogenesis
Garayoa, J.; Pastor, S.; Pinto, T.; Rius, N.; Vives, O. E-mail: pastor@ific.uv.es E-mail: nuria@ific.uv.es
2009-09-01
We consider the full Boltzmann equations for standard and soft leptogenesis, instead of the usual integrated Boltzmann equations which assume kinetic equilibrium for all species. Decays and inverse decays may be inefficient for thermalising the heavy-(s)neutrino distribution function, leading to significant deviations from kinetic equilibrium. We analyse the impact of using the full kinetic equations in the case of a previously generated lepton asymmetry, and find that the washout of this initial asymmetry due to the interactions of the right-handed neutrino is larger than when calculated via the integrated equations. We also solve the full Boltzmann equations for soft leptogenesis, where the lepton asymmetry induced by the soft SUSY-breaking terms in sneutrino decays is a purely thermal effect, since at T = 0 the asymmetry in leptons cancels the one in sleptons. In this case, we obtain that in the weak washout regime (K ∼< 1) the final lepton asymmetry can change up to a factor four with respect to previous estimates.
Modified Lattice Boltzmann method for compressible fluid simulations
Hinton, F. L.; Rosenbluth, M. N.; Wong, S. K.; Lin-Liu, Y. R.; Miller, R. L.
2001-06-01
A modified lattice Boltzmann algorithm is shown to have much better stability to growing temperature perturbations, when compared with the standard lattice Boltzmann algorithm. The damping rates of long-wavelength waves, which determine stability, are derived using a collisional equilibrium distribution function which has the property that the Euler equations are obtained exactly in the limit of zero time step. Using this equilibrium distribution function, we show that our algorithm has inherent positive hyperviscosity and hyperdiffusivity, for very small values of viscosity and thermal diffusivity, which are lacking in the standard algorithm. Short-wavelength modes are shown to be stable for temperatures greater than a lower limit. Results from a computer code are used to compare these algorithms, and to confirm the damping rate predictions made analytically. Finite amplitude sound waves in the simulated fluid steepen, as expected from gas dynamic theory.
Kikuchi, Takashi; Horioka, Kazuhiko
2009-05-15
Possible emittance growths of intense, nonuniform beams during a transport in a focusing channel are derived as a function of nonlinear field energy and space charge tune depression factors. The nonlinear field energy of the beam with thermal equilibrium distribution is estimated by considering the particle distribution across the cross section of the beam. The results show that the possible emittance growth can be suppressed by keeping the beam particle in thermal equilibrium distribution during the beam transport.
CMB spectral distortions as solutions to the Boltzmann equations
NASA Astrophysics Data System (ADS)
Ota, Atsuhisa
2017-01-01
We propose to re-interpret the cosmic microwave background spectral distortions as solutions to the Boltzmann equation. This approach makes it possible to solve the second order Boltzmann equation explicitly, with the spectral y distortion and the momentum independent second order temperature perturbation, while generation of μ distortion cannot be explained even at second order in this framework. We also extend our method to higher order Boltzmann equations systematically and find new type spectral distortions, assuming that the collision term is linear in the photon distribution functions, namely, in the Thomson scattering limit. As an example, we concretely construct solutions to the cubic order Boltzmann equation and show that the equations are closed with additional three parameters composed of a cubic order temperature perturbation and two cubic order spectral distortions. The linear Sunyaev-Zel'dovich effect whose momentum dependence is different from the usual y distortion is also discussed in the presence of the next leading order Kompaneets terms, and we show that higher order spectral distortions are also generated as a result of the diffusion process in a framework of higher order Boltzmann equations. The method may be applicable to a wider class of problems and has potential to give a general prescription to non-equilibrium physics.
Sonnad, Kiran G.; Cary, John R.
2015-04-15
A procedure to obtain a near equilibrium phase space distribution function has been derived for beams with space charge effects in a generalized periodic focusing transport channel. The method utilizes the Lie transform perturbation theory to canonically transform to slowly oscillating phase space coordinates. The procedure results in transforming the periodic focusing system to a constant focusing one, where equilibrium distributions can be found. Transforming back to the original phase space coordinates yields an equilibrium distribution function corresponding to a constant focusing system along with perturbations resulting from the periodicity in the focusing. Examples used here include linear and nonlinear alternating gradient focusing systems. It is shown that the nonlinear focusing components can be chosen such that the system is close to integrability. The equilibrium distribution functions are numerically calculated, and their properties associated with the corresponding focusing system are discussed.
Silvestre-Alcantara, Whasington; Bhuiyan, Lutful B.; Outhwaite, Christopher W.; Henderson, Douglas
2010-01-01
The properties of the singlet ion distributions at and around contact in a restricted primitive model double layer are characterized in the modified Poisson–Boltzmann theory. Comparisons are made with the corresponding exact Monte Carlo simulation data, the results from the Gouy–Chapman–Stern theory coupled to an exclusion volume term, and the mean spherical approximation. Particular emphasis is given to the behaviour of the theoretical predictions in relation to the contact value theorem involving the charge profile. The simultaneous behaviour of the coion and counterion contact values is also examined. The performance of the modified Poisson–Boltzmann theory in regard to the contact value theorems is very reasonable with the contact characteristics showing semi-quantitative or better agreement overall with the simulation results. The exclusion-volume-treated Gouy–Chapman–Stern theory reveals a fortuitous cancellation of errors, while the mean spherical approximation is poor. PMID:20664814
Transition in the Equilibrium Distribution Function of Relativistic Particles
Mendoza, M.; Araújo, N. A. M.; Succi, S.; Herrmann, H. J.
2012-01-01
We analyze a transition from single peaked to bimodal velocity distribution in a relativistic fluid under increasing temperature, in contrast with a non-relativistic gas, where only a monotonic broadening of the bell-shaped distribution is observed. Such transition results from the interplay between the raise in thermal energy and the constraint of maximum velocity imposed by the speed of light. We study the Bose-Einstein, the Fermi-Dirac, and the Maxwell-Jüttner distributions, and show that they all exhibit the same qualitative behavior. We characterize the nature of the transition in the framework of critical phenomena and show that it is either continuous or discontinuous, depending on the group velocity. We analyze the transition in one, two, and three dimensions, with special emphasis on twodimensions, for which a possible experiment in graphene, based on the measurement of the Johnson-Nyquist noise, is proposed. PMID:22937220
NASA Astrophysics Data System (ADS)
Imai, M.; Sataka, M.; Kawatsura, K.; Takahiro, K.; Komaki, K.; Shibata, H.; Sugai, H.; Nishio, K.
2009-08-01
Both equilibrium and non-equilibrium charge-state distributions for 2.0 MeV/u sulfur ions after passing through carbon foils were studied experimentally. For the equilibrium charge-state distribution, incident ions of S 7+, S 12+, S 14+ and S 16+ were injected into carbon foils 54, 98, 150 and 200 μg/cm 2 in thickness, whereas for the non-equilibrium distributions, new measurements for S 15+ and S 16+ incidences were made through carbon foils of 0.9-10 μg/cm 2 to supplement our previous experiments regarding S 6+-S 14+ incidences [M. Imai, M. Sataka, K. Kawatsura, K. Takahiro, K. Komaki, H. Shibata, H. Sugai, K. Nishio, Nucl. Instr. and Meth. B 230 (2005) 63; M. Imai, M. Sataka, K. Kawatsura, K. Takahiro, K. Komaki, H. Shibata, H. Sugai, K. Nishio, Nucl. Instr. and Meth. B 256 (2007) 11]. Mean charge states for S 6+-S 14+ incidences as functions of the penetration thickness merged at 6.9 μg/cm 2 and changed together until reaching equilibrium at around 100 μg/cm 2, while those for S 15+ and S 16+ incidences took different paths to equilibrium, which was also the case for distribution widths for S 6+-S 14+, S 15+ and S 16+ incidences. An equilibrium mean charge state of 12.68 and distribution width of 1.11 were attained with equilibrium charge distributions between 6+ and 16+.
The Approach to Equilibrium: Detailed Balance and the Master Equation
ERIC Educational Resources Information Center
Alexander, Millard H.; Hall, Gregory E.; Dagdigian, Paul J.
2011-01-01
The approach to the equilibrium (Boltzmann) distribution of populations of internal states of a molecule is governed by inelastic collisions in the gas phase and with surfaces. The set of differential equations governing the time evolution of the internal state populations is commonly called the master equation. An analytic solution to the master…
The stress distribution in shell bodies and wings as an equilibrium problem
NASA Technical Reports Server (NTRS)
Wagner, H
1937-01-01
This report treats the stress distribution in shell-shaped airplane components (fuselage, wings) as an equilibrium problem; it includes both cylindrical and non-cylindrical shells. In particular, it treats the stress distribution at the point of stress application and at cut-out points.
NASA Astrophysics Data System (ADS)
Wang, Huimin
2017-01-01
In this paper, a new lattice Boltzmann model for the Korteweg-de Vries (KdV) equation is proposed. By using the Chapman-Enskog expansion and the multi-scale time expansion, a series of partial differential equations in different time scales and several higher- order moments of equilibrium distribution functions are obtained. In order to make the scheme obey the three conservation laws of the KdV equation, two equilibrium distribution functions are used and a correlation between the first conservation law and the second conservation law is constructed. In numerical examples, the numerical results of the KdV equation obtained by this scheme are compared with those results obtained by the previous lattice Boltzmann model. Numerical experiments demonstrate this scheme can be used to reduce the truncation error of the lattice Boltzmann scheme and preserve the three conservation laws.
Numerical method based on the lattice Boltzmann model for the Fisher equation.
Yan, Guangwu; Zhang, Jianying; Dong, Yinfeng
2008-06-01
In this paper, a lattice Boltzmann model for the Fisher equation is proposed. First, the Chapman-Enskog expansion and the multiscale time expansion are used to describe higher-order moment of equilibrium distribution functions and a series of partial differential equations in different time scales. Second, the modified partial differential equation of the Fisher equation with the higher-order truncation error is obtained. Third, comparison between numerical results of the lattice Boltzmann models and exact solution is given. The numerical results agree well with the classical ones.
Solitons of the Kadomtsev-Petviashvili equation based on lattice Boltzmann model
NASA Astrophysics Data System (ADS)
Wang, Huimin
2017-01-01
In this paper, a lattice Boltzmann model for the Kadomtsev-Petviashvili equation is proposed. By using the Chapman-Enskog expansion and the multi-scale time expansion, a series of partial differential equations in different time scales are obtained. Due to the asymmetry in x direction and y direction of the equation, the moments of the equilibrium distribution function are selected are asymmetric. The numerical results demonstrate the lattice Boltzmann method is an effective method to simulate the solitons of the Kadomtsev-Petviashvili equation.
Equilibrium distribution of the wave energy in a carbyne chain
NASA Astrophysics Data System (ADS)
Kovriguine, D. A.; Nikitenkova, S. P.
2016-03-01
The steady-state energy distribution of thermal vibrations at a given ambient temperature has been investigated based on a simple mathematical model that takes into account central and noncentral interactions between carbon atoms in a one-dimensional carbyne chain. The investigation has been performed using standard asymptotic methods of nonlinear dynamics in terms of the classical mechanics. In the first-order nonlinear approximation, there have been revealed resonant wave triads that are formed at a typical nonlinearity of the system under phase matching conditions. Each resonant triad consists of one longitudinal and two transverse vibration modes. In the general case, the chain is characterized by a superposition of similar resonant triplets of different spectral scales. It has been found that the energy equipartition of nonlinear stationary waves in the carbyne chain at a given temperature completely obeys the standard Rayleigh-Jeans law due to the proportional amplitude dispersion. The possibility of spontaneous formation of three-frequency envelope solitons in carbyne has been demonstrated. Heat in the form of such solitons can propagate in a chain of carbon atoms without diffusion, like localized waves.
Uniqueness of the equilibrium of an electron plasma on magnetic surfaces
Durand de Gevigney, Benoit
2011-01-15
The equilibrium of an electron plasma on magnetic surfaces is governed by a Poisson-Boltzmann equation. The electrons follow a Boltzmann distribution on each surface and the charge density depends exponentially on the electric potential. It is a well-known property that the classical Poisson's equation, for which the charge density is an independent parameter, possesses a unique solution provided suitable boundary conditions are given. Here we show that the Poisson-Boltzmann equation describing electron plasmas on magnetic surfaces also has a unique solution.
Temperature based Restricted Boltzmann Machines
NASA Astrophysics Data System (ADS)
Li, Guoqi; Deng, Lei; Xu, Yi; Wen, Changyun; Wang, Wei; Pei, Jing; Shi, Luping
2016-01-01
Restricted Boltzmann machines (RBMs), which apply graphical models to learning probability distribution over a set of inputs, have attracted much attention recently since being proposed as building blocks of multi-layer learning systems called deep belief networks (DBNs). Note that temperature is a key factor of the Boltzmann distribution that RBMs originate from. However, none of existing schemes have considered the impact of temperature in the graphical model of DBNs. In this work, we propose temperature based restricted Boltzmann machines (TRBMs) which reveals that temperature is an essential parameter controlling the selectivity of the firing neurons in the hidden layers. We theoretically prove that the effect of temperature can be adjusted by setting the parameter of the sharpness of the logistic function in the proposed TRBMs. The performance of RBMs can be improved by adjusting the temperature parameter of TRBMs. This work provides a comprehensive insights into the deep belief networks and deep learning architectures from a physical point of view.
Temperature based Restricted Boltzmann Machines.
Li, Guoqi; Deng, Lei; Xu, Yi; Wen, Changyun; Wang, Wei; Pei, Jing; Shi, Luping
2016-01-13
Restricted Boltzmann machines (RBMs), which apply graphical models to learning probability distribution over a set of inputs, have attracted much attention recently since being proposed as building blocks of multi-layer learning systems called deep belief networks (DBNs). Note that temperature is a key factor of the Boltzmann distribution that RBMs originate from. However, none of existing schemes have considered the impact of temperature in the graphical model of DBNs. In this work, we propose temperature based restricted Boltzmann machines (TRBMs) which reveals that temperature is an essential parameter controlling the selectivity of the firing neurons in the hidden layers. We theoretically prove that the effect of temperature can be adjusted by setting the parameter of the sharpness of the logistic function in the proposed TRBMs. The performance of RBMs can be improved by adjusting the temperature parameter of TRBMs. This work provides a comprehensive insights into the deep belief networks and deep learning architectures from a physical point of view.
The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics
NASA Astrophysics Data System (ADS)
Tirnakli, Ugur; Borges, Ernesto P.
2016-03-01
As well known, Boltzmann-Gibbs statistics is the correct way of thermostatistically approaching ergodic systems. On the other hand, nontrivial ergodicity breakdown and strong correlations typically drag the system into out-of-equilibrium states where Boltzmann-Gibbs statistics fails. For a wide class of such systems, it has been shown in recent years that the correct approach is to use Tsallis statistics instead. Here we show how the dynamics of the paradigmatic conservative (area-preserving) stan-dard map exhibits, in an exceptionally clear manner, the crossing from one statistics to the other. Our results unambiguously illustrate the domains of validity of both Boltzmann-Gibbs and Tsallis statistical distributions. Since various important physical systems from particle confinement in magnetic traps to autoionization of molecular Rydberg states, through particle dynamics in accelerators and comet dynamics, can be reduced to the standard map, our results are expected to enlighten and enable an improved interpretation of diverse experimental and observational results.
The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics.
Tirnakli, Ugur; Borges, Ernesto P
2016-03-23
As well known, Boltzmann-Gibbs statistics is the correct way of thermostatistically approaching ergodic systems. On the other hand, nontrivial ergodicity breakdown and strong correlations typically drag the system into out-of-equilibrium states where Boltzmann-Gibbs statistics fails. For a wide class of such systems, it has been shown in recent years that the correct approach is to use Tsallis statistics instead. Here we show how the dynamics of the paradigmatic conservative (area-preserving) stan-dard map exhibits, in an exceptionally clear manner, the crossing from one statistics to the other. Our results unambiguously illustrate the domains of validity of both Boltzmann-Gibbs and Tsallis statistical distributions. Since various important physical systems from particle confinement in magnetic traps to autoionization of molecular Rydberg states, through particle dynamics in accelerators and comet dynamics, can be reduced to the standard map, our results are expected to enlighten and enable an improved interpretation of diverse experimental and observational results.
NASA Astrophysics Data System (ADS)
Xiao, Zhiyong
2016-12-01
Accumulation of impact craters is the major reason causing equilibrium of crater populations on airless planetary surfaces. Besides primary craters, the effect of widespread secondaries on the equilibrium of local crater populations is little studied. Here the different secondary crater populations formed by the Hokusai crater on Mercury are systematically studied, and they are compared with those on the Moon to investigate their contribution to the evolution of local crater populations. Self-secondaries cause equilibrium on continuous ejecta deposits in a short time, and the equilibrium crater population has a differential size-frequency distribution (SFD) slope of about -3. Background secondaries are abundant on Mercury, and equilibrium caused by a combination of primaries and potential background secondaries follows the same pattern on the Moon and Mercury. The spatial dispersion of fragments that form both near-field and distant secondaries is the major factor affecting the degree of mutual destruction and thus the final crater SFD. Some clustered distant secondaries on Mercury are likely formed by individual fragments considering their large spatial dispersion and identical morphology with same-sized primaries, and the SFD rollovers of these secondaries possibly reflect the inherent SFD rollovers of the impact fragments. Near-field secondaries and many other distant secondaries have morphology and spatial distribution that are consistent with being formed by clustered fragments, and mutual destruction of secondaries may be the major reason causing the observed SFD rollovers. Heterogeneous secondary impacts are a potential explanation for both different crater densities within the equilibrium diameter range and different regolith thicknesses on coeval surfaces.
NASA Astrophysics Data System (ADS)
Frank, T. D.
The virial theorem and the concept of canonical-statistical distributions represent two fundamental elements of statistical physics. We apply these concepts to hand tremor oscillations recorded from six Parkinson patients. We find that the virial theorem holds for Parkinson tremor oscillations. In contrast, we find that the concept of canonical distributions fails to a certain extent and needs to be replaced by the notion of non-canonical (i.e., canonical-dissipative) distributions. In doing so, our analysis reveals both general statistical aspects and non-equilibrium aspects of Parkinson hand tremor.
Saksena, Radhika S; Mazzeo, Marco D; Zasada, Stefan J; Coveney, Peter V
2010-08-28
We present very large-scale rheological studies of self-assembled cubic gyroid liquid crystalline phases in ternary mixtures of oil, water and amphiphilic species performed on petascale supercomputers using the lattice-Boltzmann method. These nanomaterials have found diverse applications in materials science and biotechnology, for example, in photovoltaic devices and protein crystallization. They are increasingly gaining importance as delivery vehicles for active agents in pharmaceuticals, personal care products and food technology. In many of these applications, the self-assembled structures are subject to flows of varying strengths and we endeavour to understand their rheological response with the objective of eventually predicting it under given flow conditions. Computationally, our lattice-Boltzmann simulations of ternary fluids are inherently memory- and data-intensive. Furthermore, our interest in dynamical processes necessitates remote visualization and analysis as well as the associated transfer and storage of terabytes of time-dependent data. These simulations are distributed on a high-performance grid infrastructure using the application hosting environment; we employ a novel parallel in situ visualization approach which is particularly suited for such computations on petascale resources. We present computational and I/O performance benchmarks of our application on three different petascale systems.
Importance of Pore Size Distribution of Fine-grained Sediments on Gas Hydrate Equilibrium
NASA Astrophysics Data System (ADS)
Kwon, T. H.; Kim, H. S.; Cho, G. C.; Park, T. H.
2015-12-01
Gas hydrates have been considered as a new source of natural gases. For the gas hydrate production, the gas hydrate reservoir should be depressurized below the equilibrium pressure of gas hydrates. Therefore, it is important to predict the equilibrium of gas hydrates in the reservoir conditions because it can be affected by the pore size of the host sediments due to the capillary effect. In this study, gas hydrates were synthesized in fine-grained sediment samples including a pure silt sample and a natural clayey silt sample cored from a hydrate occurrence region in Ulleung Basin, East Sea, offshore Korea. Pore size distributions of the samples were obtained by the nitrogen adsorption and desorption test and the mercury intrusion porosimetry. The equilibrium curve of gas hydrates in the fine-grained sediments were found to be significantly influenced by the clay fraction and the corresponding small pores (>50 nm in diameter). For the clayey silt sample, the equilibrium pressure was higher by ~1.4 MPa than the bulk equilibrium pressure. In most cases of oceanic gas hydrate reservoirs, sandy layers are found interbedded with fine-grained sediment layers while gas hydrates are intensively accumulated in the sandy layers. Our experiment results reveal the inhibition effect of fine-grained sediments against gas hydrate formation, in which greater driving forces (e.g., higher pressure or lower temperature) are required during natural gas migration. Therefore, gas hydrate distribution in interbedded layers of sandy and fine-grained sediments can be explained by such capillary effect induced by the pore size distribution of host sediments.
Hunt, Rosemary A R; Ludlow, R Frederick; Otto, Sijbren
2009-11-19
Multicomponent chemical systems that exhibit a network of covalent and intermolecular interactions may produce interesting and often unexpected chemical or physical behavior. The formation of aggregates is a well-recognized example and presents a particular analytical challenge. We now report the development of a numerical fitting method capable of estimating equilibrium constants for the formation of aggregates from the product distribution of a dynamic combinatorial library containing self-recognizing library members.
Mélykúti, Bence; Hespanha, João P.; Khammash, Mustafa
2014-01-01
Many biochemical reaction networks are inherently multiscale in time and in the counts of participating molecular species. A standard technique to treat different time scales in the stochastic kinetics framework is averaging or quasi-steady-state analysis: it is assumed that the fast dynamics reaches its equilibrium (stationary) distribution on a time scale where the slowly varying molecular counts are unlikely to have changed. We derive analytic equilibrium distributions for various simple biochemical systems, such as enzymatic reactions and gene regulation models. These can be directly inserted into simulations of the slow time-scale dynamics. They also provide insight into the stimulus–response of these systems. An important model for which we derive the analytic equilibrium distribution is the binding of dimer transcription factors (TFs) that first have to form from monomers. This gene regulation mechanism is compared to the cases of the binding of simple monomer TFs to one gene or to multiple copies of a gene, and to the cases of the cooperative binding of two or multiple TFs to a gene. The results apply equally to ligands binding to enzyme molecules. PMID:24920118
Clamping in Boltzmann machines.
Livesey, M
1991-01-01
A certain assumption that appears in the proof of correctness of the standard Boltzmann machine learning procedure is investigated. The assumption, called the clamping assumption, concerns the behavior of a Boltzmann machine when some of its units are clamped to a fixed state. It is argued that the clamping assumption is essentially an assertion of the time reversibility of a certain Markov chain underlying the behavior of the Boltzmann machine. As such, the clamping assumption is generally false, though it is certainly true of the Boltzmann machines themselves. The author also considers how the concept of the Boltzmann machine may be generalized while retaining the validity of the clamping assumption.
Golneshan, A A; Lahonian, M
2011-01-01
In clinical applications of magnetic fluid hyperthermia (MFH) for cancer treatment it is very important to ensure maximum damage to the tumour while protecting the normal tissue. The resultant heating pattern by magnetic nanoparticles (MNPs) in the tumour is closely related to the dispersion of MNPs. In this study the effect of MNPs dispersion on temperature distribution in a tumour and surrounding healthy tissue, during MFH, has been investigated. Accordingly, the Pennes bio-heat equation (BHE) in a spherical tissue with Neumann curved boundary condition has been resolved. The effects of blood perfusion, metabolism heat generation as well as MNPs heat dissipation in an alternating magnetic field as source term, have been considered. To solve the Pennes BHE, the three dimensional lattice Boltzmann method (LBM) has been used. To show the accuracy of the model, simulations have been compared with analytical, experimental and numerical results, reported in the literature. Then, temperature distribution within tissue has been investigated in two cases, homogeneous distribution and Gaussian distribution of specific absorption rate (SAR). Results showed that for the studied cases, unlike homogeneous distribution, Gaussian distribution of SAR is able to raise the temperature of tumour cells above the treatment temperature.
Lattice Boltzmann modeling of phonon transport
NASA Astrophysics Data System (ADS)
Guo, Yangyu; Wang, Moran
2016-06-01
A novel lattice Boltzmann scheme is proposed for phonon transport based on the phonon Boltzmann equation. Through the Chapman-Enskog expansion, the phonon lattice Boltzmann equation under the gray relaxation time approximation recovers the classical Fourier's law in the diffusive limit. The numerical parameters in the lattice Boltzmann model are therefore rigorously correlated to the bulk material properties. The new scheme does not only eliminate the fictitious phonon speed in the diagonal direction of a square lattice system in the previous lattice Boltzmann models, but also displays very robust performances in predicting both temperature and heat flux distributions consistent with analytical solutions for diverse numerical cases, including steady-state and transient, macroscale and microscale, one-dimensional and multi-dimensional phonon heat transport. This method may provide a powerful numerical tool for deep studies of nonlinear and nonlocal heat transports in nanosystems.
Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks.
Anderson, David F; Craciun, Gheorghe; Gopalkrishnan, Manoj; Wiuf, Carsten
2015-09-01
We consider the relationship between stationary distributions for stochastic models of reaction systems and Lyapunov functions for their deterministic counterparts. Specifically, we derive the well-known Lyapunov function of reaction network theory as a scaling limit of the non-equilibrium potential of the stationary distribution of stochastically modeled complex balanced systems. We extend this result to general birth-death models and demonstrate via example that similar scaling limits can yield Lyapunov functions even for models that are not complex or detailed balanced, and may even have multiple equilibria.
U.S. stock market interaction network as learned by the Boltzmann machine
Borysov, Stanislav S.; Roudi, Yasser; Balatsky, Alexander V.
2015-12-07
Here, we study historical dynamics of joint equilibrium distribution of stock returns in the U.S. stock market using the Boltzmann distribution model being parametrized by external fields and pairwise couplings. Within Boltzmann learning framework for statistical inference, we analyze historical behavior of the parameters inferred using exact and approximate learning algorithms. Since the model and inference methods require use of binary variables, effect of this mapping of continuous returns to the discrete domain is studied. The presented results show that binarization preserves the correlation structure of the market. Properties of distributions of external fields and couplings as well as the market interaction network and industry sector clustering structure are studied for different historical dates and moving window sizes. We demonstrate that the observed positive heavy tail in distribution of couplings is related to the sparse clustering structure of the market. We also show that discrepancies between the model’s parameters might be used as a precursor of financial instabilities.
U.S. stock market interaction network as learned by the Boltzmann machine
NASA Astrophysics Data System (ADS)
Borysov, Stanislav S.; Roudi, Yasser; Balatsky, Alexander V.
2015-12-01
We study historical dynamics of joint equilibrium distribution of stock returns in the U.S. stock market using the Boltzmann distribution model being parametrized by external fields and pairwise couplings. Within Boltzmann learning framework for statistical inference, we analyze historical behavior of the parameters inferred using exact and approximate learning algorithms. Since the model and inference methods require use of binary variables, effect of this mapping of continuous returns to the discrete domain is studied. The presented results show that binarization preserves the correlation structure of the market. Properties of distributions of external fields and couplings as well as the market interaction network and industry sector clustering structure are studied for different historical dates and moving window sizes. We demonstrate that the observed positive heavy tail in distribution of couplings is related to the sparse clustering structure of the market. We also show that discrepancies between the model's parameters might be used as a precursor of financial instabilities.
Zheng, Xiliang; Wang, Jin
2015-04-01
We uncovered the universal statistical laws for the biomolecular recognition/binding process. We quantified the statistical energy landscapes for binding, from which we can characterize the distributions of the binding free energy (affinity), the equilibrium constants, the kinetics and the specificity by exploring the different ligands binding with a particular receptor. The results of the analytical studies are confirmed by the microscopic flexible docking simulations. The distribution of binding affinity is Gaussian around the mean and becomes exponential near the tail. The equilibrium constants of the binding follow a log-normal distribution around the mean and a power law distribution in the tail. The intrinsic specificity for biomolecular recognition measures the degree of discrimination of native versus non-native binding and the optimization of which becomes the maximization of the ratio of the free energy gap between the native state and the average of non-native states versus the roughness measured by the variance of the free energy landscape around its mean. The intrinsic specificity obeys a Gaussian distribution near the mean and an exponential distribution near the tail. Furthermore, the kinetics of binding follows a log-normal distribution near the mean and a power law distribution at the tail. Our study provides new insights into the statistical nature of thermodynamics, kinetics and function from different ligands binding with a specific receptor or equivalently specific ligand binding with different receptors. The elucidation of distributions of the kinetics and free energy has guiding roles in studying biomolecular recognition and function through small-molecule evolution and chemical genetics.
Li, Zhihui; Ma, Qiang; Wu, Junlin; Jiang, Xinyu; Zhang, Hanxin
2014-12-09
Based on the Gas-Kinetic Unified Algorithm (GKUA) directly solving the Boltzmann model equation, the effect of rotational non-equilibrium is investigated recurring to the kinetic Rykov model with relaxation property of rotational degrees of freedom. The spin movement of diatomic molecule is described by moment of inertia, and the conservation of total angle momentum is taken as a new Boltzmann collision invariant. The molecular velocity distribution function is integrated by the weight factor on the internal energy, and the closed system of two kinetic controlling equations is obtained with inelastic and elastic collisions. The optimization selection technique of discrete velocity ordinate points and numerical quadrature rules for macroscopic flow variables with dynamic updating evolvement are developed to simulate hypersonic flows, and the gas-kinetic numerical scheme is constructed to capture the time evolution of the discretized velocity distribution functions. The gas-kinetic boundary conditions in thermodynamic non-equilibrium and numerical procedures are studied and implemented by directly acting on the velocity distribution function, and then the unified algorithm of Boltzmann model equation involving non-equilibrium effect is presented for the whole range of flow regimes. The hypersonic flows involving non-equilibrium effect are numerically simulated including the inner flows of shock wave structures in nitrogen with different Mach numbers of 1.5-Ma-25, the planar ramp flow with the whole range of Knudsen numbers of 0.0009-Kn-10 and the three-dimensional re-entering flows around tine double-cone body.
Effects of non-equilibrium particle distributions in deuterium-tritium burning
Michta, D; Graziani, F; Pruet, J; Luu, T
2009-08-18
We investigate the effects of non-equilibrium particle distributions resulting from rapid deuterium-tritium burning in plasmas using a Fokker-Planck code that incorporates small-angle Coulomb scattering, Brehmsstrahlung, Compton scattering, and thermal-nuclear burning. We find that in inertial confinement fusion environments, deviations away from Maxwellian distributions for either deuterium or tritium ions are small and result in 1% changes in the energy production rates. The deuterium and tritium effective temperatures are not equal, but differ by only about 2.5% near the time of peak burn rate. Simulations with high Z (Xe) dopants show that the dopant temperature closely tracks that of the fuel. On the other hand, fusion product ion distributions are highly non-Maxwellian, and careful treatments of energy-exchange between these ions and other particles is important for determining burn rates.
NASA Astrophysics Data System (ADS)
Chakraborty, P.; Kapusta, J. I.
2017-01-01
In simulations of high energy heavy ion collisions that employ viscous hydrodynamics, single particle distributions are distorted from their thermal equilibrium form due to gradients in the flow velocity. These are closely related to the formulas for the shear and bulk viscosities in the quasiparticle approximation. Distorted single particle distributions are now commonly used to calculate the emission of photons and dilepton pairs, and in the late stage to calculate the conversion of a continuous fluid to individual particles. We show how distortions of the single particle distribution functions due to both shear and bulk viscous effects can be done rigorously in the quasiparticle approximation and illustrate it with the linear σ model at finite temperature.
H to Zn Ionization Equilibrium for the Non-Maxwellian Electron κ-distributions: Updated Calculations
NASA Astrophysics Data System (ADS)
Dzifčáková, E.; Dudík, J.
2013-05-01
New data for the calculation of ionization and recombination rates have been published in the past few years, most of which are included in the CHIANTI database. We used these data to calculate collisional ionization and recombination rates for the non-Maxwellian κ-distributions with an enhanced number of particles in the high-energy tail, which have been detected in the solar transition region and the solar wind. Ionization equilibria for elements H to Zn are derived. The κ-distributions significantly influence both the ionization and recombination rates and widen the ion abundance peaks. In comparison with the Maxwellian distribution, the ion abundance peaks can also be shifted to lower or higher temperatures. The updated ionization equilibrium calculations result in large changes for several ions, notably Fe VIII-Fe XIV. The results are supplied in electronic form compatible with the CHIANTI database.
NASA Astrophysics Data System (ADS)
Di Troia, C.
2015-11-01
A class of parametric distribution functions was proposed in (Di Troia 2012 Plasma Phys. Control. Fusion 54 105017) as equilibrium distribution functions (EDFs) for charged particles in fusion plasmas, representing supra-thermal particles in anisotropic equilibria for Neutral Beam Injection and Ion Cyclotron Heating scenarios. Moreover, those EDFs can be used to represent also nearly isotropic equilibria for Slowing-Down alpha particles and core thermal plasma populations. Such EDFs depend on constants of motion (COMs). In axisymmetric system with no equilibrium electric field, they depend on toroidal canonical momentum {{P}φ} , kinetic energy w and magnetic moment μ. In the present work, the same EDFs are obtained from first principles and general hypothesis. The derivation is probabilistic and makes use of the Bayes’ Theorem. The bayesian argument is used to describe how the plasma is far from the prior probability distribution function (pdf), e.g. Maxwellian, based on the information obtained from magnetic moment and guiding center velocity pdf. Once the general functional form of the EDF has been settled, it is shown how to associate a modified Landau collision operator in the Fokker-Planck equation, to describe the system relaxation towards the proposed EDF.
Diffusion and near-equilibrium distribution of MRI and CT contrast agents in articular cartilage
NASA Astrophysics Data System (ADS)
Silvast, Tuomo S.; Kokkonen, Harri T.; Jurvelin, Jukka S.; Quinn, Thomas M.; Nieminen, Miika T.; Töyräs, Juha
2009-11-01
Charged contrast agents have been used both in vitro and in vivo for estimation of the fixed charge density (FCD) in articular cartilage. In the present study, the effects of molecular size and charge on the diffusion and equilibrium distribution of several magnetic resonance imaging (MRI) and computed tomography (CT) contrast agents were investigated. Full thickness cartilage disks (Ø = 4.0 mm, n = 64) were prepared from fresh bovine patellae. Contrast agent (gadopentetate: Magnevist®, gadodiamide: Omniscan™, ioxaglate: Hexabrix™ or sodium iodide: NaI) diffusion was allowed either through the articular surface or through the deep cartilage. CT imaging of the samples was conducted before contrast agent administration and after 1, 5, 9, 16, 25 and 29 h (and with three samples after 2, 3, 4 and 5 days) diffusion using a clinical peripheral quantitative computed tomography (pQCT) instrument. With all contrast agents, the diffusion through the deep cartilage was slower when compared to the diffusion through the articular surface. With ioxaglate, gadopentetate and gadodiamide it took over 29 h for diffusion to reach the near-equilibrium state. The slow diffusion of the contrast agents raise concerns regarding the validity of techniques for FCD estimation, as these contrast agents may not reach the equilibrium state that is assumed. However, since cartilage composition, i.e. deep versus superficial, had a significant effect on diffusion, imaging of the nonequilibrium diffusion process might enable more accurate assessment of cartilage integrity.
NASA Astrophysics Data System (ADS)
Suzuki, Hideyuki; Imura, Jun-Ichi; Horio, Yoshihiko; Aihara, Kazuyuki
2013-04-01
The chaotic Boltzmann machine proposed in this paper is a chaotic pseudo-billiard system that works as a Boltzmann machine. Chaotic Boltzmann machines are shown numerically to have computing abilities comparable to conventional (stochastic) Boltzmann machines. Since no randomness is required, efficient hardware implementation is expected. Moreover, the ferromagnetic phase transition of the Ising model is shown to be characterised by the largest Lyapunov exponent of the proposed system. In general, a method to relate probabilistic models to nonlinear dynamics by derandomising Gibbs sampling is presented.
Suzuki, Hideyuki; Imura, Jun-ichi; Horio, Yoshihiko; Aihara, Kazuyuki
2013-01-01
The chaotic Boltzmann machine proposed in this paper is a chaotic pseudo-billiard system that works as a Boltzmann machine. Chaotic Boltzmann machines are shown numerically to have computing abilities comparable to conventional (stochastic) Boltzmann machines. Since no randomness is required, efficient hardware implementation is expected. Moreover, the ferromagnetic phase transition of the Ising model is shown to be characterised by the largest Lyapunov exponent of the proposed system. In general, a method to relate probabilistic models to nonlinear dynamics by derandomising Gibbs sampling is presented.
Equilibrium and stability in a heliotron with anisotropic hot particle slowing-down distribution
Cooper, W. A.; Asahi, Y.; Narushima, Y.; Suzuki, Y.; Watanabe, K. Y.; Graves, J. P.; Isaev, M. Yu.
2012-10-15
The equilibrium and linear fluid Magnetohydrodynamic (MHD) stability in an inward-shifted large helical device heliotron configuration are investigated with the 3D ANIMEC and TERPSICHORE codes, respectively. A modified slowing-down distribution function is invoked to study anisotropic pressure conditions. An appropriate choice of coefficients and exponents allows the simulation of neutral beam injection in which the angle of injection is varied from parallel to perpendicular. The fluid stability analysis concentrates on the application of the Johnson-Kulsrud-Weimer energy principle. The growth rates are maximum at <{beta}>{approx}2%, decrease significantly at <{beta}>{approx}4.5%, do not vary significantly with variations of the injection angle and are similar to those predicted with a bi-Maxwellian hot particle distribution function model. Stability is predicted at <{beta}>{approx}2.5% with a sufficiently peaked energetic particle pressure profile. Electrostatic potential forms from the MHD instability necessary for guiding centre orbit following are calculated.
NASA Astrophysics Data System (ADS)
Bazow, D.; Denicol, G. S.; Heinz, U.; Martinez, M.; Noronha, J.
2016-12-01
The dissipative dynamics of an expanding massless gas with constant cross section in a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe is studied. The mathematical problem of solving the full nonlinear relativistic Boltzmann equation is recast into an infinite set of nonlinear ordinary differential equations for the moments of the one-particle distribution function. Momentum-space resolution is determined by the number of nonhydrodynamic modes included in the moment hierarchy, i.e., by the truncation order. We show that in the FLRW spacetime the nonhydrodynamic modes decouple completely from the hydrodynamic degrees of freedom. This results in the system flowing as an ideal fluid while at the same time producing entropy. The solutions to the nonlinear Boltzmann equation exhibit transient tails of the distribution function with nontrivial momentum dependence. The evolution of this tail is not correctly captured by the relaxation time approximation nor by the linearized Boltzmann equation. However, the latter probes additional high-momentum details unresolved by the relaxation time approximation. While the expansion of the FLRW spacetime is slow enough for the system to move towards (and not away from) local thermal equilibrium, it is not sufficiently slow for the system to actually ever reach complete local equilibrium. Equilibration is fastest in the relaxation time approximation, followed, in turn, by kinetic evolution with a linearized and a fully nonlinear Boltzmann collision term.
Wong, Fiona; Wania, Frank
2011-06-01
Assessing the behaviour of organic chemicals in soil is a complex task as it is governed by the physical chemical properties of the chemicals, the characteristics of the soil as well as the ambient conditions of the environment. The chemical partitioning space, defined by the air-water partition coefficient (K(AW)) and the soil organic carbon-water partition coefficient (K(OC)), was employed to visualize the equilibrium distribution of organic contaminants between the air-filled pores, the pore water and the solid phases of the bulk soil and the relative importance of the three transport processes removing contaminants from soil (evaporation, leaching and particle erosion). The partitioning properties of twenty neutral organic chemicals (i.e. herbicides, pharmaceuticals, polychlorinated biphenyls and volatile chemicals) were estimated using poly-parameter linear free energy relationships and superimposed onto these maps. This allows instantaneous estimation of the equilibrium phase distribution and mobility of neutral organic chemicals in soil. Although there is a link between the major phase and the dominant transport process, such that chemicals found in air-filled pore space are subject to evaporation, those in water-filled pore space undergo leaching and those in the sorbed phase are associated with particle erosion, the partitioning coefficient thresholds for distribution and mobility can often deviate by many orders of magnitude. In particular, even a small fraction of chemical in pore water or pore air allows for evaporation and leaching to dominate over solid phase transport. Multiple maps that represent soils that differ in the amount and type of soil organic matter, water saturation, temperature, depth of surface soil horizon, and mineral matters were evaluated.
NASA Astrophysics Data System (ADS)
Bihani, A. D.; Daigle, H.; Cook, A.; Glosser, D.; Shushtarian, A.
2015-12-01
Coexistence of three methane phases (liquid (L), gas (G), hydrate (H)) in marine gas hydrate systems may occur according to in-situ pressure, temperature, salinity and pore size. In sediments with salinity close to seawater, a discrete zone of three-phase (3P) equilibrium may occur near the base of the regional hydrate stability zone (RHSZ) due to capillary effects. The existence of a 3P zone influences the location of the bottom-simulating reflection (BSR) and has implications for methane fluxes at the base of the RHSZ. We studied hydrate stability conditions in two wells, WR313-G and WR313-H, at Walker Ridge Block 313 in the northern Gulf of Mexico. We determined pore size distributions (PSD) by constructing a synthetic nuclear magnetic resonance (NMR) relaxation time distribution. Correlations were obtained by non-linear regression on NMR, gamma ray, and bulk density logs from well KC-151 at Keathley Canyon. The correlations enabled construction of relaxation time distributions for WR313-G and WR313-H, which were used to predict PSD through comparison with mercury injection capillary pressure measurements. With the computed PSD, L+H and L+G methane solubility was determined from in-situ pressure and temperature. The intersection of the L+G and L+H curves for various pore sizes allowed calculation of the depth range of the 3P equilibrium zone. As in previous studies at Blake Ridge and Hydrate Ridge, the top of the 3P zone moves upwards with increasing water depth and overlies the bulk 3P equilibrium depth. In clays at Walker Ridge, the predicted thickness of the 3P zone is approximately 35 m, but in coarse sands it is only a few meters due to the difference in absolute pore sizes and the width of the PSD. The thick 3P zone in the clays may explain in part why the BSR is only observed in the sand layers at Walker Ridge, although other factors may influence the presence or absence of a BSR.
Bihani, Abhishek; Daigle, Hugh; Cook, Ann; Glosser, Deborah; Shushtarian, Arash
2015-12-15
Coexistence of three methane phases (liquid (L), gas (G), hydrate (H)) in marine gas hydrate systems may occur according to in-situ pressure, temperature, salinity and pore size. In sediments with salinity close to seawater, a discrete zone of three-phase (3P) equilibrium may occur near the base of the regional hydrate stability zone (RHSZ) due to capillary effects. The existence of a 3P zone influences the location of the bottom-simulating reflection (BSR) and has implications for methane fluxes at the base of the RHSZ. We studied hydrate stability conditions in two wells, WR313-G and WR313-H, at Walker Ridge Block 313 in the northern Gulf of Mexico. We determined pore size distributions (PSD) by constructing a synthetic nuclear magnetic resonance (NMR) relaxation time distribution. Correlations were obtained by non-linear regression on NMR, gamma ray, and bulk density logs from well KC-151 at Keathley Canyon. The correlations enabled construction of relaxation time distributions for WR313-G and WR313-H, which were used to predict PSD through comparison with mercury injection capillary pressure measurements. With the computed PSD, L+H and L+G methane solubility was determined from in-situ pressure and temperature. The intersection of the L+G and L+H curves for various pore sizes allowed calculation of the depth range of the 3P equilibrium zone. As in previous studies at Blake Ridge and Hydrate Ridge, the top of the 3P zone moves upwards with increasing water depth and overlies the bulk 3P equilibrium depth. In clays at Walker Ridge, the predicted thickness of the 3P zone is approximately 35 m, but in coarse sands it is only a few meters due to the difference in absolute pore sizes and the width of the PSD. The thick 3P zone in the clays may explain in part why the BSR is only observed in the sand layers at Walker Ridge, although other factors may influence the presence or absence of a BSR.
Raut, L K
1991-01-01
A study is conducted in attempts to increase the understanding of the links between macroeconomic effects and causes of population growth in formulating policy. An overlapping generations general equilibrium model is employed aggregating household decisions about fertility, savings, and investment in the human capital of children with the objective of studying intertemporal relationships among population growth, income distribution, inter-generation social mobility, skill composition of the labor force, and household income. As a result of endogenous fertility, the equilibrium path attains steady state from the second generation. Income tax transfer, child taxation, and social security taxation policies are also examined in the paper. A structural explanation is given for the inverse household income-child quantity and negative child quality-quantity relationships seen in developing countries. In a Cobb-Douglas economy, these relationships hold in the short-run, potentially working over the long-run in other economies. Overall, the model shows that group interests may hinder emergence of perfect capital markets with private initiatives. Where developing countries are concerned, these results have strong implications for population policy. A policy mix of building good quality schools, or subsidizing rural education, introducing a formal social security program, and providing high-yield, risk-free investments, banking, and insurance services to the poor is recommended.
Temperature based Restricted Boltzmann Machines
Li, Guoqi; Deng, Lei; Xu, Yi; Wen, Changyun; Wang, Wei; Pei, Jing; Shi, Luping
2016-01-01
Restricted Boltzmann machines (RBMs), which apply graphical models to learning probability distribution over a set of inputs, have attracted much attention recently since being proposed as building blocks of multi-layer learning systems called deep belief networks (DBNs). Note that temperature is a key factor of the Boltzmann distribution that RBMs originate from. However, none of existing schemes have considered the impact of temperature in the graphical model of DBNs. In this work, we propose temperature based restricted Boltzmann machines (TRBMs) which reveals that temperature is an essential parameter controlling the selectivity of the firing neurons in the hidden layers. We theoretically prove that the effect of temperature can be adjusted by setting the parameter of the sharpness of the logistic function in the proposed TRBMs. The performance of RBMs can be improved by adjusting the temperature parameter of TRBMs. This work provides a comprehensive insights into the deep belief networks and deep learning architectures from a physical point of view. PMID:26758235
Multi-valued Boltzmann machine
Lin, C.T.; Lee, C.S.G.
1995-04-01
The idea of Hopfield network is based on the Ising spin glass model in which each spin has only two possible states: up and down. This paper generalizes these ideas to multivalue case based on the XY spin glass model in which each spin can be in any direction in a plane. Simply using the gradient descent method and the analog Hopfield network, two different analog connectionist structures and their corresponding evolving rules are first designed to transform the XY spin glass model to distributed computational models. Since these two structures can easily get stuck in local minima, a multivalued Boltzmann machine is proposed which adopts the discrete planar spin glass model for the local minimum problem. The multivalued Boltzmann machine can be applied to the mobile robot navigation problem by defining proper artificial magnetic field on the traverse terrain. This new approach has shown to have several advantages over existing graph search and potential field techniques. 28 refs.
A lattice Boltzmann model for the Burgers-Fisher equation.
Zhang, Jianying; Yan, Guangwu
2010-06-01
A lattice Boltzmann model is developed for the one- and two-dimensional Burgers-Fisher equation based on the method of the higher-order moment of equilibrium distribution functions and a series of partial differential equations in different time scales. In order to obtain the two-dimensional Burgers-Fisher equation, vector sigma(j) has been used. And in order to overcome the drawbacks of "error rebound," a new assumption of additional distribution is presented, where two additional terms, in first order and second order separately, are used. Comparisons with the results obtained by other methods reveal that the numerical solutions obtained by the proposed method converge to exact solutions. The model under new assumption gives better results than that with second order assumption.
Acoustic equations of state for simple lattice Boltzmann velocity sets.
Viggen, Erlend Magnus
2014-07-01
The lattice Boltzmann (LB) method typically uses an isothermal equation of state. This is not sufficient to simulate a number of acoustic phenomena where the equation of state cannot be approximated as linear and constant. However, it is possible to implement variable equations of state by altering the LB equilibrium distribution. For simple velocity sets with velocity components ξ(iα)∈(-1,0,1) for all i, these equilibria necessarily cause error terms in the momentum equation. These error terms are shown to be either correctable or negligible at the cost of further weakening the compressibility. For the D1Q3 velocity set, such an equilibrium distribution is found and shown to be unique. Its sound propagation properties are found for both forced and free waves, with some generality beyond D1Q3. Finally, this equilibrium distribution is applied to a nonlinear acoustics simulation where both mechanisms of nonlinearity are simulated with good results. This represents an improvement on previous such simulations and proves that the compressibility of the method is still sufficiently strong even for nonlinear acoustics.
NASA Astrophysics Data System (ADS)
Barthes, Laurent; Mallet, Cécile
2010-05-01
Keywords: Rain Drop Size Distribution, Breakup, coalescence, disdrometer The study of the vertical evolution of raindrop size distributions (DSDs) during rainfall, from the freezing level isotherm to ground level, is a key to improving our understanding of the microphysics of rain. In numerous domains such as remote sensing, telecommunications, soil erosion, and the study of the rain's efficiency in 'washing' the atmosphere, the DSD plays an important role. Among the different processes affecting the evolution of DSD, breakup and coalescence are two of the most significant. Models of coalescence and breakup lead to equilibrium of the raindrop size distribution (DSD) after a fall through sufficient vertical height. At equilibrium, the DSD no longer evolves, and its shape is unique whatever the rain rate or LWC. This implies that the DSD is known, to within a multiplication constant. These models based on experimental measurements have been developed over the past 40 years. The Low and List (1982a,b) parameterization (hereinafter LL82) and the Greg M. McFarquhar (2004) model are both based on the same laboratory experiments, which lead to an equilibrium drop size distribution (EDSD) with two or three peaks, and an exponential tail with a slope of approximately Λ=65 cm-1. Numerous measurements using disdrometer collected in different climatic areas: Paris, France (Mars to October 2000), Iowa-City (April to October 2002), and Djougou (Benin June to September 2006) corresponding to 537 hours of rain period have shown that for high rain rates, close to a state of equilibrium, this slope lies between Λ=20 - 22 cm-1. This latter value is corroborated by others measurements found in the literature (Hu & Srivastava, 1995). Hu & Srivastava suggested that the Low and List parameterization may overestimate the effects of the breakup process. This hypothesis is in adequation with recent laboratory experiments (A.P. Barros 2008) in which the authors conclude that the number of
Student understanding of the Boltzmann factor
NASA Astrophysics Data System (ADS)
Smith, Trevor I.; Mountcastle, Donald B.; Thompson, John R.
2015-12-01
[This paper is part of the Focused Collection on Upper Division Physics Courses.] We present results of our investigation into student understanding of the physical significance and utility of the Boltzmann factor in several simple models. We identify various justifications, both correct and incorrect, that students use when answering written questions that require application of the Boltzmann factor. Results from written data as well as teaching interviews suggest that many students can neither recognize situations in which the Boltzmann factor is applicable nor articulate the physical significance of the Boltzmann factor as an expression for multiplicity, a fundamental quantity of statistical mechanics. The specific student difficulties seen in the written data led us to develop a guided-inquiry tutorial activity, centered around the derivation of the Boltzmann factor, for use in undergraduate statistical mechanics courses. We report on the development process of our tutorial, including data from teaching interviews and classroom observations of student discussions about the Boltzmann factor and its derivation during the tutorial development process. This additional information informed modifications that improved students' abilities to complete the tutorial during the allowed class time without sacrificing the effectiveness as we have measured it. These data also show an increase in students' appreciation of the origin and significance of the Boltzmann factor during the student discussions. Our findings provide evidence that working in groups to better understand the physical origins of the canonical probability distribution helps students gain a better understanding of when the Boltzmann factor is applicable and how to use it appropriately in answering relevant questions.
Three-dimensional lattice Boltzmann model for electrodynamics.
Mendoza, M; Muñoz, J D
2010-11-01
In this paper we introduce a three-dimensional Lattice-Boltzmann model that recovers in the continuous limit the Maxwell equations in materials. In order to build conservation equations with antisymmetric tensors, like the Faraday law, the model assigns four auxiliary vectors to each velocity vector. These auxiliary vectors, when combined with the distribution functions, give the electromagnetic fields. The evolution is driven by the usual Bhatnager-Gross-Krook (BGK) collision rule, but with a different form for the equilibrium distribution functions. This lattice Bhatnager-Gross-Krook (LBGK) model allows us to consider for both dielectrics and conductors with realistic parameters, and therefore it is adequate to simulate the most diverse electromagnetic problems, like the propagation of electromagnetic waves (both in dielectric media and in waveguides), the skin effect, the radiation pattern of a small dipole antenna and the natural frequencies of a resonant cavity, all with 2% accuracy. Actually, it shows to be one order of magnitude faster than the original Finite-difference time-domain (FDTD) formulation by Yee to reach the same accuracy. It is, therefore, a valuable alternative to simulate electromagnetic fields and opens lattice Boltzmann for a broad spectrum of new applications in electrodynamics.
Hahn, M.; Savin, D. W.
2015-02-10
We describe the influence of electron-impact multiple ionization (EIMI) on the ionization balance of collisionally ionized plasmas. Previous ionization balance calculations have largely neglected EIMI. Here, EIMI cross-section data are incorporated into calculations of both equilibrium and non-equilibrium charge-state distributions (CSDs). For equilibrium CSDs, we find that EIMI has only a small effect and can usually be ignored. However, for non-equilibrium plasmas the influence of EIMI can be important. In particular, we find that for plasmas in which the temperature oscillates there are significant differences in the CSD when including versus neglecting EIMI. These results have implications for modeling and spectroscopy of impulsively heated plasmas, such as nanoflare heating of the solar corona.
Nicholls, David C.; Dopita, Michael A.; Sutherland, Ralph S.; Kewley, Lisa J.; Palay, Ethan
2013-08-15
In this paper we develop tools for observers to use when analyzing nebular spectra for temperatures and metallicities, with two goals: to present a new, simple method to calculate equilibrium electron temperatures for collisionally excited line flux ratios, using the latest atomic data; and to adapt current methods to include the effects of possible non-equilibrium ''{kappa}'' electron energy distributions. Adopting recent collision strength data for [O III], [S III], [O II], [S II], and [N II], we find that existing methods based on older atomic data seriously overestimate the electron temperatures, even when considering purely Maxwellian statistics. If {kappa} distributions exist in H II regions and planetary nebulae as they do in solar system plasmas, it is important to investigate the observational consequences. This paper continues our previous work on the {kappa} distribution. We present simple formulaic methods that allow observers to (1) measure equilibrium electron temperatures and atomic abundances using the latest atomic data, and (2) to apply simple corrections to existing equilibrium analysis techniques to allow for possible non-equilibrium effects. These tools should lead to better consistency in temperature and abundance measurements, and a clearer understanding of the physics of H II regions and planetary nebulae.
A Tightly Coupled Non-Equilibrium Magneto-Hydrodynamic Model for Inductively Coupled RF Plasmas
2016-02-29
for public release; distribution unlimited 13. SUPPLEMENTARY NOTES Journal article published in the Journal of Applied Physics , Vol. #118, Issue #13...effects are described based on a hybrid State-to-State (StS) approach. A multi- temperature formulation is used to account for thermal non-equilibrium...allowing for non-Boltzmann distributions of their populations. Free-electrons are assumed Maxwellian at their own temperature . The governing equations
A Boltzmann treatment for the vorton excess problem
Peter, Patrick; Ringeval, Christophe E-mail: christophe.ringeval@uclouvain.be
2013-05-01
We derive and solve a Boltzmann equation governing the cosmological evolution of the number density of current carrying cosmic string loops, whose centrifugally supported equilibrium configurations are also referred to as vortons. The phase space is three-dimensional and consists of the time variable, the loop size, and a conserved quantum number. Our approach includes gravitational wave emission, a possibly finite lifetime for the vortons and works with any initial loop distribution and for any loop production function. We then show how our results generalize previous approaches on the vorton excess problem by tracking down the time evolution of the various sub-populations of current-carrying loops in a string network.
Lattice Boltzmann equation method for the Cahn-Hilliard equation
NASA Astrophysics Data System (ADS)
Zheng, Lin; Zheng, Song; Zhai, Qinglan
2015-01-01
In this paper a lattice Boltzmann equation (LBE) method is designed that is different from the previous LBE for the Cahn-Hilliard equation (CHE). The starting point of the present CHE LBE model is from the kinetic theory and the work of Lee and Liu [T. Lee and L. Liu, J. Comput. Phys. 229, 8045 (2010), 10.1016/j.jcp.2010.07.007]; however, because the CHE does not conserve the mass locally, a modified equilibrium density distribution function is introduced to treat the diffusion term in the CHE. Numerical simulations including layered Poiseuille flow, static droplet, and Rayleigh-Taylor instability have been conducted to validate the model. The results show that the predictions of the present LBE agree well with the analytical solution and other numerical results.
Force-field functor theory: classical force-fields which reproduce equilibrium quantum distributions
Babbush, Ryan; Parkhill, John; Aspuru-Guzik, Alán
2013-01-01
Feynman and Hibbs were the first to variationally determine an effective potential whose associated classical canonical ensemble approximates the exact quantum partition function. We examine the existence of a map between the local potential and an effective classical potential which matches the exact quantum equilibrium density and partition function. The usefulness of such a mapping rests in its ability to readily improve Born-Oppenheimer potentials for use with classical sampling. We show that such a map is unique and must exist. To explore the feasibility of using this result to improve classical molecular mechanics, we numerically produce a map from a library of randomly generated one-dimensional potential/effective potential pairs then evaluate its performance on independent test problems. We also apply the map to simulate liquid para-hydrogen, finding that the resulting radial pair distribution functions agree well with path integral Monte Carlo simulations. The surprising accessibility and transferability of the technique suggest a quantitative route to adapting Born-Oppenheimer potentials, with a motivation similar in spirit to the powerful ideas and approximations of density functional theory. PMID:24790954
Babbush, Ryan; Parkhill, John; Aspuru-Guzik, Alán
2013-01-01
Feynman and Hibbs were the first to variationally determine an effective potential whose associated classical canonical ensemble approximates the exact quantum partition function. We examine the existence of a map between the local potential and an effective classical potential which matches the exact quantum equilibrium density and partition function. The usefulness of such a mapping rests in its ability to readily improve Born-Oppenheimer potentials for use with classical sampling. We show that such a map is unique and must exist. To explore the feasibility of using this result to improve classical molecular mechanics, we numerically produce a map from a library of randomly generated one-dimensional potential/effective potential pairs then evaluate its performance on independent test problems. We also apply the map to simulate liquid para-hydrogen, finding that the resulting radial pair distribution functions agree well with path integral Monte Carlo simulations. The surprising accessibility and transferability of the technique suggest a quantitative route to adapting Born-Oppenheimer potentials, with a motivation similar in spirit to the powerful ideas and approximations of density functional theory.
Delmau, L.H.
2002-06-13
An extension of the model developed in FY01 for predicting equilibrium distribution ratios in the Caustic-Side Solvent Extraction (CSSX) process is presented here. Motivation for extending the model arose from the need to predict extraction performance of the recently optimized solvent composition and the desire to include additional waste components. This model involves the extraction of cesium and potassium from different cesium, potassium, and sodium media over a large range of concentrations. Those different media include a large variety of anions such as nitrate, hydroxide, nitrite, chloride, fluoride, sulfate, and carbonate. The model was defined based on several hundreds of experimental data points and predicted satisfactorily the cesium extraction from five different SRS waste simulants. This process model encompassed almost exclusively 1:1:1 metal:anion:ligand species. Fluoride, sulfate, and carbonate species were found to be very little extractable, and their main impact is reflected through their activity effects. This model gave a very good cesium and potassium extraction prediction from sodium salts, which is what is needed when trying to predict the behavior from actual waste. However, the extraction from potassium or cesium salts, and the extraction of sodium could be improved, and some additional effort was devoted to improve the thermodynamic rigor of the model. Toward this end, more detailed anion-specific models were developed based on the cesium, potassium, and sodium distribution ratios obtained with simple systems containing single anions, but it has not yet proven possible to combine those models to obtain better predictions than provided by the process model.
Quasi-Equilibrium Density Distributions of Small Dust Aggregations in the Solar Nebula
NASA Astrophysics Data System (ADS)
Sekiya, Minoru
1998-06-01
The rotational velocity of a fluid element around the midplane of the solar nebula increased as dust settled toward the midplane. The Kelvin and Helmholtz instability due to velocity difference of a dust-rich region and a dust-poor region should have occurred and the dust layer became turbulent when the Richardson number decreased below the critical value. Then, dust aggregations stirred up due to turbulent diffusion and were prevented to settle further. In this paper, the sizes of dust aggregations are assumed to be equal to or smaller than the typical radius of chondrules (∼0.3 mm). In this case, even very weak turbulence stirs up dust aggregations. Therefore a dust density distribution is considered to be self regulated so that the Richardson number is nearly equal to the critical value. The quasi-equilibrium dust density distribution is derived analytically by assuming that the Richardson number is equal to the critical value. The derived dust density at the midplane is much smaller than the critical density of the gravitational stability, if the solar composition of dust to gas ratio is assumed. On the other hand, the dust aggregations concentrate around the midplane and the dust layer becomes gravitationally unstable, if more than 97% (at 1 AU from the Sun) of the gaseous components have been dissipated from the nebula, leaving dusty components. Two alternative scenarios of planetesimal formation are proposed: planetesimals were formed by (1) mutual sticking of dust aggregations by nongravitational forces or by (2) gravitational instabilities in the nebula where the dust to gas ratio is much larger than the ratio with solar elemental abundance. Case (2) might be realized due to dissipation of the nebular gas and/or addition of dust by the bipolar outflow. In case (1), chondrule sizes do not indicate the maximum size of dust aggregations in the solar nebula.
NASA Astrophysics Data System (ADS)
Piasecki, Alison; Sessions, Alex; Peterson, Brian; Eiler, John
2016-10-01
Many previous studies have examined abundances of deuterium (D) and 13C within small organic molecules. Recent advances in analytical instrumentation add the abilities to measure site-specific and multiply substituted isotopologues of natural organics. Here we perform first-principles calculations of the equilibrium distributions of 13C and D in the volatile alkanes (including both single and multiple substitutions), as a guide to the interpretation of current measurements and as a basis for anticipating isotope effects that might be examined with future analytical techniques. The models we present illustrate several common themes of the isotopic structures of the small alkanes, including; temperature dependent enrichment of clumped isotope species, with amplitudes in the order D-D > 13C-D > 13C-13C; similarity in strength of such clumped isotope effects between different molecules (e.g., 13C-D clumping is ∼5‰ enriched at 300 K in methane, ethane and propane); a ∼10× contrast between the amplitudes of stronger adjacent substitution of two heavy isotopes vs. weaker non-adjacent substitution; temperature-dependent site-specific fractionation of D and 13C into interior positions of molecules relative to terminal methyl groups; and a relatively simple additive effect to the overall amplitude of enrichment when clumped and site specific effects combine in the same isotopologue. We suggest that the most promising tools suggested by our results are isotopic thermometers based on site-specific distribution of deuterium, which exhibits strong (∼100‰), highly temperature dependent fractionation between methyl groups and methylene carbon positions in propane (and likely other larger n-alkanes).
Local non-equilibrium thermodynamics
Jinwoo, Lee; Tanaka, Hajime
2015-01-01
Local Shannon entropy lies at the heart of modern thermodynamics, with much discussion of trajectory-dependent entropy production. When taken at both boundaries of a process in phase space, it reproduces the second law of thermodynamics over a finite time interval for small scale systems. However, given that entropy is an ensemble property, it has never been clear how one can assign such a quantity locally. Given such a fundamental omission in our knowledge, we construct a new ensemble composed of trajectories reaching an individual microstate, and show that locally defined entropy, information, and free energy are properties of the ensemble, or trajectory-independent true thermodynamic potentials. We find that the Boltzmann-Gibbs distribution and Landauer's principle can be generalized naturally as properties of the ensemble, and that trajectory-free state functions of the ensemble govern the exact mechanism of non-equilibrium relaxation. PMID:25592077
Lattice Boltzmann methods for global linear instability analysis
NASA Astrophysics Data System (ADS)
Pérez, José Miguel; Aguilar, Alfonso; Theofilis, Vassilis
2016-11-01
Modal global linear instability analysis is performed using, for the first time ever, the lattice Boltzmann method (LBM) to analyze incompressible flows with two and three inhomogeneous spatial directions. Four linearization models have been implemented in order to recover the linearized Navier-Stokes equations in the incompressible limit. Two of those models employ the single relaxation time and have been proposed previously in the literature as linearization of the collision operator of the lattice Boltzmann equation. Two additional models are derived herein for the first time by linearizing the local equilibrium probability distribution function. Instability analysis results are obtained in three benchmark problems, two in closed geometries and one in open flow, namely the square and cubic lid-driven cavity flow and flow in the wake of the circular cylinder. Comparisons with results delivered by classic spectral element methods verify the accuracy of the proposed new methodologies and point potential limitations particular to the LBM approach. The known issue of appearance of numerical instabilities when the SRT model is used in direct numerical simulations employing the LBM is shown to be reflected in a spurious global eigenmode when the SRT model is used in the instability analysis. Although this mode is absent in the multiple relaxation times model, other spurious instabilities can also arise and are documented herein. Areas of potential improvements in order to make the proposed methodology competitive with established approaches for global instability analysis are discussed.
NASA Technical Reports Server (NTRS)
Grams, G. W.; SHARDANAND
1972-01-01
The inherent errors of applying terrestrial atmospheric ozone distribution studies to the atmosphere of other planets are discussed. Limitations associated with some of the earlier treatments of photochemical equilibrium distributions of ozone in planetary atmospheres are described. A technique having more universal application is presented. Ozone concentration profiles for the Martian atmosphere based on the results of the Mariner 4 radio occultation experiment and the more recent results with Mariner 6 and Mariner 7 have been calculated using this approach.
Perfect entropy functions of the Lattice Boltzmann method
NASA Astrophysics Data System (ADS)
Karlin, I. V.; Ferrante, A.; Öttinger, H. C.
1999-07-01
In this letter, we derive entropy functions whose local equilibria are suitable to recover the Navier-Stokes equations in the framework of the Lattice Boltzmann method. For the two-dimensional nine-velocity lattice we demonstrate that such an entropy function is unique, and that the expansion of the corresponding local equilibrium is the well-known local equilibrium of Y. H. Qian et al. (Europhys. Lett., 17 (1992) 479). Based on the knowledge of entropy functions, we introduce a new version of the Lattice Boltzmann method with an H-theorem built in.
NASA Astrophysics Data System (ADS)
Kulchytskyy, Bohdan; Andriyash, Evgeny; Amin, Mohammed; Melko, Roger
The field of machine learning has been revolutionized by the recent improvements in the training of deep networks. Their architecture is based on a set of stacked layers of simpler modules. One of the most successful building blocks, known as a restricted Boltzmann machine, is an energetic model based on the classical Ising Hamiltonian. In our work, we investigate the benefits of quantum effects on the learning capacity of Boltzmann machines by extending its underlying Hamiltonian with a transverse field. For this purpose, we employ exact and stochastic training procedures on data sets with physical origins.
Boltzmann hierarchy for interacting neutrinos I: formalism
Oldengott, Isabel M.; Rampf, Cornelius; Wong, Yvonne Y.Y. E-mail: cornelius.rampf@port.ac.uk
2015-04-01
Starting from the collisional Boltzmann equation, we derive for the first time and from first principles the Boltzmann hierarchy for neutrinos including interactions with a scalar particle. Such interactions appear, for example, in majoron-like models of neutrino mass generation. We study two limits of the scalar mass: (i) An extremely massive scalar whose only role is to mediate an effective 4-fermion neutrino-neutrino interaction, and (ii) a massless scalar that can be produced in abundance and thus demands its own Boltzmann hierarchy. In contrast to, e.g., the first-order Boltzmann hierarchy for Thomson-scattering photons, our interacting neutrino/scalar Boltzmann hierarchies contain additional momentum-dependent collision terms arising from a non-negligible energy transfer in the neutrino-neutrino and neutrino-scalar interactions. This necessitates that we track each momentum mode of the phase space distributions individually, even if the particles were massless. Comparing our hierarchy with the commonly used (c{sub eff}{sup 2},c{sub vis}{sup 2})-parameterisation, we find no formal correspondence between the two approaches, which raises the question of whether the latter parameterisation even has an interpretation in terms of particle scattering. Lastly, although we have invoked majoron-like models as a motivation for our study, our treatment is in fact generally applicable to all scenarios in which the neutrino and/or other ultrarelativistic fermions interact with scalar particles.
Boltzmann-Electron Model in Aleph.
Hughes, Thomas Patrick; Hooper, Russell
2014-11-01
We apply the Boltzmann-electron model in the electrostatic, particle-in-cell, finite- element code Aleph to a plasma sheath. By assuming a Boltzmann energy distribution for the electrons, the model eliminates the need to resolve the electron plasma fre- quency, and avoids the numerical "grid instability" that can cause unphysical heating of electrons. This allows much larger timesteps to be used than with kinetic electrons. Ions are treated with the standard PIC algorithm. The Boltzmann-electron model re- quires solution of a nonlinear Poisson equation, for which we use an iterative Newton solver (NOX) from the Trilinos Project. Results for the spatial variation of density and voltage in the plasma sheath agree well with an analytic model
NASA Astrophysics Data System (ADS)
Matsuyama, Akinobu; Aiba, Nobuyuki; Yagi, Masatoshi
2015-11-01
An axisymmetric MHD equilibrium model is studied to allow the inclusion of both beam inertia and energy spectrum for runaway electron beam. Following kinetic-MHD hybrid approach, we evaluate the RE beam current from the integrals of the RE distribution function. The distribution function is here evaluated by a relativistic guiding-center trace code ETC-Rel, where we have implemented the effects of collisions, radiations, and exponential growth into the code. Because to directly treat the Dreicer mechanism in particle simulations is time consuming, the primary RE source is modeled by a Monte-Carlo weighing scheme taking into account the instantaneous generation rate. This paper applies ETC-Rel to the parametric study of the MHD equilibrium with different RE beam parameters. Kinetic effects on the MHD equilibrium appears, e.g., as enhanced Shafranov shifts due to the inertia of highly relativistic electrons. A kinetic modification to the equilibrium becomes significant if the contribution of the beam inertia - being increased with the total electron mass of multi-MeV RE populations - becomes large enough to affect the radial force balance. This work was supported in part by MEXT KAKENHI Grant No. 23561009 and 26820404.
NASA Astrophysics Data System (ADS)
Xu, Dazhi; Cao, Jianshu
2016-08-01
The concept of polaron, emerged from condense matter physics, describes the dynamical interaction of moving particle with its surrounding bosonic modes. This concept has been developed into a useful method to treat open quantum systems with a complete range of system-bath coupling strength. Especially, the polaron transformation approach shows its validity in the intermediate coupling regime, in which the Redfield equation or Fermi's golden rule will fail. In the polaron frame, the equilibrium distribution carried out by perturbative expansion presents a deviation from the canonical distribution, which is beyond the usual weak coupling assumption in thermodynamics. A polaron transformed Redfield equation (PTRE) not only reproduces the dissipative quantum dynamics but also provides an accurate and efficient way to calculate the non-equilibrium steady states. Applications of the PTRE approach to problems such as exciton diffusion, heat transport and light-harvesting energy transfer are presented.
Macroscopic model and truncation error of discrete Boltzmann method
NASA Astrophysics Data System (ADS)
Hwang, Yao-Hsin
2016-10-01
A derivation procedure to secure the macroscopically equivalent equation and its truncation error for discrete Boltzmann method is proffered in this paper. Essential presumptions of two time scales and a small parameter in the Chapman-Enskog expansion are disposed of in the present formulation. Equilibrium particle distribution function instead of its original non-equilibrium form is chosen as key variable in the derivation route. Taylor series expansion encompassing fundamental algebraic manipulations is adequate to realize the macroscopically differential counterpart. A self-contained and comprehensive practice for the linear one-dimensional convection-diffusion equation is illustrated in details. Numerical validations on the incurred truncation error in one- and two-dimensional cases with various distribution functions are conducted to verify present formulation. As shown in the computational results, excellent agreement between numerical result and theoretical prediction are found in the test problems. Straightforward extensions to more complicated systems including convection-diffusion-reaction, multi-relaxation times in collision operator as well as multi-dimensional Navier-Stokes equations are also exposed in the Appendix to point out its expediency in solving complicated flow problems.
Endo, Satoshi
2016-08-10
Narcosis occurs as a result of the accumulation of chemicals in the phospholipid membrane. The toxic threshold concentration in the membrane is thought to be relatively constant across different chemicals and species. Hence, estimating chemical concentrations in the membrane is expected to reduce the variability of narcotic critical body residue (CBR) data. In this study, a high quality CBR dataset for three aquatic species reported recently in the literature was evaluated with the internal equilibrium distribution concept. The raw wet-weight-based CBR values were converted to membrane-weight-based CBR values by assuming that the chemical is distributed in storage lipids, membranes, proteins, and water according to the respective equilibrium partition coefficients. Several sets of partition coefficients were compared for this analysis. The results were consistent with the notion that the use of a structural protein instead of serum albumin as a surrogate for the body protein fraction could reduce the variability of CBRs. Partition coefficients predicted by polyparameter linear free energy relationships (PP-LFERs) reduced the variability of CBRs as much as or even more than experimental partition coefficients did. It is suggested that CBR data for chemicals with larger structural diversity and biological species with more distinct compositions are needed to evaluate further the equilibrium distribution concept and the constant membrane threshold hypothesis.
The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics
Tirnakli, Ugur; Borges, Ernesto P.
2016-01-01
As well known, Boltzmann-Gibbs statistics is the correct way of thermostatistically approaching ergodic systems. On the other hand, nontrivial ergodicity breakdown and strong correlations typically drag the system into out-of-equilibrium states where Boltzmann-Gibbs statistics fails. For a wide class of such systems, it has been shown in recent years that the correct approach is to use Tsallis statistics instead. Here we show how the dynamics of the paradigmatic conservative (area-preserving) stan-dard map exhibits, in an exceptionally clear manner, the crossing from one statistics to the other. Our results unambiguously illustrate the domains of validity of both Boltzmann-Gibbs and Tsallis statistical distributions. Since various important physical systems from particle confinement in magnetic traps to autoionization of molecular Rydberg states, through particle dynamics in accelerators and comet dynamics, can be reduced to the standard map, our results are expected to enlighten and enable an improved interpretation of diverse experimental and observational results. PMID:27004989
Polar-coordinate lattice Boltzmann modeling of compressible flows.
Lin, Chuandong; Xu, Aiguo; Zhang, Guangcai; Li, Yingjun; Succi, Sauro
2014-01-01
We present a polar coordinate lattice Boltzmann kinetic model for compressible flows. A method to recover the continuum distribution function from the discrete distribution function is indicated. Within the model, a hybrid scheme being similar to, but different from, the operator splitting is proposed. The temporal evolution is calculated analytically, and the convection term is solved via a modified Warming-Beam (MWB) scheme. Within the MWB scheme a suitable switch function is introduced. The current model works not only for subsonic flows but also for supersonic flows. It is validated and verified via the following well-known benchmark tests: (i) the rotational flow, (ii) the stable shock tube problem, (iii) the Richtmyer-Meshkov (RM) instability, and (iv) the Kelvin-Helmholtz instability. As an original application, we studied the nonequilibrium characteristics of the system around three kinds of interfaces, the shock wave, the rarefaction wave, and the material interface, for two specific cases. In one of the two cases, the material interface is initially perturbed, and consequently the RM instability occurs. It is found that the macroscopic effects due to deviating from thermodynamic equilibrium around the material interface differ significantly from those around the mechanical interfaces. The initial perturbation at the material interface enhances the coupling of molecular motions in different degrees of freedom. The amplitude of deviation from thermodynamic equilibrium around the shock wave is much higher than those around the rarefaction wave and material interface. By comparing each component of the high-order moments and its value in equilibrium, we can draw qualitatively the main behavior of the actual distribution function. These results deepen our understanding of the mechanical and material interfaces from a more fundamental level, which is indicative for constructing macroscopic models and other kinds of kinetic models.
Flavored quantum Boltzmann equations
Cirigliano, Vincenzo; Lee, Christopher; Ramsey-Musolf, Michael J.; Tulin, Sean
2010-05-15
We derive from first principles, using nonequilibrium field theory, the quantum Boltzmann equations that describe the dynamics of flavor oscillations, collisions, and a time-dependent mass matrix in the early universe. Working to leading nontrivial order in ratios of relevant time scales, we study in detail a toy model for weak-scale baryogenesis: two scalar species that mix through a slowly varying time-dependent and CP-violating mass matrix, and interact with a thermal bath. This model clearly illustrates how the CP asymmetry arises through coherent flavor oscillations in a nontrivial background. We solve the Boltzmann equations numerically for the density matrices, investigating the impact of collisions in various regimes.
Jelić-Ivanović, Z; Spasojević-Kalimanovska, V; Topić, A; Spasić, S; Petrović, V
1994-08-01
The distribution of the alpha 1-antitrypsin (Pi) phenotypes and subtypes was investigated in a population sample of 1060 unrelated individuals from Serbia (Yugoslavia). The allele frequencies estimates were: Pi*M1: 0.702; Pi*M2: 0.183; Pi*M3: 0.088; Pi*Z: 0.013, Pi*S: 0.007; Pi*P: 0.004; Pi*F: 0.003. The observed phenotype frequencies differed very significantly from those expected assuming H.W. equilibrium (chi 2 = 49.51, p < 0.0005). The deviation from equilibrium involved the three Pi*M subtypes: an excess of Pi*M1, Pi*M2 and Pi*M3 homozygotes was found, with the corresponding decreased number of M1M2 and M1M3 heterozygotes. The possible significance of this finding is discussed.
Classical non-Markovian Boltzmann equation
Alexanian, Moorad
2014-08-01
The modeling of particle transport involves anomalous diffusion, (x²(t) ) ∝ t{sup α} with α ≠ 1, with subdiffusive transport corresponding to 0 < α < 1 and superdiffusive transport to α > 1. These anomalies give rise to fractional advection-dispersion equations with memory in space and time. The usual Boltzmann equation, with only isolated binary collisions, is Markovian and, in particular, the contributions of the three-particle distribution function are neglected. We show that the inclusion of higher-order distribution functions give rise to an exact, non-Markovian Boltzmann equation with resulting transport equations for mass, momentum, and kinetic energy with memory in both time and space. The two- and the three-particle distribution functions are considered under the assumption that the two- and the three-particle correlation functions are translationally invariant that allows us to obtain advection-dispersion equations for modeling transport in terms of spatial and temporal fractional derivatives.
Classical non-Markovian Boltzmann equation
NASA Astrophysics Data System (ADS)
Alexanian, Moorad
2014-08-01
The modeling of particle transport involves anomalous diffusion, ⟨x2(t) ⟩ ∝ tα with α ≠ 1, with subdiffusive transport corresponding to 0 < α < 1 and superdiffusive transport to α > 1. These anomalies give rise to fractional advection-dispersion equations with memory in space and time. The usual Boltzmann equation, with only isolated binary collisions, is Markovian and, in particular, the contributions of the three-particle distribution function are neglected. We show that the inclusion of higher-order distribution functions give rise to an exact, non-Markovian Boltzmann equation with resulting transport equations for mass, momentum, and kinetic energy with memory in both time and space. The two- and the three-particle distribution functions are considered under the assumption that the two- and the three-particle correlation functions are translationally invariant that allows us to obtain advection-dispersion equations for modeling transport in terms of spatial and temporal fractional derivatives.
NASA Astrophysics Data System (ADS)
Brun-Battistini, Dominique; Mondragon-Suarez, Jose Humberto; Sandoval-Villalbazo, Alfredo; García-Perciante, Ana Laura
2015-11-01
In 1936, Richard C. Tolman showed that in thermodynamic equilibrium a temperature gradient can be compensated by a gravitational potential gradient. In reference, in a linearized gravity approximation, Tolman's law was extended for inhomogeneous non-equilibrium systems, suggesting that the contribution of the gravitational field to heat flow can be seen as a cross effect. In this work this contribution to the heat flux for a dilute simple fluid in an isotropic Schwarzschild metric is analyzed. In this case, the effect of the field is contained in the covariant derivative, such that the molecules follow geodesics. The results show that the effect of the field on the heat flux does not vanish, in contrast with what is suggested by other authors. The authors acknowledge support from CONACyT through grant CB2011/167563.
Fermion particle production in semiclassical Boltzmann-Vlasov transport theory
Dawson, John F.; Mihaila, Bogdan; Cooper, Fred
2009-07-01
We present numerical solutions of the semiclassical Boltzmann-Vlasov equation for fermion particle-antiparticle production by strong electric fields in boost-invariant coordinates in (1+1) and (3+1) dimensional QED. We compare the Boltzmann-Vlasov results with those of recent quantum field theory calculations and find good agreement. We conclude that extending the Boltzmann-Vlasov approach to the case of QCD should allow us to do a thorough investigation of how backreaction affects recent results on the dependence of the transverse momentum distribution of quarks and antiquarks on a second Casimir invariant of color SU(3)
On the dispute between Boltzmann and Gibbs entropy
NASA Astrophysics Data System (ADS)
Buonsante, Pierfrancesco; Franzosi, Roberto; Smerzi, Augusto
2016-12-01
The validity of the concept of negative temperature has been recently challenged by arguing that the Boltzmann entropy (that allows negative temperatures) is inconsistent from a mathematical and statistical point of view, whereas the Gibbs entropy (that does not admit negative temperatures) provides the correct definition for the microcanonical entropy. Here we prove that the Boltzmann entropy is thermodynamically and mathematically consistent. Analytical results on two systems supporting negative temperatures illustrate the scenario we propose. In addition we numerically study a lattice system to show that negative temperature equilibrium states are accessible and obey standard statistical mechanics prediction.
Monahan, William B; Tingley, Morgan W
2012-01-01
The ability of species to respond to novel future climates is determined in part by their physiological capacity to tolerate climate change and the degree to which they have reached and continue to maintain distributional equilibrium with the environment. While broad-scale correlative climatic measurements of a species' niche are often described as estimating the fundamental niche, it is unclear how well these occupied portions actually approximate the fundamental niche per se, versus the fundamental niche that exists in environmental space, and what fitness values bounding the niche are necessary to maintain distributional equilibrium. Here, we investigate these questions by comparing physiological and correlative estimates of the thermal niche in the introduced North American house sparrow (Passer domesticus). Our results indicate that occupied portions of the fundamental niche derived from temperature correlations closely approximate the centroid of the existing fundamental niche calculated on a fitness threshold of 50% population mortality. Using these niche measures, a 75-year time series analysis (1930-2004) further shows that: (i) existing fundamental and occupied niche centroids did not undergo directional change, (ii) interannual changes in the two niche centroids were correlated, (iii) temperatures in North America moved through niche space in a net centripetal fashion, and consequently, (iv) most areas throughout the range of the house sparrow tracked the existing fundamental niche centroid with respect to at least one temperature gradient. Following introduction to a new continent, the house sparrow rapidly tracked its thermal niche and established continent-wide distributional equilibrium with respect to major temperature gradients. These dynamics were mediated in large part by the species' broad thermal physiological tolerances, high dispersal potential, competitive advantage in human-dominated landscapes, and climatically induced changes to the
U.S. stock market interaction network as learned by the Boltzmann machine
Borysov, Stanislav S.; Roudi, Yasser; Balatsky, Alexander V.
2015-12-07
Here, we study historical dynamics of joint equilibrium distribution of stock returns in the U.S. stock market using the Boltzmann distribution model being parametrized by external fields and pairwise couplings. Within Boltzmann learning framework for statistical inference, we analyze historical behavior of the parameters inferred using exact and approximate learning algorithms. Since the model and inference methods require use of binary variables, effect of this mapping of continuous returns to the discrete domain is studied. The presented results show that binarization preserves the correlation structure of the market. Properties of distributions of external fields and couplings as well as themore » market interaction network and industry sector clustering structure are studied for different historical dates and moving window sizes. We demonstrate that the observed positive heavy tail in distribution of couplings is related to the sparse clustering structure of the market. We also show that discrepancies between the model’s parameters might be used as a precursor of financial instabilities.« less
Boltzmann Solver with Adaptive Mesh in Velocity Space
Kolobov, Vladimir I.; Arslanbekov, Robert R.; Frolova, Anna A.
2011-05-20
We describe the implementation of direct Boltzmann solver with Adaptive Mesh in Velocity Space (AMVS) using quad/octree data structure. The benefits of the AMVS technique are demonstrated for the charged particle transport in weakly ionized plasmas where the collision integral is linear. We also describe the implementation of AMVS for the nonlinear Boltzmann collision integral. Test computations demonstrate both advantages and deficiencies of the current method for calculations of narrow-kernel distributions.
Efficient Asymptotic Preserving Deterministic methods for the Boltzmann Equation
2011-04-01
release, distribution unlimited 13. SUPPLEMENTARY NOTES See also ADA579248. Models and Computational Methods for Rarefied Flows (Modeles et methodes de...nonlinear collisional kinetic equation. The most well-known example is represented by the Boltzmann equation of rarefied gas dynamics (Cercignani, 1988...et al. (2010). Although the scope of our insights is wider, here we will focus mainly on the classical Boltzmann equation of rarefied gas dynamics
Equilibrium distributions and relaxation times in gaslike economic models: an analytical derivation.
Calbet, Xavier; López, José-Luis; López-Ruiz, Ricardo
2011-03-01
A step-by-step procedure to derive analytically the exact dynamical evolution equations of the probability density functions (PDFs) of well-known kinetic wealth exchange economic models is shown. This technique gives a dynamical insight into the evolution of the PDF, for example, allowing the calculation of its relaxation times. Their equilibrium PDFs can also be calculated by finding its stationary solutions. This gives as a result an integro-differential equation, which can be solved analytically in some cases and numerically in others. This should provide some guidance into the type of PDFs that can be derived from particular economic agent exchange rules or, for that matter, any other kinetic model of gases with particular collision physics.
Generalizing the Boltzmann equation in complex phase space.
Zadehgol, Abed
2016-08-01
In this work, a generalized form of the BGK-Boltzmann equation is proposed, where the velocity, position, and time can be represented by real or complex variables. The real representation leads to the conventional BGK-Boltzmann equation, which can recover the continuity and Navier-Stokes equations. We show that the complex representation yields a different set of equations, and it can also recover the conservation and Navier-Stokes equations, at low Mach numbers, provided that the imaginary component of the macroscopic mass can be neglected. We briefly review the Constant Speed Kinetic Model (CSKM), which was introduced in Zadehgol and Ashrafizaadeh [J. Comp. Phys. 274, 803 (2014)JCTPAH0021-999110.1016/j.jcp.2014.06.053] and Zadehgol [Phys. Rev. E 91, 063311 (2015)PLEEE81539-375510.1103/PhysRevE.91.063311]. The CSKM is then used as a basis to show that the complex-valued equilibrium distribution function of the present model can be identified with a simple singularity in the complex phase space. The virtual particles, in the present work, are concentrated on virtual "branes" which surround the computational nodes. Employing the Cauchy integral formula, it is shown that certain variations of the "branes," in the complex phase space, do not affect the local kinetic states. This property of the new model, which is referred to as the "apparent jumps" in the present work, is used to construct new models. The theoretical findings have been tested by simulating three benchmark flows. The results of the present simulations are in excellent agreement with the previous results reported by others.
Generalizing the Boltzmann equation in complex phase space
NASA Astrophysics Data System (ADS)
Zadehgol, Abed
2016-08-01
In this work, a generalized form of the BGK-Boltzmann equation is proposed, where the velocity, position, and time can be represented by real or complex variables. The real representation leads to the conventional BGK-Boltzmann equation, which can recover the continuity and Navier-Stokes equations. We show that the complex representation yields a different set of equations, and it can also recover the conservation and Navier-Stokes equations, at low Mach numbers, provided that the imaginary component of the macroscopic mass can be neglected. We briefly review the Constant Speed Kinetic Model (CSKM), which was introduced in Zadehgol and Ashrafizaadeh [J. Comp. Phys. 274, 803 (2014), 10.1016/j.jcp.2014.06.053] and Zadehgol [Phys. Rev. E 91, 063311 (2015), 10.1103/PhysRevE.91.063311]. The CSKM is then used as a basis to show that the complex-valued equilibrium distribution function of the present model can be identified with a simple singularity in the complex phase space. The virtual particles, in the present work, are concentrated on virtual "branes" which surround the computational nodes. Employing the Cauchy integral formula, it is shown that certain variations of the "branes," in the complex phase space, do not affect the local kinetic states. This property of the new model, which is referred to as the "apparent jumps" in the present work, is used to construct new models. The theoretical findings have been tested by simulating three benchmark flows. The results of the present simulations are in excellent agreement with the previous results reported by others.
Porter, Mark L; Coon, E T; Kang, Q; Moulton, J D; Carey, J W
2012-09-01
This work focuses on an improved multicomponent interparticle-potential lattice Boltzmann model. The model results in viscosity-independent equilibrium densities and is capable of simulating kinematic viscosity ratios greater than 1000. External forces are incorporated into the discrete Boltzmann equation, rather than through an equilibrium velocity shift as in the original Shan and Chen (hereafter, SC) model. The model also requires the derivation of a momentum conserving effective velocity, which is substituted into the equilibrium distribution function and applies to both the single- and multiple-relaxation-time formulations. Additionally, higher-order isotropy is used in the calculation of the fluid-fluid interaction forces to reduce the magnitude of spurious currents (i.e., numerical errors) in the vicinity of interfaces. First, we compare the model to the SC model for static bubble simulations. We demonstrate that the model results in viscosity-independent equilibrium bubble densities for a wide range of kinematic viscosities, which is not the case for the SC model. Furthermore, we show that the model is capable of simulating stable bubbles for kinematic viscosity ratios greater than 1000 (when higher-order isotropy is used), whereas the SC model is known to be limited to kinematic viscosity ratios on the order of 10. Next we verify the model for surface tension via Laplace's law and show that the model results in the same surface tension values for a range of kinematic viscosities and kinematic viscosity ratios of 10, 100, and 1000. The model is also verified for layered cocurrent flow though parallel plates. We show that the simulated velocity profiles preserve continuity at the interface for kinematic viscosity ratios ranging from 0.001 to 1000 and that the model accurately predicts nonwetting and wetting phase relative permeability for kinematic viscosity ratios of 0.01 to 100.
Non-equilibrium phase distribution in an Al-SiC composite
NASA Technical Reports Server (NTRS)
Nutt, S. R.; Carpenter, R. W.
1985-01-01
The phase distribution in an Al-SiC composite has been investigated using high resolution analytical electron microscopy. Particular attention was focused on Al-SiC interfaces, matrix boundaries and impurity phases which would impede the easy glide of dislocations. Small crystallites of MgO were distributed singly and in clusters along Al-SiC interfaces in all specimens. Interfacial segregation and precipitation involving alloy species apparently affected precipitation in the matrix, where the distribution of phases was found to be very heterogeneous. Matrix phases also included unusually large constituent particles and dispersoids, a consequence of the composite processing methods. The relationship between the observed microstructure and the composite mechanical behavior reported by others is discussed. The heterogeneous distribution of matrix phases is expected to result in a wide variaiton in local yield stress and local work-hardening rate within the composite.
NASA Astrophysics Data System (ADS)
Liu, Y.; He, H. T.; Zhu, C.
2014-12-01
Several important equilibrium Si isotope fractionation factors are calculated here. We use a so-called volume-variable-cluster-model (VVCM) method for solids and the "water-droplet" method for aqueous species for isotope fractionation calculation at the same quantum chemistry level. The calculation results show that several silicate minerals, such as quartz, feldspar, kaolinite, etc., all enrich heavy Si isotopes relative to aqueous H4SiO4 and can be up to 3.3‰ at 25°C, different from most field observations. Meanwhile stable organosilicon complexes can enrich even lighter Si isotopes than aqueous H4SiO4. For explaining the difference between the calculation results and field observations, we calculate the kinetic isotope effect (KIE) associated with the formation of amorphous silica, and find that amorphous silica will enrich extremely light Si isotopes. From amorphous silica to crystalline quartz, the structural adjustment & transition needs getting rid of small amount of Si to re-organize the structure. Light Si isotopes will be preferentially lost and let the final crystalline quartz with a little bit more heavy Si isotopes. However, such late-stage Si heavy isotope enrichment cannot erase the total isotopic signal, crystalline quartz still inherit much light Si isotopic composition from amorphous quartz. That is the reason for the discrepancy between the calculation results and the field observations, because the formation of amorphous quartz is under a non-equilibrium process but theoretical calculations are for equilibrium isotope fractionations. With accurate equilibrium fractionation factors provided here, Si isotope distributions in earth surface environments including soil, groundwater and plants can be further interpreted. We find that δ30Si variations in soil are mainly driven by secondary minerals precipitation and adsorption. Also, bulk soil δ30Si maybe have a parabolic distribution with soil age, with a minimum value at where allophane is
Berzak, L; Jones, A D; Kaita, R; Kozub, T; Logan, N; Majeski, R; Menard, J; Zakharov, L
2010-10-01
The lithium tokamak experiment (LTX) is a modest-sized spherical tokamak (R(0)=0.4 m and a=0.26 m) designed to investigate the low-recycling lithium wall operating regime for magnetically confined plasmas. LTX will reach this regime through a lithium-coated shell internal to the vacuum vessel, conformal to the plasma last-closed-flux surface, and heated to 300-400 °C. This structure is highly conductive and not axisymmetric. The three-dimensional nature of the shell causes the eddy currents and magnetic fields to be three-dimensional as well. In order to analyze the plasma equilibrium in the presence of three-dimensional eddy currents, an extensive array of unique magnetic diagnostics has been implemented. Sensors are designed to survive high temperatures and incidental contact with lithium and provide data on toroidal asymmetries as well as full coverage of the poloidal cross-section. The magnetic array has been utilized to determine the effects of nonaxisymmetric eddy currents and to model the start-up phase of LTX. Measurements from the magnetic array, coupled with two-dimensional field component modeling, have allowed a suitable field null and initial plasma current to be produced. For full magnetic reconstructions, a three-dimensional electromagnetic model of the vacuum vessel and shell is under development.
Kaita, R.; Kozub, T.; Logan, N.; Majeski, R.; Menard, J.; Zakharov, L.
2010-12-10
The lithium tokamak experiment LTX is a modest-sized spherical tokamak R0=0.4 m and a =0.26 m designed to investigate the low-recycling lithium wall operating regime for magnetically confined plasmas. LTX will reach this regime through a lithium-coated shell internal to the vacuum vessel, conformal to the plasma last-closed-flux surface, and heated to 300-400 oC. This structure is highly conductive and not axisymmetric. The three-dimensional nature of the shell causes the eddy currents and magnetic fields to be three-dimensional as well. In order to analyze the plasma equilibrium in the presence of three-dimensional eddy currents, an extensive array of unique magnetic diagnostics has been implemented. Sensors are designed to survive high temperatures and incidental contact with lithium and provide data on toroidal asymmetries as well as full coverage of the poloidal cross-section. The magnetic array has been utilized to determine the effects of nonaxisymmetric eddy currents and to model the start-up phase of LTX. Measurements from the magnetic array, coupled with two-dimensional field component modeling, have allowed a suitable field null and initial plasma current to be produced. For full magnetic reconstructions, a three-dimensional electromagnetic model of the vacuum vessel and shell is under development.
The temperature and size distribution of large water clusters from a non-equilibrium model
Gimelshein, N.; Gimelshein, S.; Pradzynski, C. C.; Zeuch, T.; Buck, U.
2015-06-28
A hybrid Lagrangian-Eulerian approach is used to examine the properties of water clusters formed in neon-water vapor mixtures expanding through microscale conical nozzles. Experimental size distributions were reliably determined by the sodium doping technique in a molecular beam machine. The comparison of computed size distributions and experimental data shows satisfactory agreement, especially for (H{sub 2}O){sub n} clusters with n larger than 50. Thus validated simulations provide size selected cluster temperature profiles in and outside the nozzle. This information is used for an in-depth analysis of the crystallization and water cluster aggregation dynamics of recently reported supersonic jet expansion experiments.
Stability and stabilization of the lattice Boltzmann method.
Brownlee, R A; Gorban, A N; Levesley, J
2007-03-01
We revisit the classical stability versus accuracy dilemma for the lattice Boltzmann methods (LBM). Our goal is a stable method of second-order accuracy for fluid dynamics based on the lattice Bhatnager-Gross-Krook method (LBGK). The LBGK scheme can be recognized as a discrete dynamical system generated by free flight and entropic involution. In this framework the stability and accuracy analysis are more natural. We find the necessary and sufficient conditions for second-order accurate fluid dynamics modeling. In particular, it is proven that in order to guarantee second-order accuracy the distribution should belong to a distinguished surface--the invariant film (up to second order in the time step). This surface is the trajectory of the (quasi)equilibrium distribution surface under free flight. The main instability mechanisms are identified. The simplest recipes for stabilization add no artificial dissipation (up to second order) and provide second-order accuracy of the method. Two other prescriptions add some artificial dissipation locally and prevent the system from loss of positivity and local blowup. Demonstration of the proposed stable LBGK schemes are provided by the numerical simulation of a one-dimensional (1D) shock tube and the unsteady 2D flow around a square cylinder up to Reynolds number Re approximately 20,000.
Noronha, Jorge; Denicol, Gabriel S.
2015-12-30
In this paper we obtain an analytical solution of the relativistic Boltzmann equation under the relaxation time approximation that describes the out-of-equilibrium dynamics of a radially expanding massless gas. This solution is found by mapping this expanding system in flat spacetime to a static flow in the curved spacetime AdS2 Ⓧ S2. We further derive explicit analytic expressions for the momentum dependence of the single-particle distribution function as well as for the spatial dependence of its moments. We find that this dissipative system has the ability to flow as a perfect fluid even though its entropy density does not matchmore » the equilibrium form. The nonequilibrium contribution to the entropy density is shown to be due to higher-order scalar moments (which possess no hydrodynamical interpretation) of the Boltzmann equation that can remain out of equilibrium but do not couple to the energy-momentum tensor of the system. Furthermore, in this system the slowly moving hydrodynamic degrees of freedom can exhibit true perfect fluidity while being totally decoupled from the fast moving, nonhydrodynamical microscopic degrees of freedom that lead to entropy production.« less
An Equilibrium Model of Urban Population and the Distribution of Income. Discussion Paper 355-76.
ERIC Educational Resources Information Center
Yinger, John; Danziger, Sheldon
The relationship between the level of income and the population of an urban area is a familiar concern in urban economics. Existing models of the relationship between income levels and urban population are considered to assume that there is a homogeneous labor force and, hence, a world in which there is no inequality in the size distribution of…
Equilibrium distribution of Fe, Ni, Sb, and Sn between liquid Cu and a CaO-rich slag
NASA Astrophysics Data System (ADS)
Gortais, J.; Hodaj, F.; Allibert, M.; Welter, J. M.
1994-10-01
Equilibrium measurements of the distribution of Fe, Ni, Sb, and Sn between a liquid Cu-O solution and a CaF2-CaO-MgO-SiO2 were carried out at 1500 K in a magnesia crucible. The results show that the studied solutes were in the states Fe(III), Ni(II), Sb(III), and Sn(IV), in the slag, for metal O contents ranging from 100 ppm to saturation at 2.1 pct. The Cu oxide solubility in the slag was also measured in absence of the solute elements. Its maximum solubility is about 4 ± 1 mass pct Cu2O. The compositions at equilibrium allow determination of the activity coefficients (referred to pure oxide) of the four solute oxides in the slag. These values, expressed in round figures to take into account the experimental uncertainties, are 10 for Fe2O3, 20 for NiO, 10 for SnO2, 1.6 10-2 for SbO1.5, and 60 for Cu2O.
NASA Technical Reports Server (NTRS)
Decreau, P. M. E.; Carpenter, D.; Chappell, C. R.; Green, J.; Waite, J. H., Jr.
1986-01-01
Very low-energy trapped ions, mostly protons, have been observed in a region of moderate density characteristic of the plasmapause boundary and of the plasmaspheric bulge. The present paper is concerned with an examination of the latitudinal structure of the bulge under quasi-steady conditions and the conditions of the recovery phase. Details regarding the data base are considered along with observations of the morphology and dynamics of the bulge, the latitudinal density distribution in the expanded bulge, the convection scenario during the replenishment phase, and latitudinal effects on plasma characteristics during plasmasphere refilling. The data utilized have been mainly provided by the DE 1 and GEOS 2 spacecraft traveling in two perpendicular planes. It is found that the bulge is a dynamic region, where no reasonable interpretation of the observed density distribution can be achieved without taking into account the mechanism of magnetospheric convection.
NASA Technical Reports Server (NTRS)
Wickholm, D.; Bickel, W. S.
1976-01-01
The paper describes an experiment consisting of the acceleration of N(+) and N2(+) ions to energies between 0.25 and 1.75 MeV and their injection through a thin carbon foil, whereupon they were charge-state analyzed with an electrostatic analyzer. A foil-covered electrically suppressed Faraday cup, connected to a stepping motor, moved in the plane of the dispersed beams. The Faraday cup current, which was proportional to the number of incident ions, was sent to a current digitizer and computer programmed as a multiscaler. The energy-dependent charge-state fractions, the mean charge and the distribution width were calculated. It was shown that for incident atoms, the charge state distribution appeared to be spread over more charge states, while for the incident molecules, there was a greater fraction of charge states near the mean charge.
1979-01-01
ELECTE! Mg++ and K+ Distribution in Frog Muscle and Egg: B A Disproof of the Donnan Theory of Membrane B Equilibrium Applied to the Living Cells GILBERT...19107 J ABSTRACT 1. We studied the equilibrium distribution of Mg** in the form of chlo- ride and pulfate at two temperatures (5* and 25°C) in frog ...vicinity of 90 jmoles/g/ fresh muscle cells. 4. We observed a similar rectilinear distribution of Mg** in frog ovarian eggs. As in muscle tissues, no major
Sterner, R.W.; Lahey, R.T. Jr.
1983-07-01
Subchannel measurements were performed in order to determine the equilibrium quality and mass flux distribution in a four rod bundle, using air/water flow. An isokinetic technique was used to sample the flow in the center, side and corner subchannels of this test section. Flow rates of the air and water in each sampled subchannel were measured. Experiments were performed for two test-section-average mass fluxes (0.333x10/sup 6/ and 0.666x10/sup 6/ lb/sub m//h-ft/sup 2/), and the test-section-average quality was varied from 0% to 0.54% for each mass flux. Single-phase liquid, bubbly, slug and churn-turbulent two-phase flow regimes were achieved. The observed data trends agreed with previous diabatic measurements in which the center subchannel had the highest quality and mass flux, while the corner subchannel had the lowest.
Delmau, LH
2001-12-18
A multivariate mathematical model describing the extraction of cesium from different mixtures of sodium hydroxide, sodium nitrate, sodium chloride, and sodium nitrite containing potassium at variable concentrations has been established. It was determined based on the cesium, potassium, and sodium distribution ratios obtained with simple systems containing single salts. These experimental data were modeled to obtain the formation constants of complexes formed in the organic phase based on specified concentrations of components in both organic and aqueous phases. The model was applied to five different SRS waste simulants, and the corresponding cesium extraction results were predicted satisfactorily, thus validating the model.
NASA Astrophysics Data System (ADS)
Li, Fangjun; Dyt, Chris; Griffiths, Cedric
2006-05-01
In the light of global warming and sea level rise there are many coastal beaches that suffer from erosion. Beach nourishment has become a common practice to maintain the sediment balance on a shore-face. In this paper, a three-dimensional numerical model for evaluating long-term impact of beach nourishment projects has been developed. The model addresses the longstanding complex issue of coastal morphology and sediment grain size distribution from an unconventional angle, which exploits the strong links between grain size distribution and the prevailing transport direction of each sediment constituent under 'average' wave and storm action. The present model predicts the redistribution of nourished sediment according to the subtle clues implied by equilibrium distribution curves and latest coastal wave transformation theories. After verification against recent field observations in Terschelling, The Netherlands, the model was used to predict long-term effects of different beach nourishment strategies. It was found that: (a) given the source sediment available in Terschelling the tactics of large volume and less frequent implementation are better than otherwise; and (b) from a pure engineering point of view, waterline nourishment outperforms offshore trough nourishment. The model offers an additional tool for coastal engineers to evaluate the feasibility, effectiveness and the optimization of dumping locations for beach nourishment projects. It is also a useful tool for stratigraphic modelling of shallow-marine sedimentation in conjunction with sea level changes.
A gas-kinetic BGK scheme for semiclassical Boltzmann hydrodynamic transport
NASA Astrophysics Data System (ADS)
Shi, Yu-Hsin; Yang, J. Y.
2008-11-01
A class of gas-kinetic BGK schemes for solving quantum hydrodynamic transport based on the semiclassical Boltzmann equation with the relaxation time approximation is presented. The derivation is a generalization to the development of Xu [K. Xu, A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method, from gas-kinetic theory, J. Comput. Phys. 171 (2001) 289-335] for the classical gas. Both Bose-Einstein and Fermi-Dirac gases are considered. Some new features due to the quantum equilibrium distributions are delineated. The first-order Chapman-Enskog expansion of the quantum BGK-Boltzmann equation is derived. The coefficients of shear viscosity and thermal conductivity of a quantum gas are given. The van Leer's limiter is used to interpolate and construct the distribution on interface to achieve second-order accuracy. The present quantum gas-kinetic BGK scheme recovers the Xu's scheme when the classical limit is taken. Several one-dimensional quantum gas flows in a shock tube are computed to illustrate the present method.
Equilibrium quality and mass flux distributions in an adiabatic three-subchannel test section
Yadigaroglu, G.; Maganas, A.
1995-12-01
An experiment was designed to measure the fully developed quality and mass flux distributions in an adiabatic three-subchannel test section. The three subchannels had the geometrical characteristics of the corner, side, and interior subchannels of a boiling water reactor (BWR-5) rod bundle. Data collected with Refrigerant-114 at pressures ranging from 7 to 14 bars, simulating operation with water in the range 55 to 103 bars are reported. The average mass flux and quality in the test section were in the ranges 1,300 to 1,750 kg/m{sup 2} {center_dot} s and {minus}0.03 to 0.25, respectively. The data are analyzed and presented in various forms.
Information geometry of Boltzmann machines.
Amari, S; Kurata, K; Nagaoka, H
1992-01-01
A Boltzmann machine is a network of stochastic neurons. The set of all the Boltzmann machines with a fixed topology forms a geometric manifold of high dimension, where modifiable synaptic weights of connections play the role of a coordinate system to specify networks. A learning trajectory, for example, is a curve in this manifold. It is important to study the geometry of the neural manifold, rather than the behavior of a single network, in order to know the capabilities and limitations of neural networks of a fixed topology. Using the new theory of information geometry, a natural invariant Riemannian metric and a dual pair of affine connections on the Boltzmann neural network manifold are established. The meaning of geometrical structures is elucidated from the stochastic and the statistical point of view. This leads to a natural modification of the Boltzmann machine learning rule.
Padowski, Jeannie M; Pollack, Gary M
2011-12-02
Active efflux transport processes at the blood-brain barrier (BBB), such as P-glycoprotein (P-gp)-mediated efflux, can limit brain uptake of therapeutics. Accurate determination of the consequent impact on brain uptake is assumed to require sampling post-attainment of brain-to-blood distribution equilibrium. Because this approach is not always feasible, understanding the relationship between apparent degree of efflux (e.g., calculated BBB P-gp effect) and the fraction of time remaining until distribution equilibrium is achieved (FTDE) would be advantageous. This study employed simulation strategies to explore this relationship in the simplest relevant system (absence of protein binding, saturable uptake, or metabolism at the BBB). Concentration-time profiles were simulated with a 4-compartment system (blood, peripheral tissues, BBB endothelium and brain parenchyma). A unidirectional endothelium-to-blood rate constant, PS(e), represented P-gp-mediated efflux. A parameter space was selected to simulate an 18-fold P-gp effect, (K(p,brain) at distribution equilibrium in the absence [K(p,brain)=82] vs. presence [K(p,brain)=4.5] of P-gp-mediated flux), as observed for paclitaxel in P-gp-deficient vs. P-gp-competent mice. Hypothetical compounds with different P-gp effects, peripheral compartment distribution kinetics, or times to achieve distribution equilibrium were simulated by perturbing the values of relevant model parameters. P-gp effects calculated prior to attainment of distribution equilibrium may be substantially erroneous. However, reasonably accurate estimates can be obtained relatively early in the net distributional phase (under 20% error at FTDE>0.36 or 0.11 for bolus or infusion administration, respectively). Potential errors associated with non-equilibrium calculations are dependent on both P-gp-mediated and P-gp-independent components of flux across the BBB.
Chen, Yunjie; Roux, Benoît
2015-01-14
A family of hybrid simulation methods that combines the advantages of Monte Carlo (MC) with the strengths of classical molecular dynamics (MD) consists in carrying out short non-equilibrium MD (neMD) trajectories to generate new configurations that are subsequently accepted or rejected via an MC process. In the simplest case where a deterministic dynamic propagator is used to generate the neMD trajectories, the familiar Metropolis acceptance criterion based on the change in the total energy ΔE, min[1, exp( − βΔE)], guarantees that the hybrid algorithm will yield the equilibrium Boltzmann distribution. However, the functional form of the acceptance probability is more complex when the non-equilibrium switching process is generated via a non-deterministic stochastic dissipative propagator coupled to a heat bath. Here, we clarify the conditions under which the Metropolis criterion remains valid to rigorously yield a proper equilibrium Boltzmann distribution within hybrid neMD-MC algorithm.
Three-dimensional lattice Boltzmann model for magnetic reconnection
Mendoza, M.; Munoz, J. D.
2008-02-15
We develop a three-dimensional (3D) lattice Boltzmann model that recovers in the continuous limit the two-fluids theory for plasmas, and consequently includes the generalized Ohm's law. The model reproduces the magnetic reconnection process just by giving the right initial equilibrium conditions in the magnetotail, without any assumption on the resistivity in the diffusive region. In this model, the plasma is handled similar to two fluids with an interaction term, each one with distribution functions associated to a cubic lattice with 19 velocities (D3Q19). The electromagnetic fields are considered as a third fluid with an external force on a cubic lattice with 13 velocities (D3Q13). The model can simulate either viscous fluids in the incompressible limit or nonviscous compressible fluids, and successfully reproduces both the Hartmann flow and the magnetic reconnection in the magnetotail. The reconnection rate in the magnetotail obtained with this model lies between R=0.062 and R=0.073, in good agreement with the observations.
Crystallographic Lattice Boltzmann Method
NASA Astrophysics Data System (ADS)
Namburi, Manjusha; Krithivasan, Siddharth; Ansumali, Santosh
2016-06-01
Current approaches to Direct Numerical Simulation (DNS) are computationally quite expensive for most realistic scientific and engineering applications of Fluid Dynamics such as automobiles or atmospheric flows. The Lattice Boltzmann Method (LBM), with its simplified kinetic descriptions, has emerged as an important tool for simulating hydrodynamics. In a heterogeneous computing environment, it is often preferred due to its flexibility and better parallel scaling. However, direct simulation of realistic applications, without the use of turbulence models, remains a distant dream even with highly efficient methods such as LBM. In LBM, a fictitious lattice with suitable isotropy in the velocity space is considered to recover Navier-Stokes hydrodynamics in macroscopic limit. The same lattice is mapped onto a cartesian grid for spatial discretization of the kinetic equation. In this paper, we present an inverted argument of the LBM, by making spatial discretization as the central theme. We argue that the optimal spatial discretization for LBM is a Body Centered Cubic (BCC) arrangement of grid points. We illustrate an order-of-magnitude gain in efficiency for LBM and thus a significant progress towards feasibility of DNS for realistic flows.
Crystallographic Lattice Boltzmann Method
Namburi, Manjusha; Krithivasan, Siddharth; Ansumali, Santosh
2016-01-01
Current approaches to Direct Numerical Simulation (DNS) are computationally quite expensive for most realistic scientific and engineering applications of Fluid Dynamics such as automobiles or atmospheric flows. The Lattice Boltzmann Method (LBM), with its simplified kinetic descriptions, has emerged as an important tool for simulating hydrodynamics. In a heterogeneous computing environment, it is often preferred due to its flexibility and better parallel scaling. However, direct simulation of realistic applications, without the use of turbulence models, remains a distant dream even with highly efficient methods such as LBM. In LBM, a fictitious lattice with suitable isotropy in the velocity space is considered to recover Navier-Stokes hydrodynamics in macroscopic limit. The same lattice is mapped onto a cartesian grid for spatial discretization of the kinetic equation. In this paper, we present an inverted argument of the LBM, by making spatial discretization as the central theme. We argue that the optimal spatial discretization for LBM is a Body Centered Cubic (BCC) arrangement of grid points. We illustrate an order-of-magnitude gain in efficiency for LBM and thus a significant progress towards feasibility of DNS for realistic flows. PMID:27251098
NASA Astrophysics Data System (ADS)
Kukulka, T.; Brunner, K.
2015-05-01
This paper is the first of a two part series that investigates passive buoyant tracers in the ocean surface boundary layer. The first part examines the influence of equilibrium wind-waves on vertical tracer distributions, based on large eddy simulations (LES) of the wave-averaged Navier-Stokes equation. The second part applies the model to investigate observations of buoyant microplastic marine debris, which has emerged as a major ocean pollutant. The LES model captures both Langmuir turbulence (LT) and enhanced turbulent kinetic energy input due to breaking waves (BW) by imposing equilibrium wind-wave statistics for a range of wind and wave conditions. Concentration profiles of LES agree well with analytic solutions obtained for an eddy diffusivity profile that is constant near the surface and transitions into the K-Profile Parameterization (KPP) profile shape at greater depth. For a range of wind and wave conditions, the eddy diffusivity normalized by the product of water-side friction velocity and mixed layer depth, h, mainly depends on a single nondimensional parameter, the peak wavelength (which is related to Stokes drift decay depth) normalized by h. For smaller wave ages, BW critically enhances near-surface mixing, while LT effects are relatively small. For greater wave ages, both BW and LT contribute to elevated near-surface mixing, and LT significantly increases turbulent transport at greater depth. We identify a range of realistic wind and wave conditions for which only Langmuir (and not BW or shear driven) turbulence is capable of deeply submerging buoyant tracers.
NASA Astrophysics Data System (ADS)
Chau, J. F.; Or, D.; Jones, S.; Sukop, M.
2004-05-01
Liquid distribution in unsaturated porous media under different gravitational forces and resulting gaseous diffusion coefficients were investigated to enhance understanding of plant growth conditions in microgravity. Different fluid behavior in plant growth media under microgravity conditions as compared to earth presents a challenge to plant growth in long duration space exploration missions. Our primary objective was to provide qualitative description and quantitative measures of the role of reduced gravity on hydraulic and gaseous transport properties in simulated porous media. We implemented a multi-phase lattice Boltzmann code for equilibrium distribution of liquid in an idealized two-dimensional porous medium under microgravity and "normal" gravity conditions. The information was then used to provide boundary conditions for simulation of gaseous diffusion through the equilibrium domains (considering diffusion through liquid phase negligibly small). The models were tested by comparison with several analytical solutions to the diffusion equation, with excellent results. The relative diffusion coefficient for both series of simulations (with and without gravity) as functions of air-filled porosity was in good agreement with established models of Millington-Quirk. Liquid distribution under earth's gravity featured increased water content at the lower part of the medium relative to the distribution in reduced gravity, which resulted in decreased gas diffusion through a vertically oriented column of a porous medium. Simulation results for larger domains under various orientations will be presented.
Ling, G N; Niu, Z; Ochsenfeld, M
1993-01-01
We determined the equilibrium distribution of twenty-one nonmetabolized nonelectrolytes in frog muscle cells. In all cases, plots of the equilibrium intracellular concentrations of a solute in the cell water against the external concentrations of the solute yielded straight lines in agreement with the prediction of such a rectilinear plot by the polarized multilayer (PM) theory. The slopes of these straight lines yield the equilibrium distribution coefficients or q-value of that solute. It was shown that, again in agreement with the PM theory, the q-values of fourteen nonelectrolytes vary with the molecular volumes of the nonelectrolytes, obeying the "size rule", i.e., the larger the solute, the lower its q-value. The q-values of the remaining seven nonelectrolytes also decrease with their molecular volumes but on a separate curve. These q-value vs. molecular volume plots (q-v plots) show strong resemblance to similar q-v plots of solutes in dialysis sacs containing proteins and polymers assuming the fully-extended conformation (extrovert models) but no, or only weak, resemblance to q-v plots of solutions containing native globular proteins (introvert models). These findings also support the PM theory, according to which some protein(s) pervasively present in cells are in the fully-extended conformation; and that these fully extended cell protein(s) polarize(s) in multilayers all or virtually all cell water. The relationship between the q-values of the nonelectrolytes and the solutes' respective molecular volume are described by two sets of theoretical curves, calculated from an equation introduced in the preceding paper. Both curves were computed on the basis of the same exclusion intensity (Uvp = 126 cal/mole). This factor measures the extra water-to-water interaction of the polarized water which acts to keep solute out of the cell water in degree according to the size of the solute. The two curves are computed on the basis of two different values of U(s), which
Privacy-preserving restricted boltzmann machine.
Li, Yu; Zhang, Yuan; Ji, Yue
2014-01-01
With the arrival of the big data era, it is predicted that distributed data mining will lead to an information technology revolution. To motivate different institutes to collaborate with each other, the crucial issue is to eliminate their concerns regarding data privacy. In this paper, we propose a privacy-preserving method for training a restricted boltzmann machine (RBM). The RBM can be got without revealing their private data to each other when using our privacy-preserving method. We provide a correctness and efficiency analysis of our algorithms. The comparative experiment shows that the accuracy is very close to the original RBM model.
NASA Astrophysics Data System (ADS)
Lanthaler, S.; Pfefferlé, D.; Graves, J. P.; Cooper, W. A.
2017-04-01
An improved set of guiding-centre equations, expanded to one order higher in Larmor radius than usually written for guiding-centre codes, are derived for curvilinear flux coordinates and implemented into the orbit following code VENUS-LEVIS. Aside from greatly improving the correspondence between guiding-centre and full particle trajectories, the most important effect of the additional Larmor radius corrections is to modify the definition of the guiding-centre’s parallel velocity via the so-called Baños drift. The correct treatment of the guiding-centre push-forward with the Baños term leads to an anisotropic shift in the phase-space distribution of guiding-centres, consistent with the well-known magnetization term. The consequence of these higher order terms are quantified in three cases where energetic ions are usually followed with standard guiding-centre equations: (1) neutral beam injection in a MAST-like low aspect-ratio spherical equilibrium where the fast ion driven current is significantly larger with respect to previous calculations, (2) fast ion losses due to resonant magnetic perturbations where a lower lost fraction and a better confinement is confirmed, (3) alpha particles in the ripple field of the European DEMO where the effect is found to be marginal.
NASA Astrophysics Data System (ADS)
Tao, Ruichen; Hayashi, Takehiro; Kagami, Manabu; Kobayashi, Shigeru; Yasukawa, Manabu; Yang, Hui; Robinson, David; Baghsiahi, Hadi; Fernández, F. Aníbal; Selviah, David R.
2015-03-01
A stable reproducible optical standard source for measuring multimode optical fiber attenuation is required as recent round robin measurements of such fibers at several international companies and national standards organizations showed significant variation when using a source having only the encircled flux in the near field emerging from it defined. The paper presents and compares the far field modal power distributions for (i) 2 km and 3 km step-index multimode Hard Plastic Cladding Fibers, HPCF, (SI-MMF) with 200 μm silica core diameter, 0.37 numerical aperture (NA) and polymer cladding, (ii) a 10 m silica graded-index multimode fiber (GI-MMF) with 50 μm core diameter and 0.2 NA, and (ii) a near field Encircled Flux Mode Convertor or "modcon". A free space method for measuring the far field using a Lightemitting diode (LED) centered at 850 nm wavelength with 40 nm 10 dB-bandwidth and a charge-coupled device (CCD) camera is compared with a f-theta multi-element lens based far field pattern (FFP) system. Mandrels of different diameter and different numbers of turns of the fiber around them were used to achieve an equilibrium mode distribution (EMD) for the GI-MMF. The paper defines encircled angular flux (EAF) as the fraction of the total optical power radiating from a multimode optical fiber core within a certain solid angle in the far field. The paper calculates the EAF when the solid angle increases from the far field centroid.
NASA Astrophysics Data System (ADS)
Nomura, Yasunori
2015-10-01
Understanding the observed arrow of time is equivalent, under general assumptions, to explaining why Boltzmann brains do not overwhelm ordinary observers. It is usually thought that this provides a condition on the decay rate of every cosmologically accessible de Sitter vacuum, and that this condition is determined by the production rate of Boltzmann brains calculated using semiclassical theory built on each such vacuum. We argue, based on a recently developed picture of microscopic quantum gravitational degrees of freedom, that this thinking needs to be modified. In particular, depending on the structure of the fundamental theory, the decay rate of a de Sitter vacuum may not have to satisfy any condition except for the one imposed by the Poincaré recurrence. The framework discussed here also addresses the question of whether a Minkowski vacuum may produce Boltzmann brains.
Bhowmick, Amiya Ranjan; Bandyopadhyay, Subhadip; Rana, Sourav; Bhattacharya, Sabyasachi
2016-01-01
The stochastic versions of the logistic and extended logistic growth models are applied successfully to explain many real-life population dynamics and share a central body of literature in stochastic modeling of ecological systems. To understand the randomness in the population dynamics of the underlying processes completely, it is important to have a clear idea about the quasi-equilibrium distribution and its moments. Bartlett et al. (1960) took a pioneering attempt for estimating the moments of the quasi-equilibrium distribution of the stochastic logistic model. Matis and Kiffe (1996) obtain a set of more accurate and elegant approximations for the mean, variance and skewness of the quasi-equilibrium distribution of the same model using cumulant truncation method. The method is extended for stochastic power law logistic family by the same and several other authors (Nasell, 2003; Singh and Hespanha, 2007). Cumulant truncation and some alternative methods e.g. saddle point approximation, derivative matching approach can be applied if the powers involved in the extended logistic set up are integers, although plenty of evidence is available for non-integer powers in many practical situations (Sibly et al., 2005). In this paper, we develop a set of new approximations for mean, variance and skewness of the quasi-equilibrium distribution under more general family of growth curves, which is applicable for both integer and non-integer powers. The deterministic counterpart of this family of models captures both monotonic and non-monotonic behavior of the per capita growth rate, of which theta-logistic is a special case. The approximations accurately estimate the first three order moments of the quasi-equilibrium distribution. The proposed method is illustrated with simulated data and real data from global population dynamics database.
Navier-Stokes Dynamics by a Discrete Boltzmann Model
NASA Technical Reports Server (NTRS)
Rubinstein, Robet
2010-01-01
This work investigates the possibility of particle-based algorithms for the Navier-Stokes equations and higher order continuum approximations of the Boltzmann equation; such algorithms would generalize the well-known Pullin scheme for the Euler equations. One such method is proposed in the context of a discrete velocity model of the Boltzmann equation. Preliminary results on shock structure are consistent with the expectation that the shock should be much broader than the near discontinuity predicted by the Pullin scheme, yet narrower than the prediction of the Boltzmann equation. We discuss the extension of this essentially deterministic method to a stochastic particle method that, like DSMC, samples the distribution function rather than resolving it completely.
Williams, David; Gorski, Jack
1972-01-01
Kinetic and equilibrium binding studies indicate that the process by which the complex of estradiol-binding protein is transferred to the cell nuclei is very rapid and is readily reversible in intact cells; that is, the cytosol and nuclear binding sites are in a rapidly reversible equilibrium. Binding of the hormone appears to shift this equilibrium such that a large percent of the filled binding sites become associated with the nuclear fraction. A model is presented to show that the quantity of filled nuclear binding sites present at any estradiol concentration can be determined strictly by the initial binding between the hormone and the cytosol binding sites. PMID:4508334
Lattice Boltzmann solver of Rossler equation
NASA Astrophysics Data System (ADS)
Yan, Guangwu; Ruan, Li
2000-06-01
We proposed a lattice Boltzmann model for the Rossler equation. Using a method of multiscales in the lattice Boltzmann model, we get the diffusion reaction as a special case. If the diffusion effect disappeared, we can obtain the lattice Boltzmann solution of the Rossler equation on the mesescopic scale. The numerical results show the method can be used to simulate Rossler equation.
Asymmetric Boltzmann machines.
Apolloni, B; Bertoni, A; Campadelli, P; de Falco, D
1991-01-01
We study asymmetric stochastic networks from two points of view: combinatorial optimization and learning algorithms based on relative entropy minimization. We show that there are non trivial classes of asymmetric networks which admit a Lyapunov function L under deterministic parallel evolution and prove that the stochastic augmentation of such networks amounts to a stochastic search for global minima of L. The problem of minimizing L for a totally antisymmetric parallel network is shown to be associated to an NP-complete decision problem. The study of entropic learning for general asymmetric networks, performed in the non equilibrium, time dependent formalism, leads to a Hebbian rule based on time averages over the past history of the system. The general algorithm for asymmetric networks is tested on a feed-forward architecture.
Lattice Boltzmann method simulations of Stokes number effects on particle motion in a channel flow
NASA Astrophysics Data System (ADS)
Zhang, Lenan; Jebakumar, Anand Samuel; Abraham, John
2016-06-01
In a recent experimental study by Lau and Nathan ["Influence of Stokes number on the velocity and concentration distributions in particle-laden jets," J. Fluid Mech. 757, 432 (2014)], it was found that particles in a turbulent pipe flow tend to migrate preferentially toward the wall or the axis depending on their Stokes number (St). Particles with a higher St (>10) are concentrated near the axis while those with lower St (<1) move toward the walls. Jebakumar et al. ["Lattice Boltzmann method simulations of Stokes number effects on particle trajectories in a wall-bounded flow," Comput. Fluids 124, 208 (2016)] have carried out simulations of a particle in a laminar channel flow to investigate this behavior. In their work, they report a similar behavior where particles with low St migrate toward the wall and oscillate about a mean position near the wall while those with high St oscillate about the channel center plane. They have explained this behavior in terms of the Saffman lift, Magnus lift, and wall repulsion forces acting on the particle. The present work extends the previous work done by Jebakumar et al. and aims to study the behavior of particles at intermediate St ranging from 10 to 20. It is in this range where the equilibrium position of the particle changes from near the wall to the axis and the particle starts oscillating about the axis. The Lattice Boltzmann method is employed to carry out this study. It is shown that the change in mean equilibrium position is related to increasing oscillations of the particle with mean position near the wall which results in the particle moving past the center plane to the opposite side. The responsible mechanisms are explained in detail.
Noronha, Jorge; Denicol, Gabriel S.
2015-12-30
In this paper we obtain an analytical solution of the relativistic Boltzmann equation under the relaxation time approximation that describes the out-of-equilibrium dynamics of a radially expanding massless gas. This solution is found by mapping this expanding system in flat spacetime to a static flow in the curved spacetime AdS_{2} Ⓧ S_{2}. We further derive explicit analytic expressions for the momentum dependence of the single-particle distribution function as well as for the spatial dependence of its moments. We find that this dissipative system has the ability to flow as a perfect fluid even though its entropy density does not match the equilibrium form. The nonequilibrium contribution to the entropy density is shown to be due to higher-order scalar moments (which possess no hydrodynamical interpretation) of the Boltzmann equation that can remain out of equilibrium but do not couple to the energy-momentum tensor of the system. Furthermore, in this system the slowly moving hydrodynamic degrees of freedom can exhibit true perfect fluidity while being totally decoupled from the fast moving, nonhydrodynamical microscopic degrees of freedom that lead to entropy production.
NASA Astrophysics Data System (ADS)
Hong, Y.; Kirschbaum, D. B.; Fukuoka, H.
2011-12-01
The key to advancing the predictability of rainfall-triggered landslides is to use physically based slope-stability models that simulate the dynamical response of the subsurface moisture to spatiotemporal variability of rainfall in complex terrains. An early warning system applying such physical models has been developed to predict rainfall-induced shallow landslides over Java Island in Indonesia and Honduras. The prototyped early warning system integrates three major components: (1) a susceptibility mapping or hotspot identification component based on a land surface geospatial database (topographical information, maps of soil properties, and local landslide inventory etc.); (2) a satellite-based precipitation monitoring system (http://trmm.gsfc.nasa.gov) and a precipitation forecasting model (i.e. Weather Research Forecast); and (3) a physically-based, rainfall-induced landslide prediction model SLIDE (SLope-Infiltration-Distributed Equilibrium). The system utilizes the modified physical model to calculate a Factor of Safety (FS) that accounts for the contribution of rainfall infiltration and partial saturation to the shear strength of the soil in topographically complex terrains. The system's prediction performance has been evaluated using a local landslide inventory. In Java Island, Indonesia, evaluation of SLIDE modeling results by local news reports shows that the system successfully predicted landslides in correspondence to the time of occurrence of the real landslide events. Further study of SLIDE is implemented in Honduras where Hurricane Mitch triggered widespread landslides in 1998. Results shows within the approximately 1,200 square kilometers study areas, the values of hit rates reached as high as 78% and 75%, while the error indices were 35% and 49%. Despite positive model performance, the SLIDE model is limited in the early warning system by several assumptions including, using general parameter calibration rather than in situ tests and neglecting
Interface-capturing lattice Boltzmann equation model for two-phase flows
NASA Astrophysics Data System (ADS)
Lou, Qin; Guo, Zhaoli
2015-01-01
In this work, an interface-capturing lattice Boltzmann equation (LBE) model is proposed for two-phase flows. In the model, a Lax-Wendroff propagation scheme and a properly chosen equilibrium distribution function are employed. The Lax-Wendroff scheme is used to provide an adjustable Courant-Friedrichs-Lewy (CFL) number, and the equilibrium distribution is presented to remove the dependence of the relaxation time on the CFL number. As a result, the interface can be captured accurately by decreasing the CFL number. A theoretical expression is derived for the chemical potential gradient by solving the LBE directly for a two-phase system with a flat interface. The result shows that the gradient of the chemical potential is proportional to the square of the CFL number, which explains why the proposed model is able to capture the interface naturally with a small CFL number, and why large interface error exists in the standard LBE model. Numerical tests, including a one-dimensional flat interface problem, a two-dimensional circular droplet problem, and a three-dimensional spherical droplet problem, demonstrate that the proposed LBE model performs well and can capture a sharp interface with a suitable CFL number.
Interface-capturing lattice Boltzmann equation model for two-phase flows.
Lou, Qin; Guo, Zhaoli
2015-01-01
In this work, an interface-capturing lattice Boltzmann equation (LBE) model is proposed for two-phase flows. In the model, a Lax-Wendroff propagation scheme and a properly chosen equilibrium distribution function are employed. The Lax-Wendroff scheme is used to provide an adjustable Courant-Friedrichs-Lewy (CFL) number, and the equilibrium distribution is presented to remove the dependence of the relaxation time on the CFL number. As a result, the interface can be captured accurately by decreasing the CFL number. A theoretical expression is derived for the chemical potential gradient by solving the LBE directly for a two-phase system with a flat interface. The result shows that the gradient of the chemical potential is proportional to the square of the CFL number, which explains why the proposed model is able to capture the interface naturally with a small CFL number, and why large interface error exists in the standard LBE model. Numerical tests, including a one-dimensional flat interface problem, a two-dimensional circular droplet problem, and a three-dimensional spherical droplet problem, demonstrate that the proposed LBE model performs well and can capture a sharp interface with a suitable CFL number.
Convergence Properties of High-order Boltzmann Machines.
Lozano, J Antonio; Graña, Manuel; d'Anjou, Alicia; Albizuri, F Xabier
1996-12-01
The high-order Boltzmann machine (HOBM) approximates probability distributions defined on a set of binary variables, through a learning algorithm that uses Monte Carlo methods. The approximation distribution is a normalized exponential of a consensus function formed by high-degree terms and the structure of the HOBM is given by the set of weighted connections. We prove the convexity of the Kullback-Leibler divergence between the distribution to learn and the approximation distribution of the HOBM. We prove the convergence of the learning algorithm to the strict global minimum of the divergence, which corresponds to the maximum likelihood estimate of the connection weights, establishing the uniqueness of the solution. These theoretical results do not hold in the conventional Boltzmann machine, where the consensus function has first and second-degree terms and hidden units are used. Copyright 1996 Elsevier Science Ltd.
Podolsky electromagnetism at finite temperature: Implications on the Stefan-Boltzmann law
Bonin, C. A.; Bufalo, R.; Pimentel, B. M.; Zambrano, G. E. R.
2010-01-15
In this work we study Podolsky electromagnetism in thermodynamic equilibrium. We show that a Podolsky mass-dependent modification to the Stefan-Boltzmann law is induced and we use experimental data to limit the possible values for this free parameter.
Reduction of the temperature jump in the immersed boundary-thermal lattice Boltzmann method
NASA Astrophysics Data System (ADS)
Seta, Takeshi; Hayashi, Kosuke; Tomiyama, Akio
2015-11-01
We analytically and numerically investigate the boundary errors computed by the immersed boundary-thermal lattice Boltzmann method (IB-TLBM) with the two-relaxation-time (TRT) collision operator. In the linear collision operator of the TRT, we decompose the distribution function into symmetric and antisymmetric components and define the relaxation parameters for each part. We derive the theoretical relation between the relaxation parameters for the symmetric and antisymmetric parts of the distribution function so as to eliminate the temperature jump. The simple TRT collision operator succeeds in reducing the temperature jump occurring at the high relaxation time in the IB-TLBM calculation. The porous plate problem numerically and analytically demonstrate that the velocity squared terms should be neglected in the equilibrium distribution function in order to eliminate the effect of the advection velocity on the temperature jump in the IB-TLBMs. The passive scalar model without the velocity squared terms more accurately calculates the incompressible temperature equation in the IB-TLBMs, compared to the double distribution model, which is based on the relation of the distribution function gk = (ek - u)2fk / 2 . We apply the passive scalar model without the velocity squared terms to the simulation of the natural convection between a hot circular cylinder and a cold square enclosure. The proposed method adequately sets the boundary values and provides reasonable average Nusselt numbers and maximum absolute values of the stream function.
2009-09-01
and Fe hydroxides) until an approximate equilibrium is achieved between particulate and aqueous phases as: 2dC (2) where Cp2 = the soil...adsorbed inorganic P pool (M·M-1; Barrow 1983; Van Riemsdijk et al. 1984). In general, Cp2 represents a small fraction of the total adsorbed inorganic P...3) where ρ = the soil density (M·L-3). Cd2 and Cp2 are related to an equilibrium partition coefficient as: 2 2p dC k C (4) where kd2 = the
NASA Astrophysics Data System (ADS)
Bagchi, Debarshee; Tsallis, Constantino
2017-04-01
The relaxation to equilibrium of two long-range-interacting Fermi-Pasta-Ulam-like models (β type) in thermal contact is numerically studied. These systems, with different sizes and energy densities, are coupled to each other by a few thermal contacts which are short-range harmonic springs. By using the kinetic definition of temperature, we compute the time evolution of temperature and energy density of the two systems. Eventually, for some time t >teq, the temperature and energy density of the coupled system equilibrate to values consistent with standard Boltzmann-Gibbs thermostatistics. The equilibration time teq depends on the system size N as teq ∼Nγ where γ ≃ 1.8. We compute the velocity distribution P (v) of the oscillators of the two systems during the relaxation process. We find that P (v) is non-Gaussian and is remarkably close to a q-Gaussian distribution for all times before thermal equilibrium is reached. During the relaxation process we observe q > 1 while close to t =teq the value of q converges to unity and P (v) approaches a Gaussian. Thus the relaxation phenomenon in long-ranged systems connected by a thermal contact can be generically described as a crossover from q-statistics to Boltzmann-Gibbs statistics.
Mamikhin, S V; Manakhov, D V; Shcheglov, A I
2014-01-01
The additional study of the distribution of radioactive isotopes of caesium and strontium and their chemical analogues in the above-ground components of pine in the remote from the accident period was carried out. The results of the research confirmed the existence of analogy in the distribution of these elements on the components of this type of wood vegetation in the quasi-equilibrium (relatively radionuclides) condition. Also shown is the selective possibility of using the data on the ash content of the components of forest stands of pine and oak as an information analogue.
Boltzmann babies in the proper time measure
NASA Astrophysics Data System (ADS)
Bousso, Raphael; Freivogel, Ben; Yang, I.-Sheng
2008-05-01
After commenting briefly on the role of the typicality assumption in science, we advocate a phenomenological approach to the cosmological measure problem. Like any other theory, a measure should be simple, general, well defined, and consistent with observation. This allows us to proceed by elimination. As an example, we consider the proper time cutoff on a geodesic congruence. It predicts that typical observers are quantum fluctuations in the early universe, or Boltzmann babies. We sharpen this well-known youngness problem by taking into account the expansion and open spatial geometry of pocket universes. Moreover, we relate the youngness problem directly to the probability distribution for observables, such as the temperature of the cosmic background radiation. We consider a number of modifications of the proper time measure, but find none that would make it compatible with observation.
Boltzmann babies in the proper time measure
Bousso, Raphael; Freivogel, Ben; Yang, I-S.
2008-05-15
After commenting briefly on the role of the typicality assumption in science, we advocate a phenomenological approach to the cosmological measure problem. Like any other theory, a measure should be simple, general, well defined, and consistent with observation. This allows us to proceed by elimination. As an example, we consider the proper time cutoff on a geodesic congruence. It predicts that typical observers are quantum fluctuations in the early universe, or Boltzmann babies. We sharpen this well-known youngness problem by taking into account the expansion and open spatial geometry of pocket universes. Moreover, we relate the youngness problem directly to the probability distribution for observables, such as the temperature of the cosmic background radiation. We consider a number of modifications of the proper time measure, but find none that would make it compatible with observation.
Boltzmann babies in the proper time measure
Bousso, Raphael; Bousso, Raphael; Freivogel, Ben; Yang, I-Sheng
2007-12-20
After commenting briefly on the role of the typicality assumption in science, we advocate a phenomenological approach to the cosmological measure problem. Like any other theory, a measure should be simple, general, well defined, and consistent with observation. This allows us to proceed by elimination. As an example, we consider the proper time cutoff on a geodesic congruence. It predicts that typical observers are quantum fluctuations in the early universe, or Boltzmann babies. We sharpen this well-known youngness problem by taking into account the expansion and open spatial geometry of pocket universes. Moreover, we relate the youngness problem directly to the probability distribution for observables, such as the temperature of the cosmic background radiation. We consider a number of modifications of the proper time measure, but find none that would make it compatible with observation.
NASA Astrophysics Data System (ADS)
Feldman, Michal; Tennenholtz, Moshe
We introduce partition equilibrium and study its existence in resource selection games (RSG). In partition equilibrium the agents are partitioned into coalitions, and only deviations by the prescribed coalitions are considered. This is in difference to the classical concept of strong equilibrium according to which any subset of the agents may deviate. In resource selection games, each agent selects a resource from a set of resources, and its payoff is an increasing (or non-decreasing) function of the number of agents selecting its resource. While it has been shown that strong equilibrium exists in resource selection games, these games do not possess super-strong equilibrium, in which a fruitful deviation benefits at least one deviator without hurting any other deviator, even in the case of two identical resources with increasing cost functions. Similarly, strong equilibrium does not exist for that restricted two identical resources setting when the game is played repeatedly. We prove that for any given partition there exists a super-strong equilibrium for resource selection games of identical resources with increasing cost functions; we also show similar existence results for a variety of other classes of resource selection games. For the case of repeated games we identify partitions that guarantee the existence of strong equilibrium. Together, our work introduces a natural concept, which turns out to lead to positive and applicable results in one of the basic domains studied in the literature.
NASA Astrophysics Data System (ADS)
Niiyama, Tomoaki; Shimizu, Yasushi; Kobayashi, Taizo R.; Okushima, Teruaki; Ikeda, Kensuke S.
2009-05-01
We investigate numerically and analytically the effects of conservation of total translational and angular momentum on the distribution of kinetic energy among particles in microcanonical particle systems with small number of degrees of freedom, specifically microclusters. Molecular dynamics simulations of microclusters with constant total energy and momenta, using Lennard-Jones, Morse, and Coulomb plus Born-Mayer-type potentials, show that the distribution of kinetic energy among particles can be inhomogeneous and depend on particle mass and position even in thermal equilibrium. Statistical analysis using a microcanonical measure taking into account of the additional conserved quantities gives theoretical expressions for kinetic energy as a function of the mass and position of a particle with only O(1/N2) deviation from the Maxwell-Boltzmann distribution. These expressions fit numerical results well. Finally, we propose an intuitive interpretation for the inhomogeneity of the kinetic energy distributions.
Return of the Boltzmann brains
Page, Don N.
2008-09-15
Linde in J. Cosmol. Astropart. Phys. 01 (2007) 022 shows that some (though not all) versions of the global (volume-weighted) description avoid the 'Boltzmann brain' problem raised by Page [Phys. Rev. D 78, 063535 (2008)] if the universe does not have a decay time less than 20 Gyr. Here I give an apparently natural version of the volume-weighted description in which the problem persists, highlighting the ambiguity of taking the ratios of infinite volumes that appear to arise from eternal inflation.
Equilibrium distribution of dissolved sulphur species in water at 25°C and 1 atm total pressure
Garrels, R.M.; Naeser, C.R.
1958-01-01
The Eh-pH diagrams for the equilibrium concentrations in aqueous solution at 25°C of native sulphur and all the various sulphur-containing ions and acids from which the ions are produced have been constructed for systems having a total sulphur concentration of 0.1 molar. The composite of these diagrams indicates that elemental sulphur, H2S, HS− HSO4− and SO4 are the species that predominate in the environments that might be found in nature. This indication is in agreement with the composition of all sulphur-containing minerals.
NASA Astrophysics Data System (ADS)
Morgan, George; London, David
2002-12-01
This study examines the effects of increasing supersaturation, attained by single-step liquidus undercooling (ΔT), on the partitioning of barium and cesium between potassic alkali feldspar (Afs) and hydrous granitic liquid at 200 MPa. The investigation is motivated by trace-element distribution patterns in granitic pegmatites which cannot be simulated by fractionation models using "equilibrium" partition coefficients, and thus its purpose is to assess if, how, and why partition coefficients for compatible and incompatible trace elements may vary when crystal growth commences far from the crystal-melt equilibrium boundary. Barium expands the liquidus stability field of potassic feldspar to higher temperatures, such that liquidi for the Ba-rich ( 0.5 wt% BaO) compositions used are 100 °C higher than for Ba-absent analogues. At low degrees of undercooling (ΔT 50 °C), values of DBaAfs/m. ( 10-20) fall within the range of previous investigations, as do values of DCsAfs/m. (<=0.10) from experiments at all temperatures. Progressively greater undercooling is manifested in the run products by increasingly skeletal to cuneiform crystal morphologies, increased compositional zonation of Afs, and the development of compositional boundary layers in glass. Whereas the partitioning behavior of Cs (incompatible) is not measurably affected, strong undercooling apparently causes the partitioning of Ba (highly compatible) to deviate from equilibrium behavior. Feldspars produced by strong undercooling (ΔT>=100 °C) are heterogeneous, such that DBaAfs/m. versus K/K+Na varies linearly between the average value at 850 °C and the equilibrium value appropriate to the temperature of growth. Hence, high supersaturation accompanying undercooling produces feldspar compositions by isothermal growth which record a vestige of the liquid line of descent (i.e., an ontogeny within zoned crystals which approximately tracks the feldspar liquidus from high temperature to the final low temperature
Harding, Stephen E; Gillis, Richard B; Adams, Gary G
2016-01-01
Molecular weights (molar masses), molecular weight distributions, dissociation constants and other interaction parameters are fundamental characteristics of proteins, nucleic acids, polysaccharides and glycoconjugates in solution. Sedimentation equilibrium analytical ultracentrifugation provides a powerful method with no supplementary immobilization, columns or membranes required. It is a particularly powerful tool when used in conjunction with its sister technique, namely sedimentation velocity. Here, we describe key approaches now available and their application to the characterization of antibodies, polysaccharides and glycoconjugates. We indicate how major complications, such as thermodynamic non-ideality, can now be routinely dealt with, thanks to a great extent to the extensive contribution of Professor Don Winzor over several decades of research.
Recent advances in lattice Boltzmann methods
Chen, S.; Doolen, G.D.; He, X.; Nie, X.; Zhang, R.
1998-12-31
In this paper, the authors briefly present the basic principles of lattice Boltzmann method and summarize recent advances of the method, including the application of the lattice Boltzmann method for fluid flows in MEMS and simulation of the multiphase mixing and turbulence.
Student Understanding of the Boltzmann Factor
ERIC Educational Resources Information Center
Smith, Trevor I.; Mountcastle, Donald B.; Thompson, John R.
2015-01-01
We present results of our investigation into student understanding of the physical significance and utility of the Boltzmann factor in several simple models. We identify various justifications, both correct and incorrect, that students use when answering written questions that require application of the Boltzmann factor. Results from written data…
Global existence proof for relativistic Boltzmann equation
Dudynski, M. ); Ekiel-Jezewska, M.L. )
1992-02-01
The existence and causality of solutions to the relativistic Boltzmann equation in L[sup 1] and in L[sub loc][sup 1] are proved. The solutions are shown to satisfy physically natural a priori bounds, time-independent in L[sup 1]. The results rely upon new techniques developed for the nonrelativistic Boltzmann equation by DiPerna and Lions.
Spectral Classification Using Restricted Boltzmann Machine
NASA Astrophysics Data System (ADS)
Fuqiang, Chen; Yan, Wu; Yude, Bu; Guodong, Zhao
2014-01-01
In this study, a novel machine learning algorithm, restricted Boltzmann machine, is introduced. The algorithm is applied for the spectral classification in astronomy. Restricted Boltzmann machine is a bipartite generative graphical model with two separate layers (one visible layer and one hidden layer), which can extract higher level features to represent the original data. Despite generative, restricted Boltzmann machine can be used for classification when modified with a free energy and a soft-max function. Before spectral classification, the original data are binarised according to some rule. Then, we resort to the binary restricted Boltzmann machine to classify cataclysmic variables and non-cataclysmic variables (one half of all the given data for training and the other half for testing). The experiment result shows state-of-the-art accuracy of 100%, which indicates the efficiency of the binary restricted Boltzmann machine algorithm.
Ainsworth, Nathan G; Grijalva, Prof. Santiago
2013-01-01
This paper discusses a proposed frequency restoration controller which operates as an outer loop to frequency droop for voltage-source inverters. By quasi-equilibrium analysis, we show that the proposed controller is able to provide arbitrarily small steady-state frequency error while maintaing power sharing between inverters without need for communication or centralized control. We derive rate of convergence, discuss design considerations (including a fundamental trade-off that must be made in design), present a design procedure to meet a maximum frequency error requirement, and show simulation results verifying our analysis and design method. The proposed controller will allow flexible plug-and-play inverter-based networks to meet a specified maximum frequency error requirement.
An upwind, kinetic flux-vector splitting method for flows in chemical and thermal non-equilibrium
NASA Technical Reports Server (NTRS)
Eppard, W. M.; Grossman, B.
1993-01-01
We have developed new upwind kinetic difference schemes for flows with non-equilibrium thermodynamics and chemistry. These schemes are derived from the Boltzmann equation with the resulting Euler schemes developed as moments of the discretized Boltzmann scheme with a locally Maxwellian velocity distribution. Splitting the velocity distribution at the Boltzmann level is seen to result in a flux-split Euler scheme and is called Kinetic Flux Vector Splitting (KFVS). Extensions to flows with finite-rate chemistry and vibrational relaxation is accomplished utilizing nonequilibrium kinetic theory. Computational examples are presented comparing KFVS with the schemes of Van Leer and Roe for a quasi-one-dimensional flow through a supersonic diffuser, inviscid flow through two-dimensional inlet, and viscous flow over a cone at zero angle-of-attack. Calculations are also shown for the transonic flow over a bump in a channel and the transonic flow over an NACA 0012 airfoil. The results show that even though the KFVS scheme is a Riemann solver at the kinetic level, its behavior at the Euler level is more similar to the existing flux-vector splitting algorithms than to the flux-difference splitting scheme of Roe.
Non-Boltzmann Ensembles and Monte Carlo Simulations
NASA Astrophysics Data System (ADS)
Murthy, K. P. N.
2016-10-01
Boltzmann sampling based on Metropolis algorithm has been extensively used for simulating a canonical ensemble and for calculating macroscopic properties of a closed system at desired temperatures. An estimate of a mechanical property, like energy, of an equilibrium system, is made by averaging over a large number microstates generated by Boltzmann Monte Carlo methods. This is possible because we can assign a numerical value for energy to each microstate. However, a thermal property like entropy, is not easily accessible to these methods. The reason is simple. We can not assign a numerical value for entropy, to a microstate. Entropy is not a property associated with any single microstate. It is a collective property of all the microstates. Toward calculating entropy and other thermal properties, a non-Boltzmann Monte Carlo technique called Umbrella sampling was proposed some forty years ago. Umbrella sampling has since undergone several metamorphoses and we have now, multi-canonical Monte Carlo, entropic sampling, flat histogram methods, Wang-Landau algorithm etc. This class of methods generates non-Boltzmann ensembles which are un-physical. However, physical quantities can be calculated as follows. First un-weight a microstates of the entropic ensemble; then re-weight it to the desired physical ensemble. Carry out weighted average over the entropic ensemble to estimate physical quantities. In this talk I shall tell you of the most recent non- Boltzmann Monte Carlo method and show how to calculate free energy for a few systems. We first consider estimation of free energy as a function of energy at different temperatures to characterize phase transition in an hairpin DNA in the presence of an unzipping force. Next we consider free energy as a function of order parameter and to this end we estimate density of states g(E, M), as a function of both energy E, and order parameter M. This is carried out in two stages. We estimate g(E) in the first stage. Employing g
NASA Technical Reports Server (NTRS)
Chen, Yih-Kang
1992-01-01
Effect of flow field properties on the heating distribution over a 140 deg blunt cone was determined for a Martian atmosphere using Euler, Navier-Stokes (NS), viscous shock layer (VSL), and reacting boundary layer (BLIMPK) equations. The effect of gas kinetics on the flow field and the surface heating distribution were investigated. Gas models with nine species and nine reactions were implemented into the codes. Effects of surface catalysis on the heating distribution were studied using a surface kinetics model having five reactions.
Analysis of multifragmentation in a Boltzmann-Langevin approach
Zhang, F.; Suraud, E.
1995-06-01
By using the Boltzmann-Langevin equation, which incorporates dynamical fluctuations beyond usual transport theories, we simulate the {sup 40}Ca+{sup 40}Ca reaction system at different beam energies 20, 60, and 90 MeV/nucleon for different impact parameters. Dynamical fluctuations become larger and larger with increasing bombarding energy and the system can reach densities corresponding to the unstable region of the nuclear matter equation of state at energies above 60 MeV/nucleon. By coupling the Boltzmann-Langevin equation with a coalescence model in the late stages of the reaction, we obtain the distribution of the intermediate mass fragments in each event. From the correlation analysis of these fragments, we recover some trends of recent multifragmentation data. A critical behavior analysis is also provided.
A Boltzmann model for rod alignment and schooling fish
NASA Astrophysics Data System (ADS)
Carlen, Eric; Carvalho, Maria C.; Degond, Pierre; Wennberg, Bernt
2015-06-01
We consider a Boltzmann model introduced by Bertin, Droz and Grégoire as a binary interaction model of the Vicsek alignment interaction. This model considers particles lying on the circle. Pairs of particles interact by trying to reach their mid-point (on the circle) up to some noise. We study the equilibria of this Boltzmann model and we rigorously show the existence of a pitchfork bifurcation when a parameter measuring the inverse of the noise intensity crosses a critical threshold. The analysis is carried over rigorously when there are only finitely many non-zero Fourier modes of the noise distribution. In this case, we can show that the critical exponent of the bifurcation is exactly 1/2. In the case of an infinite number of non-zero Fourier modes, a similar behavior can be formally obtained thanks to a method relying on integer partitions first proposed by Ben-Naïm and Krapivsky.
High performance computing with a conservative spectral Boltzmann solver
NASA Astrophysics Data System (ADS)
Haack, Jeffrey R.; Gamba, Irene M.
2012-11-01
We present new results building on the conservative deterministic spectral method for the space inhomogeneous Boltzmann equation developed by Gamba and Tharkabhushaman. This approach is a two-step process that acts on the weak form of the Boltzmann equation, and uses the machinery of the Fourier transform to reformulate the collisional integral into a weighted convolution in Fourier space. A constrained optimization problem is solved to preserve the mass, momentum, and energy of the resulting distribution. We extend this method to second order accuracy in space and time, and explore how to leverage the structure of the collisional formulation for high performance computing environments. The locality in space of the collisional term provides a straightforward memory decomposition, and we perform some initial scaling tests on high performance computing resources. We also use the improved computational power of this method to investigate a boundary-layer generated shock problem that cannot be described by classical hydrodynamics.
Representational power of restricted boltzmann machines and deep belief networks.
Le Roux, Nicolas; Bengio, Yoshua
2008-06-01
Deep belief networks (DBN) are generative neural network models with many layers of hidden explanatory factors, recently introduced by Hinton, Osindero, and Teh (2006) along with a greedy layer-wise unsupervised learning algorithm. The building block of a DBN is a probabilistic model called a restricted Boltzmann machine (RBM), used to represent one layer of the model. Restricted Boltzmann machines are interesting because inference is easy in them and because they have been successfully used as building blocks for training deeper models. We first prove that adding hidden units yields strictly improved modeling power, while a second theorem shows that RBMs are universal approximators of discrete distributions. We then study the question of whether DBNs with more layers are strictly more powerful in terms of representational power. This suggests a new and less greedy criterion for training RBMs within DBNs.
Conservative deterministic spectral Boltzmann solver near the grazing collisions limit
NASA Astrophysics Data System (ADS)
Haack, Jeffrey R.; Gamba, Irene M.
2012-11-01
We present new results building on the conservative deterministic spectral method for the space homogeneous Boltzmann equation developed by Gamba and Tharkabhushaman. This approach is a two-step process that acts on the weak form of the Boltzmann equation, and uses the machinery of the Fourier transform to reformulate the collisional integral into a weighted convolution in Fourier space. A constrained optimization problem is solved to preserve the mass, momentum, and energy of the resulting distribution. Within this framework we have extended the formulation to the case of more general case of collision operators with anisotropic scattering mechanisms, which requires a new formulation of the convolution weights. We also derive the grazing collisions limit for the method, and show that it is consistent with the Fokker-Planck-Landau equations as the grazing collisions parameter goes to zero.
NASA Astrophysics Data System (ADS)
De Souza, Roger A.; Metlenko, Veronika; Park, Daesung; Weirich, Thomas E.
2012-05-01
18O/16O exchange and subsequent time-of-flight secondary ion mass spectrometry (ToF-SIMS) analysis was employed to investigate the transport of oxygen, and thus the behavior of oxygen vacancies, in [nominally undoped, (100) oriented] single-crystal SrTiO3 substrates. Isotope exchange anneals were performed as a function of temperature, 948 < T/K < 1123, at an oxygen activity aO2 = 0.50 and as a function of oxygen activity, 0.01 < aO2 < 0.70, at T = 1073 K. All isotope profiles show the same characteristic form: an initial drop over tens of nanometers close to the surface, which is attributed to an equilibrium space-charge layer depleted of oxygen vacancies, followed by a profile extending several microns into the solid, which is attributed to diffusion in a homogeneous bulk phase. The entire isotope profile can be described quantitatively by a numerical solution to the diffusion equation with a position-dependent diffusion coefficient; the description yields the tracer diffusion coefficient in the bulk D*(∞), the surface exchange coefficient ks*, and the space-charge potential Φ0. All D*(∞) data are consistent with nominally undoped SrTiO3 substrates being weakly acceptor doped; the activation enthalpy for the migration of oxygen vacancies in bulk SrTiO3 is found to be ΔHmig,V ≈ 0.6 eV. The surface termination of the SrTiO3 substrates was seen to affect significantly the surface exchange coefficient ks*. Values of Φ0 obtained as a function of T and aO2 are approximately 0.5 V, indicating strong depletion of oxygen vacancies within the equilibrium surface space-charge layers. Thermodynamic modeling indicates that space-charge formation at the TiO2-terminated (100) surface is driven by the Gibbs formation energy of oxygen vacancies at the interface being lower than in the bulk.
An Infinite Restricted Boltzmann Machine.
Côté, Marc-Alexandre; Larochelle, Hugo
2016-07-01
We present a mathematical construction for the restricted Boltzmann machine (RBM) that does not require specifying the number of hidden units. In fact, the hidden layer size is adaptive and can grow during training. This is obtained by first extending the RBM to be sensitive to the ordering of its hidden units. Then, with a carefully chosen definition of the energy function, we show that the limit of infinitely many hidden units is well defined. As with RBM, approximate maximum likelihood training can be performed, resulting in an algorithm that naturally and adaptively adds trained hidden units during learning. We empirically study the behavior of this infinite RBM, showing that its performance is competitive to that of the RBM, while not requiring the tuning of a hidden layer size.
Approximate learning algorithm in Boltzmann machines.
Yasuda, Muneki; Tanaka, Kazuyuki
2009-11-01
Boltzmann machines can be regarded as Markov random fields. For binary cases, they are equivalent to the Ising spin model in statistical mechanics. Learning systems in Boltzmann machines are one of the NP-hard problems. Thus, in general we have to use approximate methods to construct practical learning algorithms in this context. In this letter, we propose new and practical learning algorithms for Boltzmann machines by using the belief propagation algorithm and the linear response approximation, which are often referred as advanced mean field methods. Finally, we show the validity of our algorithm using numerical experiments.
Global classical solutions of the Boltzmann equation with long-range interactions.
Gressman, Philip T; Strain, Robert M
2010-03-30
This is a brief announcement of our recent proof of global existence and rapid decay to equilibrium of classical solutions to the Boltzmann equation without any angular cutoff, that is, for long-range interactions. We consider perturbations of the Maxwellian equilibrium states and include the physical cross-sections arising from an inverse-power intermolecular potential r(-(p-1)) with p > 2, and more generally. We present here a mathematical framework for unique global in time solutions for all of these potentials. We consider it remarkable that this equation, derived by Boltzmann (1) in 1872 and Maxwell (2) in 1867, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the effects due to grazing collisions.
NASA Astrophysics Data System (ADS)
Wania, F.; Lei, Y. D.; Wang, C.; Abbatt, J. P. D.; Goss, K.-U.
2015-03-01
Many atmospheric and chemical variables influence the partitioning equilibrium between gas phase and condensed phases of compounds implicated in the formation of secondary organic aerosol (SOA). The large number of factors and their interaction makes it often difficult to assess their relative importance and concerted impact. Here we introduce a two-dimensional space which maps regions of dominant atmospheric phase distribution within a coordinate system defined by equilibrium partition coefficients between the gas phase, an aqueous phase and a water-insoluble organic matter (WIOM) phase. Placing compounds formed from the oxidation of n-alkanes, terpenes and mono-aromatic hydrocarbons on the maps based on their predicted partitioning properties allows for a simple graphical assessment of their equilibrium phase distribution behaviour. Specifically, it allows for the simultaneous visualisation and quantitative comparison of the impact on phase distribution of changes in atmospheric parameters (such as temperature, salinity, WIOM-phase polarity, organic aerosol load, and liquid water content) and chemical properties (such as oxidation state, molecular size, functionalisation, and dimerisation). The graphical analysis reveals that the addition of hydroxyl, carbonyl and carboxyl groups increases the affinity of aliphatic, alicyclic and aromatic hydrocarbons for the aqueous phase more rapidly than their affinity for WIOM, suggesting that the aqueous phase may often be relevant even for substances that are considerably larger than the C2 and C3 compounds that are typically believed to be associated with aqueous SOA. In particular, the maps identify some compounds that contribute to SOA formation if partitioning to both WIOM and aqueous phase is considered but would remain in the gas phase if either condensed phase were neglected. For example, many semi-volatile α-pinene oxidation products will contribute to aqueous SOA under the conditions of high liquid water content
NASA Astrophysics Data System (ADS)
Wania, F.; Lei, Y. D.; Wang, C.; Abbatt, J. P. D.; Goss, K.-U.
2014-10-01
Many atmospheric and chemical variables influence the partitioning equilibrium between gas phase and condensed phases of compounds implicated in the formation of secondary organic aerosol (SOA). The large number of factors and their interaction makes it often difficult to assess their relative importance and concerted impact. Here we introduce a two-dimensional space, which maps regions of dominant atmospheric phase distribution within a coordinate system defined by equilibrium partitioning coefficients between the gas phase, an aqueous phase and a water insoluble organic matter (WIOM) phase. Placing compounds formed from the oxidation of n-alkanes, terpenes and mono-aromatic hydrocarbons on the maps based on their predicted partitioning properties allows for a simple graphical assessment of their equilibrium phase distribution behaviour. Specifically, it allows for the simultaneous visualization and quantitative comparison of the impact on phase distribution of changes in atmospheric parameters (such as temperature, salinity, WIOM phase polarity, organic aerosol load, and liquid water content), and chemical properties (such as oxidation state, molecular size, functionalization, and dimerisation). The graphical analysis reveals that the addition of hydroxyl, carbonyl and carboxyl groups increases the affinity of aliphatic, alicyclic and aromatic hydrocarbons for the aqueous phase more rapidly than their affinity for WIOM, suggesting that the aqueous phase may often be relevant even for substances that are considerably larger than the C2 and C3 compounds that are typically believed to be associated with aqueous SOA. In particular, the maps identify some compounds that contribute to SOA formation if partitioning to both WIOM and aqueous phase is considered, but would remain in the gas phase if either condensed phase were neglected. For example, many semi-volatile α-pinene oxidation products will contribute to aqueous SOA under the high liquid water content
Hypersonic Shock Wave Computations Using the Generalized Boltzmann Equation
NASA Astrophysics Data System (ADS)
Agarwal, Ramesh; Chen, Rui; Cheremisin, Felix G.
2006-11-01
Hypersonic shock structure in diatomic gases is computed by solving the Generalized Boltzmann Equation (GBE), where the internal and translational degrees of freedom are considered in the framework of quantum and classical mechanics respectively [1]. The computational framework available for the standard Boltzmann equation [2] is extended by including both the rotational and vibrational degrees of freedom in the GBE. There are two main difficulties encountered in computation of high Mach number flows of diatomic gases with internal degrees of freedom: (1) a large velocity domain is needed for accurate numerical description of the distribution function resulting in enormous computational effort in calculation of the collision integral, and (2) about 50 energy levels are needed for accurate representation of the rotational spectrum of the gas. Our methodology addresses these problems, and as a result the efficiency of calculations has increased by several orders of magnitude. The code has been validated by computing the shock structure in Nitrogen for Mach numbers up to 25 including the translational and rotational degrees of freedom. [1] Beylich, A., ``An Interlaced System for Nitrogen Gas,'' Proc. of CECAM Workshop, ENS de Lyon, France, 2000. [2] Cheremisin, F., ``Solution of the Boltzmann Kinetic Equation for High Speed Flows of a Rarefied Gas,'' Proc. of the 24th Int. Symp. on Rarefied Gas Dynamics, Bari, Italy, 2004.
Filter-matrix lattice Boltzmann model for microchannel gas flows.
Zhuo, Congshan; Zhong, Chengwen
2013-11-01
The lattice Boltzmann method has been shown to be successful for microscale gas flows, and it has attracted significant research interest. In this paper, the recently proposed filter-matrix lattice Boltzmann (FMLB) model is first applied to study the microchannel gas flows, in which a Bosanquet-type effective viscosity is used to capture the flow behaviors in the transition regime. A kinetic boundary condition, the combined bounce-back and specular-reflection scheme with the second-order slip scheme, is also designed for the FMLB model. By analyzing a unidirectional flow, the slip velocity and the discrete effects related to the boundary condition are derived within the FMLB model, and a revised scheme is presented to overcome such effects, which have also been validated through numerical simulations. To gain an accurate simulation in a wide range of Knudsen numbers, covering the slip and the entire transition flow regimes, a set of slip coefficients with an introduced fitting function is adopted in the revised second-order slip boundary condition. The periodic and pressure-driven microchannel flows have been investigated by the present model in this study. The numerical results, including the velocity profile and the mass flow rate, as well as the nonlinear pressure distribution along the channel, agree fairly well with the solutions of the linearized Boltzmann equation, the direct simulation Monte Carlo results, the experimental data, and the previous results of the multiple effective relaxation lattice Boltzmann model. Also, the present results of the velocity profile and the mass flow rate show that the present model with the fitting function can yield improved predictions for the microchannel gas flow with higher Knudsen numbers in the transition flow regime.
Lattice Boltzmann approach for complex nonequilibrium flows.
Montessori, A; Prestininzi, P; La Rocca, M; Succi, S
2015-10-01
We present a lattice Boltzmann realization of Grad's extended hydrodynamic approach to nonequilibrium flows. This is achieved by using higher-order isotropic lattices coupled with a higher-order regularization procedure. The method is assessed for flow across parallel plates and three-dimensional flows in porous media, showing excellent agreement of the mass flow with analytical and numerical solutions of the Boltzmann equation across the full range of Knudsen numbers, from the hydrodynamic regime to ballistic motion.
Alternating minimization and Boltzmann machine learning.
Byrne, W
1992-01-01
Training a Boltzmann machine with hidden units is appropriately treated in information geometry using the information divergence and the technique of alternating minimization. The resulting algorithm is shown to be closely related to gradient descent Boltzmann machine learning rules, and the close relationship of both to the EM algorithm is described. An iterative proportional fitting procedure for training machines without hidden units is described and incorporated into the alternating minimization algorithm.
Poisson-Boltzmann-Nernst-Planck model
Zheng Qiong; Wei Guowei
2011-05-21
The Poisson-Nernst-Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. It provides qualitative explanation and increasingly quantitative predictions of experimental measurements for the ion transport problems in many areas such as semiconductor devices, nanofluidic systems, and biological systems, despite many limitations. While the PNP model gives a good prediction of the ion transport phenomenon for chemical, physical, and biological systems, the number of equations to be solved and the number of diffusion coefficient profiles to be determined for the calculation directly depend on the number of ion species in the system, since each ion species corresponds to one Nernst-Planck equation and one position-dependent diffusion coefficient profile. In a complex system with multiple ion species, the PNP can be computationally expensive and parameter demanding, as experimental measurements of diffusion coefficient profiles are generally quite limited for most confined regions such as ion channels, nanostructures and nanopores. We propose an alternative model to reduce number of Nernst-Planck equations to be solved in complex chemical and biological systems with multiple ion species by substituting Nernst-Planck equations with Boltzmann distributions of ion concentrations. As such, we solve the coupled Poisson-Boltzmann and Nernst-Planck (PBNP) equations, instead of the PNP equations. The proposed PBNP equations are derived from a total energy functional by using the variational principle. We design a number of computational techniques, including the Dirichlet to Neumann mapping, the matched interface and boundary, and relaxation based iterative procedure, to ensure efficient solution of the proposed PBNP equations. Two protein molecules, cytochrome c551 and Gramicidin A, are employed to validate the proposed model under a wide range of bulk ion concentrations and external
Poisson-Boltzmann-Nernst-Planck model
NASA Astrophysics Data System (ADS)
Zheng, Qiong; Wei, Guo-Wei
2011-05-01
The Poisson-Nernst-Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. It provides qualitative explanation and increasingly quantitative predictions of experimental measurements for the ion transport problems in many areas such as semiconductor devices, nanofluidic systems, and biological systems, despite many limitations. While the PNP model gives a good prediction of the ion transport phenomenon for chemical, physical, and biological systems, the number of equations to be solved and the number of diffusion coefficient profiles to be determined for the calculation directly depend on the number of ion species in the system, since each ion species corresponds to one Nernst-Planck equation and one position-dependent diffusion coefficient profile. In a complex system with multiple ion species, the PNP can be computationally expensive and parameter demanding, as experimental measurements of diffusion coefficient profiles are generally quite limited for most confined regions such as ion channels, nanostructures and nanopores. We propose an alternative model to reduce number of Nernst-Planck equations to be solved in complex chemical and biological systems with multiple ion species by substituting Nernst-Planck equations with Boltzmann distributions of ion concentrations. As such, we solve the coupled Poisson-Boltzmann and Nernst-Planck (PBNP) equations, instead of the PNP equations. The proposed PBNP equations are derived from a total energy functional by using the variational principle. We design a number of computational techniques, including the Dirichlet to Neumann mapping, the matched interface and boundary, and relaxation based iterative procedure, to ensure efficient solution of the proposed PBNP equations. Two protein molecules, cytochrome c551 and Gramicidin A, are employed to validate the proposed model under a wide range of bulk ion concentrations and external
1982-09-01
that for variouis standard types of equilibria* they hold. In particular, if one uses the teaporary equilibrium framework one can use the standard ...T, the integral converges toward f’ia(da) f fU(b~dc)6(a,b,c)T( asdm ) A B C which is fR (da) f d(lib,c) U0 T (cab) A BxC Me converse Is obvious
Lattice Boltzmann algorithm for continuum multicomponent flow
NASA Astrophysics Data System (ADS)
Halliday, I.; Hollis, A. P.; Care, C. M.
2007-08-01
We present a multicomponent lattice Boltzmann simulation for continuum fluid mechanics, paying particular attention to the component segregation part of the underlying algorithm. In the principal result of this paper, the dynamics of a component index, or phase field, is obtained for a segregation method after U. D’Ortona [Phys. Rev. E 51, 3718 (1995)], due to Latva-Kokko and Rothman [Phys. Rev. E 71 056702 (2005)]. The said dynamics accord with a simulation designed to address multicomponent flow in the continuum approximation and underwrite improved simulation performance in two main ways: (i) by reducing the interfacial microcurrent activity considerably and (ii) by facilitating simulational access to regimes of flow with a low capillary number and drop Reynolds number [I. Halliday, R. Law, C. M. Care, and A. Hollis, Phys. Rev. E 73, 056708 (2006)]. The component segregation method studied, used in conjunction with Lishchuk’s method [S. V. Lishchuk, C. M. Care, and I. Halliday, Phys. Rev. E 67, 036701 (2003)], produces an interface, which is distributed in terms of its component index; however, the hydrodynamic boundary conditions which emerge are shown to support the notion of a sharp, unstructured, continuum interface.
Equilibrium cluster fluids: pair interactions via inverse design.
Jadrich, R B; Bollinger, J A; Lindquist, B A; Truskett, T M
2015-12-28
Inverse methods of statistical mechanics are becoming productive tools in the design of materials with specific microstructures or properties. While initial studies have focused on solid-state design targets (e.g., assembly of colloidal superlattices), one can alternatively design fluid states with desired morphologies. This work addresses the latter and demonstrates how a simple iterative Boltzmann inversion strategy can be used to determine the isotropic pair potential that reproduces the radial distribution function of a fluid of amorphous clusters with prescribed size. The inverse designed pair potential of this "ideal" cluster fluid, with its broad attractive well and narrow repulsive barrier at larger separations, is qualitatively different from the so-called SALR form most commonly associated with equilibrium cluster formation in colloids, which features short-range attractive (SA) and long-range repulsive (LR) contributions. These differences reflect alternative mechanisms for promoting cluster formation with an isotropic pair potential, and they in turn produce structured fluids with qualitatively different static and dynamic properties. Specifically, equilibrium simulations show that the amorphous clusters resulting from the inverse designed potentials display more uniformity in size and shape, and they also show greater spatial and temporal resolution than those resulting from SALR interactions.
Improved lattice Boltzmann modeling of binary flow based on the conservative Allen-Cahn equation
NASA Astrophysics Data System (ADS)
Ren, Feng; Song, Baowei; Sukop, Michael C.; Hu, Haibao
2016-08-01
The primary and key task of binary fluid flow modeling is to track the interface with good accuracy, which is usually challenging due to the sharp-interface limit and numerical dispersion. This article concentrates on further development of the conservative Allen-Cahn equation (ACE) [Geier et al., Phys. Rev. E 91, 063309 (2015), 10.1103/PhysRevE.91.063309] under the framework of the lattice Boltzmann method (LBM), with incorporation of the incompressible hydrodynamic equations [Liang et al., Phys. Rev. E 89, 053320 (2014), 10.1103/PhysRevE.89.053320]. Utilizing a modified equilibrium distribution function and an additional source term, this model is capable of correctly recovering the conservative ACE through the Chapman-Enskog analysis. We also simulate four phase-tracking benchmark cases, including one three-dimensional case; all show good accuracy as well as low numerical dispersion. By coupling the incompressible hydrodynamic equations, we also simulate layered Poiseuille flow and the Rayleigh-Taylor instability, illustrating satisfying performance in dealing with complex flow problems, e.g., high viscosity ratio, high density ratio, and high Reynolds number situations. The present work provides a reliable and efficient solution for binary flow modeling.
On pressure and velocity boundary conditions for the lattice Boltzmann BGK model
Zou, Q. |; He, X.
1997-06-01
Pressure (density) and velocity boundary conditions are studied for 2-D and 3-D lattice Boltzmann BGK models (LBGK) and a new method to specify these conditions is proposed. These conditions are constructed in consistency with the wall boundary condition, based on the idea of bounceback of the non-equilibrium distribution. When these conditions are used together with the incompressible LBGK model [J. Stat. Phys. {bold 81}, 35 (1995)] the simulation results recover the analytical solution of the plane Poiseuille flow driven by a pressure (density) difference. The half-way wall bounceback boundary condition is also used with the pressure (density) inlet/outlet conditions proposed in this paper and in Phys. Fluids {bold 8}, 2527 (1996) to study 2-D Poiseuille flow and 3-D square duct flow. The numerical results are approximately second-order accurate. The magnitude of the error of the half-way wall bounceback boundary condition is comparable with that of other published boundary conditions and it has better stability behavior. {copyright} {ital 1997 American Institute of Physics.}
Large-scale parallel lattice Boltzmann-cellular automaton model of two-dimensional dendritic growth
NASA Astrophysics Data System (ADS)
Jelinek, Bohumir; Eshraghi, Mohsen; Felicelli, Sergio; Peters, John F.
2014-03-01
An extremely scalable lattice Boltzmann (LB)-cellular automaton (CA) model for simulations of two-dimensional (2D) dendritic solidification under forced convection is presented. The model incorporates effects of phase change, solute diffusion, melt convection, and heat transport. The LB model represents the diffusion, convection, and heat transfer phenomena. The dendrite growth is driven by a difference between actual and equilibrium liquid composition at the solid-liquid interface. The CA technique is deployed to track the new interface cells. The computer program was parallelized using the Message Passing Interface (MPI) technique. Parallel scaling of the algorithm was studied and major scalability bottlenecks were identified. Efficiency loss attributable to the high memory bandwidth requirement of the algorithm was observed when using multiple cores per processor. Parallel writing of the output variables of interest was implemented in the binary Hierarchical Data Format 5 (HDF5) to improve the output performance, and to simplify visualization. Calculations were carried out in single precision arithmetic without significant loss in accuracy, resulting in 50% reduction of memory and computational time requirements. The presented solidification model shows a very good scalability up to centimeter size domains, including more than ten million of dendrites. Catalogue identifier: AEQZ_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEQZ_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, UK Licensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 29,767 No. of bytes in distributed program, including test data, etc.: 3131,367 Distribution format: tar.gz Programming language: Fortran 90. Computer: Linux PC and clusters. Operating system: Linux. Has the code been vectorized or parallelized?: Yes. Program is parallelized using MPI
The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential
NASA Astrophysics Data System (ADS)
Liu, Shuangqian; Yang, Xiongfeng
2017-01-01
Boundary effects are central to the dynamics of the dilute particles governed by the Boltzmann equation. In this paper, we study both the diffuse reflection and the specular reflection boundary value problems for the Boltzmann equation with a soft potential, in which the collision kernel is ruled by the inverse power law. For the diffuse reflection boundary condition, based on an L 2 argument and its interplay with intricate {L^∞} analysis for the linearized Boltzmann equation, we first establish the global existence and then obtain the exponential decay in {L^∞} space for the nonlinear Boltzmann equation in general classes of bounded domain. It turns out that the zero lower bound of the collision frequency and the singularity of the collision kernel lead to some new difficulties for achieving the a priori {L^∞} estimates and time decay rates of the solution. In the course of the proof, we capture some new properties of the probability integrals along the stochastic cycles and improve the {L^2-L^∞} theory to give a more direct approach to overcome those difficulties. As to the specular reflection condition, our key contribution is to develop a new time-velocity weighted {L^∞} theory so that we could deal with the greater difficulties stemming from the complicated velocity relations among the specular cycles and the zero lower bound of the collision frequency. From this new point, we are also able to prove that the solutions of the linearized Boltzmann equation tend to equilibrium exponentially in {L^∞} space with the aid of the L 2 theory and a bootstrap argument. These methods, in the latter case, can be applied to the Boltzmann equation with soft potential for all other types of boundary condition.
NASA Astrophysics Data System (ADS)
Briant, Marc; Einav, Amit
2016-06-01
The Boltzmann-Nordheim equation is a modification of the Boltzmann equation, based on physical considerations, that describes the dynamics of the distribution of particles in a quantum gas composed of bosons or fermions. In this work we investigate the Cauchy theory of the spatially homogeneous Boltzmann-Nordheim equation for bosons, in dimension d≥slant 3. We show existence and uniqueness locally in time for any initial data in L^∞ (1+| v| ^s) with finite mass and energy, for a suitable s, as well as the instantaneous creation of moments of all order.
NASA Astrophysics Data System (ADS)
Ginzburg, Irina; Roux, Laetitia; Silva, Goncalo
2015-10-01
This work demonstrates that in advection-diffusion Lattice Boltzmann schemes, the local mass-conserving boundary rules, such as bounce-back and local specular reflection, may modify the transport coefficients predicted by the Chapman-Enskog expansion when they enforce to zero not only the normal, but also the tangential boundary flux. In order to accommodate it to the bulk solution, the system develops a Knudsen-layer correction to the non-equilibrium part of the population solution. Two principal secondary effects-(i) decrease in the diffusion coefficient, and (ii) retardation of the average advection velocity, obtained in a closed analytical form, are proportional, respectively, to freely assigned diagonal weights for equilibrium mass and velocity terms. In addition, due to their transverse velocity gradients, the boundary layers affect the longitudinal diffusion coefficient similarly to Taylor dispersion, as they grow as the square of the Péclet number. These numerical artifacts can be eliminated or reduced by a proper space distribution of the free-tunable collision eigenvalue in two-relaxation-time schemes.
NASA Astrophysics Data System (ADS)
Nauenberg, Michael
2004-03-01
In 1916 Einstein introduced the first rules for a quantum theory of electromagnetic radiation and applied them to a model of matter in thermal equilibrium with radiation to derive Planck's black-body formula. Einstein's treatment is extended here to time-dependent stochastic variables, which leads to a master equation for the probability distribution that describes the irreversible approach of his model to thermal equilibrium and elucidates aspects of the foundations of statistical mechanics. An analytic solution of the master equation is obtained in the Fokker-Planck approximation, which is in excellent agreement with numerical results. It is shown that the equilibrium probability distribution is proportional to the total number of microstates for a given configuration, in accordance with Boltzmann's fundamental postulate of equal a priori probabilities. Although the counting of these configurations depends on the particle statistics, the corresponding probability is determined here by the dynamics which are embodied in Einstein's quantum transition probabilities for the emission and absorption of radiation. In a special limit, it is shown that the photons in Einstein's model can act as a thermal bath for the evolution of the atoms toward the canonical equilibrium distribution. In this limit, the present model is mathematically equivalent to an extended version of the Ehrenfests's "dog-flea" model.
Non-equilibrium cation distribution and enhanced spin disorder in hollow CoFe2O4 nanoparticles.
Jaffari, G Hassnain; Ceylan, A; Bui, Holt P; Beebe, Thomas P; Ozcan, S; Shah, S Ismat
2012-08-22
We present magnetic properties of hollow and solid CoFe(2)O(4) nanoparticles that were obtained by annealing of Co(33)Fe(67)/CoFe(2)O(4) (core/shell) nanoparticles. Hollow nanoparticles were polycrystalline whereas the solid nanoparticles were mostly single crystal. Electronic structure studies were performed by photoemission which revealed that particles with hollow morphology have a higher degree of inversion compared to solid nanoparticles and the bulk counterpart. Electronic structure and the magnetic measurements show that particles have uncompensated spins. Quantitative comparison of saturation magnetization (M(S )), assuming bulk Néel type spin structure with cationic distribution, calculated from quantitative XPS analysis, is presented. The thickness of uncompensated spins is calculated to be significantly large for particles with hollow morphology compared to solid nanoparticles. Both morphologies show a lack of saturation up to 7 T. Moreover magnetic irreversibility exists up to 7 T of cooling fields for the entire temperature range (10-300 K). These effects are due to the large bulk anisotropy constant of CoFe(2)O(4) which is the highest among the cubic spinel ferrites. The effect of the uncompensated spins for hollow nanoparticles was investigated by cooling the sample in large fields of up to 9 T. The magnitude of horizontal shift resulting from the unidirectional anisotropy was more than three times larger than that of solid nanoparticles. As an indication signature of uncompensated spin structure, 11% vertical shift for hollow nanoparticles is observed, whereas solid nanoparticles do not show a similar shift. Deconvolution of the hysteresis response recorded at 300 K reveals the presence of a significant paramagnetic component for particles with hollow morphology which further confirms enhanced spin disorder.
NASA Astrophysics Data System (ADS)
Asinari, Pietro
2010-10-01
The homogeneous isotropic Boltzmann equation (HIBE) is a fundamental dynamic model for many applications in thermodynamics, econophysics and sociodynamics. Despite recent hardware improvements, the solution of the Boltzmann equation remains extremely challenging from the computational point of view, in particular by deterministic methods (free of stochastic noise). This work aims to improve a deterministic direct method recently proposed [V.V. Aristov, Kluwer Academic Publishers, 2001] for solving the HIBE with a generic collisional kernel and, in particular, for taking care of the late dynamics of the relaxation towards the equilibrium. Essentially (a) the original problem is reformulated in terms of particle kinetic energy (exact particle number and energy conservation during microscopic collisions) and (b) the computation of the relaxation rates is improved by the DVM-like correction, where DVM stands for Discrete Velocity Model (ensuring that the macroscopic conservation laws are exactly satisfied). Both these corrections make possible to derive very accurate reference solutions for this test case. Moreover this work aims to distribute an open-source program (called HOMISBOLTZ), which can be redistributed and/or modified for dealing with different applications, under the terms of the GNU General Public License. The program has been purposely designed in order to be minimal, not only with regards to the reduced number of lines (less than 1000), but also with regards to the coding style (as simple as possible). Program summaryProgram title: HOMISBOLTZ Catalogue identifier: AEGN_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGN_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License No. of lines in distributed program, including test data, etc.: 23 340 No. of bytes in distributed program, including test data, etc.: 7 635 236 Distribution format: tar
Relativity, nonextensivity, and extended power law distributions.
Silva, R; Lima, J A S
2005-11-01
A proof of the relativistic theorem by including nonextensive effects is given. As it happens in the nonrelativistic limit, the molecular chaos hypothesis advanced by Boltzmann does not remain valid, and the second law of thermodynamics combined with a duality transformation implies that the parameter lies on the interval [0,2]. It is also proven that the collisional equilibrium states (null entropy source term) are described by the relativistic power law extension of the exponential Juttner distribution which reduces, in the nonrelativistic domain, to the Tsallis power law function. As a simple illustration of the basic approach, we derive the relativistic nonextensive equilibrium distribution for a dilute charged gas under the action of an electromagnetic field . Such results reduce to the standard ones in the extensive limit, thereby showing that the nonextensive entropic framework can be harmonized with the space-time ideas contained in the special relativity theory.
A tightly coupled non-equilibrium model for inductively coupled radio-frequency plasmas
Munafò, A. Alfuhaid, S. A. Panesi, M.; Cambier, J.-L.
2015-10-07
The objective of the present work is the development of a tightly coupled magneto-hydrodynamic model for inductively coupled radio-frequency plasmas. Non Local Thermodynamic Equilibrium (NLTE) effects are described based on a hybrid State-to-State approach. A multi-temperature formulation is used to account for thermal non-equilibrium between translation of heavy-particles and vibration of molecules. Excited electronic states of atoms are instead treated as separate pseudo-species, allowing for non-Boltzmann distributions of their populations. Free-electrons are assumed Maxwellian at their own temperature. The governing equations for the electro-magnetic field and the gas properties (e.g., chemical composition and temperatures) are written as a coupled system of time-dependent conservation laws. Steady-state solutions are obtained by means of an implicit Finite Volume method. The results obtained in both LTE and NLTE conditions over a broad spectrum of operating conditions demonstrate the robustness of the proposed coupled numerical method. The analysis of chemical composition and temperature distributions along the torch radius shows that: (i) the use of the LTE assumption may lead to an inaccurate prediction of the thermo-chemical state of the gas, and (ii) non-equilibrium phenomena play a significant role close the walls, due to the combined effects of Ohmic heating and macroscopic gradients.
Mamonov, Artem B.; Bhatt, Divesh; Cashman, Derek J.; Ding, Ying; Zuckerman, Daniel M.
2009-01-01
We introduce “library based Monte Carlo” (LBMC) simulation, which performs Boltzmann sampling of molecular systems based on pre-calculated statistical libraries of molecular-fragment configurations, energies, and interactions. The library for each fragment can be Boltzmann distributed and thus account for all correlations internal to the fragment. LBMC can be applied to both atomistic and coarse-grained models, as we demonstrate in this “proof of principle” report. We first verify the approach in a toy model and in implicitly solvated poly-alanine systems. We next study five proteins, up to 309 residues in size. Based on atomistic equilibrium libraries of peptide-plane configurations, the proteins are modeled with fully atomistic backbones and simplified Gō-like interactions among residues. We show that full equilibrium sampling can be obtained in days to weeks on a single processor, suggesting that more accurate models are well within reach. For the future, LBMC provides a convenient platform for constructing adjustable or mixed-resolution models: the configurations of all atoms can be stored at no run-time cost, while an arbitrary subset of interactions is “turned on.” PMID:19594147
How good is the Lattice Boltzmann method?
NASA Astrophysics Data System (ADS)
Kocheemoolayil, Joseph; Barad, Michael; Kiris, Cetin
2016-11-01
Conflicting opinions exist in literature regarding how efficient the lattice Boltzmann method is relative to high-order finite difference approximations of the Navier-Stokes equations on Cartesian meshes, especially at high Mach numbers. We address the question from the pragmatic viewpoint of a practitioner. Dispersion, dissipation and aliasing errors of various lattice Boltzmann models are systematically quantified. The number of floating point operations and memory required for a desired accuracy level are carefully compared for the two numerical methods. Turbulent kinetic energy budgets for several standard test cases such as the decaying Taylor-Green vortex problem are used to evaluate how effective the stabilization mechanisms necessary for lattice Boltzmann method at high Reynolds numbers are. Detailed comments regarding the cyclomatic complexity of the underlying software, scalability of the underlying algorithm on state-of-the-art high-performance computing platforms and wall clock times and relative accuracy for selected simulations conducted using the two approaches are also made.
Computing association probabilities using parallel Boltzmann machines.
Iltis, R A; Ting, P Y
1993-01-01
A new computational method is presented for solving the data association problem using parallel Boltzmann machines. It is shown that the association probabilities can be computed with arbitrarily small errors if a sufficient number of parallel Boltzmann machines are available. The probability beta(i)(j) that the i th measurement emanated from the jth target can be obtained simply by observing the relative frequency with which neuron v(i,j) in a two-dimensional network is on throughout the layers. Some simple tracking examples comparing the performance of the Boltzmann algorithm to the exact data association solution and with the performance of an alternative parallel method using the Hopfield neural network are also presented.
Fast Lattice Boltzmann Solver for Relativistic Hydrodynamics
Mendoza, M.; Herrmann, H. J.; Boghosian, B. M.; Succi, S.
2010-07-02
A lattice Boltzmann formulation for relativistic fluids is presented and numerically validated through quantitative comparison with recent hydrodynamic simulations of relativistic fluids. In order to illustrate its capability to handle complex geometries, the scheme is also applied to the case of a three-dimensional relativistic shock wave, generated by a supernova explosion, impacting on a massive interstellar cloud. This formulation opens up the possibility of exporting the proven advantages of lattice Boltzmann methods, namely, computational efficiency and easy handling of complex geometries, to the context of (mildly) relativistic fluid dynamics at large, from quark-gluon plasmas up to supernovae with relativistic outflows.
Analytic solutions of the relativistic Boltzmann equation
NASA Astrophysics Data System (ADS)
Hatta, Yoshitaka; Martinez, Mauricio; Xiao, Bo-Wen
2015-04-01
We present new analytic solutions to the relativistic Boltzmann equation within the relaxation time approximation. We first obtain spherically expanding solutions which are the kinetic counterparts of the exact solutions of the Israel-Stewart equation in the literature. This allows us to compare the solutions of the kinetic and hydrodynamic equations at an analytical level. We then derive a novel boost-invariant solution of the Boltzmann equation which has an unconventional dependence on the proper time. The existence of such a solution is also suggested in second-order hydrodynamics and fluid-gravity correspondence.
Simulation of Blood Flow at Vessel Bifurcation by Lattice Boltzmann Method
NASA Astrophysics Data System (ADS)
Kang, Xiu-Ying; Liu, Da-He; Zhou, Jing; Jin, Yong-Juan
2005-11-01
The application of the lattice Boltzmann method to the large vessel bifurcation blood flow is investigated in a wide range of Reynolds numbers. The velocity, shear stress and pressure distributions at the bifurcation are presented in detail. The flow separation zones revealed with increase of Reynolds number are located in the areas of the daughter branches distal to the outer corners of the bifurcation where some deposition of particular blood components might occur to form arteriosclerosis. The results also demonstrate that the lattice Boltzmann method is adaptive to simulating the flow in larger vessels under a high Reynolds number.
iAPBS: a programming interface to Adaptive Poisson-Boltzmann Solver
Konecny, Robert; Baker, Nathan A.; McCammon, J. A.
2012-07-26
The Adaptive Poisson-Boltzmann Solver (APBS) is a state-of-the-art suite for performing Poisson-Boltzmann electrostatic calculations on biomolecules. The iAPBS package provides a modular programmatic interface to the APBS library of electrostatic calculation routines. The iAPBS interface library can be linked with a Fortran or C/C++ program thus making all of the APBS functionality available from within the application. Several application modules for popular molecular dynamics simulation packages -- Amber, NAMD and CHARMM are distributed with iAPBS allowing users of these packages to perform implicit solvent electrostatic calculations with APBS.
The solution of the relaxation problem for the Boltzmann equation by the integral iteration method
NASA Technical Reports Server (NTRS)
Limar, Y. F.
1972-01-01
The Boltzmann equation is considered in terms of the problem of relaxation of some initial distribution function which depends only on velocities, to Maxwell's distribution function. The Boltzmann equation is given for the relaxation problem in which the distribution function f(t, u, v) is time dependent and is also dependent on two other variables u and v (the velocities of rigid spherical molecules). An iteration process is discussed in which the velocity space u, v is subdivided into squares, the distribution function in each square being approximated by the second-order surface from the values of the distribution function at nine points. The set of all of these points forms a network of u, v values at the nodes of which the distribution function can be found.
NASA Astrophysics Data System (ADS)
Hong, Y.; Zhang, K.; Gourley, J. J.
2015-12-01
Floods and landslides account for the large number of natural hazards and affect more people than many other types of natural disasters around the world. This study proposed a coupled hydrological-geotechnical model iCRESLIDE (Integration of Coupled Routing and Excess Storage and SLope-Infiltration-Distributed Equilibrium). The iCRESLIDE is designed to remedy the discrepancy of the original landslide model (SLIDE) by coupling with a hydrological model (CREST) and building an integrated system for predicting cascading storm-flood-landslides using remote sensing and geospatial datasets. This coupled system is implemented and evaluated in Macon County, North Carolina, where Hurricane Ivan triggered widespread landslides in September 2004 during the hurricane season. Model simulations from iCRESLIDE show its reliability to predict landslides occurrence (location and time). Receiver Operating Characteristic (ROC) analysis demonstrate that the iCRESLIDE has higher global accuracy (0.750) and higher sensitivity (11.36%) compared to the original SLIDE model. Such improved predictive performance demonstrates the advantage of coupling hydrological-geotechnical models, which calls more attentions and deserves further investigations in order to develop a not only geotechnical sound but also hydrological sensitive system for landslides early warning at regional scale. This talk will also present early results of the NFL (National-Flash-Landslide) Monitoring and Prediction system under development at the NOAA/OU National Weather Center.
NASA Astrophysics Data System (ADS)
Ahmed, Khaliq; Fӧger, Karl
2017-03-01
The SOFC is well-established as a high-efficiency energy conversion technology with demonstrations of micro-CHP systems delivering 60% net electrical efficiency [1]. However, there are key challenges in the path to commercialization. Foremost among them is stack durability. Operating at high temperatures, the SOFC invariably suffers from thermally induced material degradation. This is compounded by thermal stresses within the SOFC stack which are generated from a number of interacting factors. Modelling is used as a tool for predicting undesirable temperature and current density gradients. For an internal reforming SOFC, fidelity of the model is strongly linked to the representation of the fuel reforming reactions, which dictate species concentrations and net heat release. It is critical for simulation of these profiles that the set of reaction rate expressions applicable for the particular anode catalyst are chosen in the model. A relatively wide spectrum of kinetic correlations has been reported in the literature. This work presents a comparative analysis of the internal distribution of temperature, current, voltage and compositions on a SOFC anode, using various combinations of reaction kinetics and equilibrium expressions for the reactions. The results highlight the significance of the fuel reforming chemistry and kinetics in the prediction of cell performance.
Connection Between the Lattice Boltzmann Equation and the Beam Scheme
NASA Technical Reports Server (NTRS)
Xu, Kun; Luo, Li-Shi
1999-01-01
In this paper we analyze and compare the lattice Boltzmann equation with the beam scheme in details. We notice the similarity and differences between the lattice Boltzmann equation and the beam scheme. We show that the accuracy of the lattice Boltzmann equation is indeed second order in space. We discuss the advantages and limitations of lattice Boltzmann equation and the beam scheme. Based on our analysis, we propose an improved multi-dimensional beam scheme.
The Influence of Trapped Ions and Non-equilibrium EDF on Dust Particle Charging
Sukhinin, G. I.; Fedoseev, A. V.; Antipov, S. N.; Petrov, O. F.; Fortov, V. E.
2008-09-07
Dust particles charging in a low-pressure glow discharge was investigated theoretically with the help of model for trapped and free ions coupled with the self-consistent solution of Poisson equation for electric potential. Non-equilibrium (non-Maxwellian) character of electron energy distribution function depending on gas pressure and electric field was also taken into account on the basis of the solution of kinetic Boltzmann equation. The results were compared with the experimental measurements of dust particle charge depending on gas pressure. It was shown that the calculated effective charge, i.e. the difference of the dust particle charge and trapped ion charge, is in a fairly good agreement with the experimental data.
Master equation for a chemical wave front with perturbation of local equilibrium.
Dziekan, P; Lemarchand, A; Nowakowski, B
2011-08-28
In order to develop a stochastic description of gaseous reaction-diffusion systems, which includes a reaction-induced departure from local equilibrium, we derive a modified expression of the master equation from analytical calculations based on the Boltzmann equation. We apply the method to a chemical wave front of Fisher-Kolmogorov-Petrovsky-Piskunov type, whose propagation speed is known to be sensitive to small perturbations. The results of the modified master equation are compared successfully with microscopic simulations of the particle dynamics using the direct simulation Monte Carlo method. The modified master equation constitutes an efficient tool at the mesoscopic scale, which incorporates the nonequilibrium effect without need of determining the particle velocity distribution function.
An improved lattice Boltzmann scheme for multiphase fluid with multi-range interactions
Maquignon, Nicolas; Duchateau, Julien; Roussel, Gilles; Rousselle, François; Renaud, Christophe
2014-10-06
Modeling of fluids with liquid to gas phase transition has become important for understanding many environmental or industrial processes. Such simulations need new techniques, because traditional solvers are often limited. The Lattice Boltzmann Model (LBM) allows simulate complex fluids, because its mesoscopic nature gives possibility to incorporate additional physics in comparison to usual methods. In this work, an improved lattice Boltzmann model for phase transition flow will be introduced. First, the state of art for Shan and Chen (SC) type of LBM will be reminded. Then, link to real thermodynamics will be established with Maxwell equal areas construction. Convergence to isothermal liquid vapor equilibrium will be shown and discussed. Inclusion of an equation of state for real fluid and better incorporation of force term is presented. Multi-range interactions have been used for SC model, but it hasn't been yet applied to real fluid with non-ideal equation of state. In this work, we evaluate this model when it is applied to real liquid-vapor equilibrium. We show that important differences are found for evaluation of gas density. In order to recover thermodynamic consistency, we use a new scheme for calculation of force term, which is a combination of multi range model and numerical weighting used by Gong and Cheng. We show the superiority of our new model by studying convergence to equilibrium values over a large temperature range. We prove that spurious velocities remaining at equilibrium are decreased.
An improved lattice Boltzmann scheme for multiphase fluid with multi-range interactions
NASA Astrophysics Data System (ADS)
Maquignon, Nicolas; Duchateau, Julien; Roussel, Gilles; Rousselle, François; Renaud, Christophe
2014-10-01
Modeling of fluids with liquid to gas phase transition has become important for understanding many environmental or industrial processes. Such simulations need new techniques, because traditional solvers are often limited. The Lattice Boltzmann Model (LBM) allows simulate complex fluids, because its mesoscopic nature gives possibility to incorporate additional physics in comparison to usual methods. In this work, an improved lattice Boltzmann model for phase transition flow will be introduced. First, the state of art for Shan & Chen [1] [2] (SC) type of LBM will be reminded. Then, link to real thermodynamics will be established with Maxwell equal areas construction. Convergence to isothermal liquid vapor equilibrium will be shown and discussed. Inclusion of an equation of state for real fluid and better incorporation of force term is presented [4] [5]. Multi-range interactions have been used for SC model [8], but it hasn't been yet applied to real fluid with non-ideal equation of state. In this work, we evaluate this model when it is applied to real liquid-vapor equilibrium. We show that important differences are found for evaluation of gas density. In order to recover thermodynamic consistency, we use a new scheme for calculation of force term, which is a combination of multi range model and numerical weighting used by Gong & Cheng [6] [7]. We show the superiority of our new model by studying convergence to equilibrium values over a large temperature range. We prove that spurious velocities remaining at equilibrium are decreased.
NASA Technical Reports Server (NTRS)
Chau, Jessica Furrer; Or, Dani; Sukop, Michael C.; Steinberg, S. L. (Principal Investigator)
2005-01-01
Liquid distributions in unsaturated porous media under different gravitational accelerations and corresponding macroscopic gaseous diffusion coefficients were investigated to enhance understanding of plant growth conditions in microgravity. We used a single-component, multiphase lattice Boltzmann code to simulate liquid configurations in two-dimensional porous media at varying water contents for different gravity conditions and measured gas diffusion through the media using a multicomponent lattice Boltzmann code. The relative diffusion coefficients (D rel) for simulations with and without gravity as functions of air-filled porosity were in good agreement with measured data and established models. We found significant differences in liquid configuration in porous media, leading to reductions in D rel of up to 25% under zero gravity. The study highlights potential applications of the lattice Boltzmann method for rapid and cost-effective evaluation of alternative plant growth media designs under variable gravity.
NASA Astrophysics Data System (ADS)
Chau, Jessica Furrer; Or, Dani; Sukop, Michael C.
2005-08-01
Liquid distributions in unsaturated porous media under different gravitational accelerations and corresponding macroscopic gaseous diffusion coefficients were investigated to enhance understanding of plant growth conditions in microgravity. We used a single-component, multiphase lattice Boltzmann code to simulate liquid configurations in two-dimensional porous media at varying water contents for different gravity conditions and measured gas diffusion through the media using a multicomponent lattice Boltzmann code. The relative diffusion coefficients (Drel) for simulations with and without gravity as functions of air-filled porosity were in good agreement with measured data and established models. We found significant differences in liquid configuration in porous media, leading to reductions in Drel of up to 25% under zero gravity. The study highlights potential applications of the lattice Boltzmann method for rapid and cost-effective evaluation of alternative plant growth media designs under variable gravity.
Phase segregation via Vlasov-Boltzmann particle dynamics
Bastea, S
1999-01-19
background, Vlasov potential. If the repulsive potential between the two species is sufficiently weak and long ranged (so no new inter-particle correlations are introduced), such an algorithm contains the essential ingredients of the Vlasov-Boltzmann kinetics. The structure of the interface separating the two phases coexisting inside the miscibility gap is related to the dominating coarsening mechanism. We compared the equilibrium interface profiles that result directly from the Vlasov-Boltzmann equations with the profiles obtained in simulations and found very good agreement. Our model and computational scheme provide a convenient framework for the study of another important problem, the influence of phase segrega- tion on an initially prescribed hydrodynamical flow. The approach to phase segregation kinetics described here takes advan- tage of an important analytical tool available in nonequilibrium physics, the Boltzmann equation, and has a computational simplicity that should make it useful for other interesting applications.
Lattice Boltzmann equation method for multiple immiscible continuum fluids.
Spencer, T J; Halliday, I; Care, C M
2010-12-01
This paper generalizes the two-component algorithm of Sec. , extending it, in Sec. , to describe N>2 mutually immiscible fluids in the isothermal continuum regime. Each fluid has an independent interfacial tension. While retaining all its computational advantages, we remove entirely the empiricism associated with contact behavior in our previous multiple immiscible fluid models [M. M. Dupin, Phys. Rev. E 73, 055701(R) (2006); Med. Eng. Phys. 28, 13 (2006)] while solidifying the physical foundations. Moreover, the model relies upon a fluid-fluid segregation which is simpler, computationally faster, more free of artifacts (i.e., the interfacial microcurrent), and upon an interface-inducing force distribution which is analytic. The method is completely symmetric between any numbers of immiscible fluids and stable over a wide range of directly input interfacial tension. We present data on the steady-state properties of multiple interface model, which are in good agreement with theory [R. E. Johnson and S. S. Sadhal, Annu. Rev. Fluid Mech. 17, 289 (1985)], specifically on the shapes of multidrop systems. Section is an analysis of the kinetic and continuum-scale descriptions of the underlying two-component lattice Boltzmann model for immiscible fluids, extendable to more than two immiscible fluids. This extension requires (i) the use of a more local kinetic equation perturbation which is (ii) free from a reliance on measured interfacial curvature. It should be noted that viewed simply as a two-component method, the continuum algorithm is inferior to our previous methods, reported by Lishchuk [Phys. Rev. E 67, 036701 (2003)] and Halliday [Phys. Rev. E 76, 026708 (2007)]. Greater stability and parameter range is achieved in multiple drop simulations by using the forced multi-relaxation-time lattice Boltzmann method developed, along with (for completeness) a forced exactly incompressible Bhatnagar-Gross-Krook lattice Boltzmann model, in the Appendix. These appended schemes
From Newton's Law to the Linear Boltzmann Equation Without Cut-Off
NASA Astrophysics Data System (ADS)
Ayi, Nathalie
2017-03-01
We provide a rigorous derivation of the linear Boltzmann equation without cut-off starting from a system of particles interacting via a potential with infinite range as the number of particles N goes to infinity under the Boltzmann-Grad scaling. More particularly, we will describe the motion of a tagged particle in a gas close to global equilibrium. The main difficulty in our context is that, due to the infinite range of the potential, a non-integrable singularity appears in the angular collision kernel, making no longer valid the single-use of Lanford's strategy. Our proof relies then on a combination of Lanford's strategy, of tools developed recently by Bodineau, Gallagher and Saint-Raymond to study the collision process, and of new duality arguments to study the additional terms associated with the long-range interaction, leading to some explicit weak estimates.
From Newton's Law to the Linear Boltzmann Equation Without Cut-Off
NASA Astrophysics Data System (ADS)
Ayi, Nathalie
2017-01-01
We provide a rigorous derivation of the linear Boltzmann equation without cut-off starting from a system of particles interacting via a potential with infinite range as the number of particles N goes to infinity under the Boltzmann-Grad scaling. More particularly, we will describe the motion of a tagged particle in a gas close to global equilibrium. The main difficulty in our context is that, due to the infinite range of the potential, a non-integrable singularity appears in the angular collision kernel, making no longer valid the single-use of Lanford's strategy. Our proof relies then on a combination of Lanford's strategy, of tools developed recently by Bodineau, Gallagher and Saint-Raymond to study the collision process, and of new duality arguments to study the additional terms associated with the long-range interaction, leading to some explicit weak estimates.
Generalized Boltzmann formalism for oscillating neutrinos
Strack, P.; Burrows, A.
2005-05-01
In the standard approaches to neutrino transport in the simulation of core-collapse supernovas, one will often start from the classical Boltzmann equation for the neutrino's spatial, temporal, and spectral evolution. For each neutrino species, and its antiparticle, the classical density in phase space, or the associated specific intensity, will be calculated as a function of time. The neutrino radiation is coupled to matter by source and sink terms on the 'right-hand side' of the transport equation and together with the equations of hydrodynamics this set of coupled partial differential equations for classical densities describes, in principle, the evolution of core collapse and explosion. However, with the possibility of neutrino oscillations between species, a purely quantum-physical effect, how to generalize this set of Boltzmann equations for classical quantities to reflect oscillation physics has not been clear. To date, the formalisms developed have retained the character of quantum operator physics involving complex quantities and have not been suitable for easy incorporation into standard supernova codes. In this paper, we derive generalized Boltzmann equations for quasiclassical, real-valued phase-space densities that retain all the standard oscillation phenomenology, including the matter-enhanced resonant flavor conversion (Mikheev-Smirnov-Wolfenstein effect), neutrino self-interactions, and the interplay between decohering matter coupling and flavor oscillations. With this formalism, any code(s) that can now handle the solution of the classical Boltzmann or transport equation can easily be generalized to include neutrino oscillations in a quantum-physically consistent fashion.
Entropic lattice Boltzmann model for Burgers's equation.
Boghosian, Bruce M; Love, Peter; Yepez, Jeffrey
2004-08-15
Entropic lattice Boltzmann models are discrete-velocity models of hydrodynamics that possess a Lyapunov function. This feature makes them useful as nonlinearly stable numerical methods for integrating hydrodynamic equations. Over the last few years, such models have been successfully developed for the Navier-Stokes equations in two and three dimensions, and have been proposed as a new category of subgrid model of turbulence. In the present work we develop an entropic lattice Boltzmann model for Burgers's equation in one spatial dimension. In addition to its pedagogical value as a simple example of such a model, our result is actually a very effective way to simulate Burgers's equation in one dimension. At moderate to high values of viscosity, we confirm that it exhibits no trace of instability. At very small values of viscosity, however, we report the existence of oscillations of bounded amplitude in the vicinity of the shock, where gradient scale lengths become comparable with the grid size. As the viscosity decreases, the amplitude at which these oscillations saturate tends to increase. This indicates that, in spite of their nonlinear stability, entropic lattice Boltzmann models may become inaccurate when the ratio of gradient scale length to grid spacing becomes too small. Similar inaccuracies may limit the utility of the entropic lattice Boltzmann paradigm as a subgrid model of Navier-Stokes turbulence.
An Updated Equilibrium Machine
NASA Astrophysics Data System (ADS)
Schultz, Emeric
2008-08-01
A device that can demonstrate equilibrium, kinetic, and thermodynamic concepts is described. The device consists of a leaf blower attached to a plastic container divided into two chambers by a barrier of variable size and form. Styrofoam balls can be exchanged across the barrier when the leaf blower is turned on and various air pressures are applied. Equilibrium can be approached from different distributions of balls in the container under different conditions. The Le Châtelier principle can be demonstrated. Kinetic concepts can be demonstrated by changing the nature of the barrier, either changing the height or by having various sized holes in the barrier. Thermodynamic concepts can be demonstrated by taping over some or all of the openings and restricting air flow into container on either side of the barrier.
Chen, Yunjie; Roux, Benoît
2014-09-21
Hybrid schemes combining the strength of molecular dynamics (MD) and Metropolis Monte Carlo (MC) offer a promising avenue to improve the sampling efficiency of computer simulations of complex systems. A number of recently proposed hybrid methods consider new configurations generated by driving the system via a non-equilibrium MD (neMD) trajectory, which are subsequently treated as putative candidates for Metropolis MC acceptance or rejection. To obey microscopic detailed balance, it is necessary to alter the momentum of the system at the beginning and/or the end of the neMD trajectory. This strict rule then guarantees that the random walk in configurational space generated by such hybrid neMD-MC algorithm will yield the proper equilibrium Boltzmann distribution. While a number of different constructs are possible, the most commonly used prescription has been to simply reverse the momenta of all the particles at the end of the neMD trajectory ("one-end momentum reversal"). Surprisingly, it is shown here that the choice of momentum reversal prescription can have a considerable effect on the rate of convergence of the hybrid neMD-MC algorithm, with the simple one-end momentum reversal encountering particularly acute problems. In these neMD-MC simulations, different regions of configurational space end up being essentially isolated from one another due to a very small transition rate between regions. In the worst-case scenario, it is almost as if the configurational space does not constitute a single communicating class that can be sampled efficiently by the algorithm, and extremely long neMD-MC simulations are needed to obtain proper equilibrium probability distributions. To address this issue, a novel momentum reversal prescription, symmetrized with respect to both the beginning and the end of the neMD trajectory ("symmetric two-ends momentum reversal"), is introduced. Illustrative simulations demonstrate that the hybrid neMD-MC algorithm robustly yields a correct
Mixed quantum-classical equilibrium in global flux surface hopping
Sifain, Andrew E.; Wang, Linjun; Prezhdo, Oleg V.
2015-06-14
Global flux surface hopping (GFSH) generalizes fewest switches surface hopping (FSSH)—one of the most popular approaches to nonadiabatic molecular dynamics—for processes exhibiting superexchange. We show that GFSH satisfies detailed balance and leads to thermodynamic equilibrium with accuracy similar to FSSH. This feature is particularly important when studying electron-vibrational relaxation and phonon-assisted transport. By studying the dynamics in a three-level quantum system coupled to a classical atom in contact with a classical bath, we demonstrate that both FSSH and GFSH achieve the Boltzmann state populations. Thermal equilibrium is attained significantly faster with GFSH, since it accurately represents the superexchange process. GFSH converges closer to the Boltzmann averages than FSSH and exhibits significantly smaller statistical errors.
Nanoscale air bearing modeling via lattice Boltzmann method
NASA Astrophysics Data System (ADS)
Kim, Woo Tae; Jhon, Myung S.; Zhou, Yong; Staroselsky, Ilya; Chen, Hudong
2005-05-01
As spacing between the two solid surfaces in operating condition becomes much smaller than the mean free path of the air, continuum-based Navier-Stokes equation is no longer valid and one has to use a modified Reynolds equation (MRE) in simulating high Knudsen number air bearing. This MRE, which stems from the linearized Boltzmann transport equation with Bhatnagar-Gross-Krook approximation via the appropriate choice of the boundary condition, has the advantages of calculating the pressure distribution in a nanoscale confined gaseous system. In this paper, we provide a methodology based on the lattice Boltzmann method (LBM), which could enhance the computational capability of nanoscale confined gaseous system by calculating both velocity and pressure fields simultaneously. The advantage of transient and parallel nature makes this LBM an attractive tool for the next generation air bearing design. Furthermore, LBM is suitable for hybridization with lubricant morphology as well as multiscale modeling including entire disk flow analysis. We demonstrate the feasibility of this LBM by using first-order slip model as a case study. Hybridization with database established by Kang et al. [S.-C. Kang, R. M. Crone, and M. S. Jhon, J. Appl. Phys. 85, 5594 (1999)] can be performed via the similar procedure reported here to develop the state-of-the-art slider design software.
Application of Lattice Boltzmann Methods in Complex Mass Transfer Systems
NASA Astrophysics Data System (ADS)
Sun, Ning
Lattice Boltzmann Method (LBM) is a novel computational fluid dynamics method that can easily handle complex and dynamic boundaries, couple local or interfacial interactions/reactions, and be easily parallelized allowing for simulation of large systems. While most of the current studies in LBM mainly focus on fluid dynamics, however, the inherent power of this method makes it an ideal candidate for the study of mass transfer systems involving complex/dynamic microstructures and local reactions. In this thesis, LBM is introduced to be an alternative computational method for the study of electrochemical energy storage systems (Li-ion batteries (LIBs) and electric double layer capacitors (EDLCs)) and transdermal drug design on mesoscopic scale. Based on traditional LBM, the following in-depth studies have been carried out: (1) For EDLCs, the simulation of diffuse charge dynamics is carried out for both the charge and the discharge processes on 2D systems of complex random electrode geometries (pure random, random spheres and random fibers). Steric effect of concentrated solutions is considered by using modified Poisson-Nernst-Plank (MPNP) equations and compared with regular Poisson-Nernst-Plank (PNP) systems. The effects of electrode microstructures (electrode density, electrode filler morphology, filler size, etc.) on the net charge distribution and charge/discharge time are studied in detail. The influence of applied potential during discharging process is also discussed. (2) For the study of dendrite formation on the anode of LIBs, it is shown that the Lattice Boltzmann model can capture all the experimentally observed features of microstructure evolution at the anode, from smooth to mossy to dendritic. The mechanism of dendrite formation process in mesoscopic scale is discussed in detail and compared with the traditional Sand's time theories. It shows that dendrite formation is closely related to the inhomogeneous reactively at the electrode-electrolyte interface
Numerical investigations of low-density nozzle flow by solving the Boltzmann equation
NASA Technical Reports Server (NTRS)
Deng, Zheng-Tao; Liaw, Goang-Shin; Chou, Lynn Chen
1995-01-01
A two-dimensional finite-difference code to solve the BGK-Boltzmann equation has been developed. The solution procedure consists of three steps: (1) transforming the BGK-Boltzmann equation into two simultaneous partial differential equations by taking moments of the distribution function with respect to the molecular velocity u(sub z), with weighting factors 1 and u(sub z)(sup 2); (2) solving the transformed equations in the physical space based on the time-marching technique and the four-stage Runge-Kutta time integration, for a given discrete-ordinate. The Roe's second-order upwind difference scheme is used to discretize the convective terms and the collision terms are treated as source terms; and (3) using the newly calculated distribution functions at each point in the physical space to calculate the macroscopic flow parameters by the modified Gaussian quadrature formula. Repeating steps 2 and 3, the time-marching procedure stops when the convergent criteria is reached. A low-density nozzle flow field has been calculated by this newly developed code. The BGK Boltzmann solution and experimental data show excellent agreement. It demonstrated that numerical solutions of the BGK-Boltzmann equation are ready to be experimentally validated.
Determining equilibrium constants for dimerization reactions from molecular dynamics simulations.
De Jong, Djurre H; Schäfer, Lars V; De Vries, Alex H; Marrink, Siewert J; Berendsen, Herman J C; Grubmüller, Helmut
2011-07-15
With today's available computer power, free energy calculations from equilibrium molecular dynamics simulations "via counting" become feasible for an increasing number of reactions. An example is the dimerization reaction of transmembrane alpha-helices. If an extended simulation of the two helices covers sufficiently many dimerization and dissociation events, their binding free energy is readily derived from the fraction of time during which the two helices are observed in dimeric form. Exactly how the correct value for the free energy is to be calculated, however, is unclear, and indeed several different and contradictory approaches have been used. In particular, results obtained via Boltzmann statistics differ from those determined via the law of mass action. Here, we develop a theory that resolves this discrepancy. We show that for simulation systems containing two molecules, the dimerization free energy is given by a formula of the form ΔG ∝ ln(P(1) /P(0) ). Our theory is also applicable to high concentrations that typically have to be used in molecular dynamics simulations to keep the simulation system small, where the textbook dilute approximations fail. It also covers simulations with an arbitrary number of monomers and dimers and provides rigorous error estimates. Comparison with test simulations of a simple Lennard Jones system with various particle numbers as well as with reference free energy values obtained from radial distribution functions show full agreement for both binding free energies and dimerization statistics.
State densities and ionization equilibrium of atoms in dense plasmas
NASA Astrophysics Data System (ADS)
Shimamura, Isao; Fujimoto, Takashi
1990-08-01
The semiclassical Bohr-Sommerfeld quantization condition is used to derive an approximate analytical expression for the state density of the hydrogen atom in a dense plasma. An ion-sphere model with an infinitely high potential wall is assumed. The expression leads to a universal curve that spans all values of the electron density. The curve is continuous and smooth over the entire energy range, starting from the hydrogenic state density for low-lying bound states and approaching the plane-wave state density in the high-energy limit of the continuum. The number of bound states is approximately proportional to the inverse of the square root of the electron density. Integration of the state density over the Boltzmann distribution of the electronic energy results in an ionization equilibrium relation, leading to modified Saha's equation. The correction factor for this modified equation is a function of both the electron temperature and the electron density, and is expressed as a universal function of the ion coupling parameter.
Lattice Boltzmann model for traffic flow.
Meng, Jianping; Qian, Yuehong; Li, Xingli; Dai, Shiqiang
2008-03-01
Mesoscopic models for traffic flows are usually difficult to be employed because of the appearance of integro-differential terms in the models. In this work, a lattice Boltzmann model for traffic flow is introduced on the basis of the existing kinetics models by using the Bhatnagar-Gross-Krook-type approximation interaction term in the Boltzmann equation and discretizing it in time and phase space. The so-obtained model is simple while the relevant parameters are physically meaningful. Together with its discrete feature, the model can be easily used to investigate numerically the behavior of traffic flows. In consequence, the macroscopic dynamics of the model is derived using the Taylor and Chapman-Enskog expansions. For validating the model, numerical simulations are conducted under the periodic boundary conditions. It is found that the model could reasonably reproduce the fundamental diagram. Moreover, certain interesting physical phenomena can be captured by the model, such as the metastability and stop-and-go phenomena.
Lattice Boltzmann model for wave propagation.
Zhang, Jianying; Yan, Guangwu; Shi, Xiubo
2009-08-01
A lattice Boltzmann model for two-dimensional wave equation is proposed by using the higher-order moment method. The higher-order moment method is based on the solution of a series of partial differential equations obtained by using multiscale technique and Chapman-Enskog expansion. In order to obtain the lattice Boltzmann model for the wave equation with higher-order accuracy of truncation errors, we removed the second-order dissipation term and the third-order dispersion term by employing the moments up to fourth order. The reversibility in time appears owing to the absence of the second-order dissipation term and the third-order dispersion term. As numerical examples, some classical examples, such as interference, diffraction, and wave passing through a convex lens, are simulated. The numerical results show that this model can be used to simulate wave propagation.
The Boltzmann equation in the difference formulation
Szoke, Abraham; Brooks III, Eugene D.
2015-05-06
First we recall the assumptions that are needed for the validity of the Boltzmann equation and for the validity of the compressible Euler equations. We then present the difference formulation of these equations and make a connection with the time-honored Chapman - Enskog expansion. We discuss the hydrodynamic limit and calculate the thermal conductivity of a monatomic gas, using a simplified approximation for the collision term. Our formulation is more consistent and simpler than the traditional derivation.
Nonlocal Boltzmann theory of plasma channels
NASA Astrophysics Data System (ADS)
Yu, S. S.; Melendez, R. E.
1983-01-01
The mathematical framework for the Lawrence Livermore National Lab. (LLNL) code NUTS is developed. This code is designed to study the evolution of an electron beam generated plasma channel at all pressures. The Boltzmann treatment of the secondary electrons presented include all inertial, nonlocal, electric and magnetic effects, as well as effects of atomic collisions. Field equations are advanced simultaneously and self-consistently with the evolving plasma currents.
Lattice Boltzmann model for simulation of magnetohydrodynamics
NASA Technical Reports Server (NTRS)
Chen, Shiyi; Chen, Hudong; Martinez, Daniel; Matthaeus, William
1991-01-01
A numerical method, based on a discrete Boltzmann equation, is presented for solving the equations of magnetohydrodynamics (MHD). The algorithm provides advantages similar to the cellular automaton method in that it is local and easily adapted to parallel computing environments. Because of much lower noise levels and less stringent requirements on lattice size, the method appears to be more competitive with traditional solution methods. Examples show that the model accurately reproduces both linear and nonlinear MHD phenomena.
On boundary conditions in lattice Boltzmann methods
Chen, S.; Martinez, D. |; Mei, R.
1996-09-01
A lattice Boltzmann boundary condition for simulation of fluid flow using simple extrapolation is proposed. Numerical simulations, including two-dimensional Poiseuille flow, unsteady Couette flow, lid-driven square cavity flow, and flow over a column of cylinders for a range of Reynolds numbers, are carried out, showing that this scheme is of second order accuracy in space discretization. Applications of the method to other boundary conditions, including pressure condition and flux condition are discussed. {copyright} {ital 1996 American Institute of Physics.}
Entropic Lattice Boltzmann Models and Quantum Computation
2008-04-01
cellular automata, quantum cellular automata, action principles, periodic orbits, turbulence U U U UL 8 Bruce M. Boghosian (617) 627–3054 Contents 1...thereof . . 6 2.5 Lattice Boltzmann algorithm for periodic unstable orbits . . . . . . . . . . . . . . . . . . . . . 7 3 Personnel Supported 7 3.1 2005...continue to work on it in the remaining period of this grant. There are reasons for optimism in the search for quantum circuits described above. First, if
A lattice-Boltzmann scheme of the Navier-Stokes equations on a 3D cuboid lattice
NASA Astrophysics Data System (ADS)
Min, Haoda; Peng, Cheng; Wang, Lian-Ping
2015-11-01
The standard lattice-Boltzmann method (LBM) for fluid flow simulation is based on a square (in 2D) or cubic (in 3D) lattice grids. Recently, two new lattice Boltzmann schemes have been developed on a 2D rectangular grid using the MRT (multiple-relaxation-time) collision model, by adding a free parameter in the definition of moments or by extending the equilibrium moments. Here we developed a lattice Boltzmann model on 3D cuboid lattice, namely, a lattice grid with different grid lengths in different spatial directions. We designed our MRT-LBM model by matching the moment equations from the Chapman-Enskog expansion with the Navier-Stokes equations. The model guarantees correct hydrodynamics. A second-order term is added to the equilibrium moments in order to restore the isotropy of viscosity on a cuboid lattice. The form and the coefficients of the extended equilibrium moments are determined through an inverse design process. An additional benefit of the model is that the viscosity can be adjusted independent of the stress-moment relaxation parameter, thus improving the numerical stability of the model. The resulting cuboid MRT-LBM model is then validated through benchmark simulations using laminar channel flow, turbulent channel flow, and the 3D Taylor-Green vortex flow.
NASA Astrophysics Data System (ADS)
Feigin, Alexander; Belikovich, Mikhail; Kulikov, Mikhail
2016-04-01
Atomic oxygen and hydrogen are known to be among key components for the photochemistry and energy balance of the Earth's atmosphere between approximately 80 and 100 km altitude (mesopause region). Therefore, obtaining information about the vertical distributions of O and H concentrations is an important task in studies of this region. Solving of this problem is rather difficult due to the absence of regular methods which allow one to direct measurements of distributions of these components in mesosphere. However, indirect methods used to retrieve O and H distributions from the satellite-based measurements of the OH and O2(1D) airglow emission, as well as the data of IR and microwave O3 measurements have a sufficiently long development history. These methods are rooted in the use of the condition of photochemical equilibrium of ozone density in the range of altitudes from 50 to 100 km. A significant factor is that an insufficient volume of such measurement data forces researchers to use approximate ("truncated") photochemical-equilibrium conditions. In particular, it is assumed that in the daytime the ozone production reaction is perfectly balanced by ozone photodissociation, whereas during the night the only ozone sink is the reaction of ozone with atomic hydrogen, which, in its turn, leads to formation of excited OH and airglow emission of the latter. The presentation analyzes applicability of the photochemical-equilibrium conditions both in the total and truncated forms for description of the spatio-temporal evolution of mesospheric ozone during a year. The analysis is based on year-long time series generated by a 3D chemical transport model, which reproduces correctly various types of atmosphere dynamics in the range of altitudes from 50 to 100 km. These data are used to determine statistics of the ratio between the correct (calculated dynamically) distributions of the O3 density and its uncontracted and truncated equilibrium values for the conditions of the
Consistent lattice Boltzmann equations for phase transitions
NASA Astrophysics Data System (ADS)
Siebert, D. N.; Philippi, P. C.; Mattila, K. K.
2014-11-01
Unlike conventional computational fluid dynamics methods, the lattice Boltzmann method (LBM) describes the dynamic behavior of fluids in a mesoscopic scale based on discrete forms of kinetic equations. In this scale, complex macroscopic phenomena like the formation and collapse of interfaces can be naturally described as related to source terms incorporated into the kinetic equations. In this context, a novel athermal lattice Boltzmann scheme for the simulation of phase transition is proposed. The continuous kinetic model obtained from the Liouville equation using the mean-field interaction force approach is shown to be consistent with diffuse interface model using the Helmholtz free energy. Density profiles, interface thickness, and surface tension are analytically derived for a plane liquid-vapor interface. A discrete form of the kinetic equation is then obtained by applying the quadrature method based on prescribed abscissas together with a third-order scheme for the discretization of the streaming or advection term in the Boltzmann equation. Spatial derivatives in the source terms are approximated with high-order schemes. The numerical validation of the method is performed by measuring the speed of sound as well as by retrieving the coexistence curve and the interface density profiles. The appearance of spurious currents near the interface is investigated. The simulations are performed with the equations of state of Van der Waals, Redlich-Kwong, Redlich-Kwong-Soave, Peng-Robinson, and Carnahan-Starling.
Lattice Boltzmann method and channel flow
NASA Astrophysics Data System (ADS)
Stensholt, Sigvat; Mongstad Hope, Sigmund
2016-07-01
Lattice Boltzmann methods are presented at an introductory level with a focus on fairly simple simulations that can be used to test and illustrate the model’s capabilities. Two scenarios are presented. The first is a simple laminar flow in a straight channel driven by a pressure gradient (Poiseuille flow). The second is a more complex, including a wedge where Moffatt vortices may be induced if the wedge is deep enough. Simulations of the Poiseuille flow scenario accurately capture the theoretical velocity profile. The experiment shows the location of the fluid-wall boundary and the effects viscosity has on the velocity and convergence time. The numerical capabilities of the lattice Boltzmann model are tested further by simulating the more complex Moffatt vortex scenario. The method reproduces with high accuracy the theoretical predction that Moffat vortices will not form in a wedge if the vertex angle exceeds 146°. Practical issues limitations of the lattice Boltzmann method are discussed. In particular the accuracy of the bounce-back boundary condition is first order dependent on the grid resolution.
Hybrid lattice Boltzmann method on overlapping grids
NASA Astrophysics Data System (ADS)
Di Ilio, G.; Chiappini, D.; Ubertini, S.; Bella, G.; Succi, S.
2017-01-01
In this work, a hybrid lattice Boltzmann method (HLBM) is proposed, where the standard lattice Boltzmann implementation based on the Bhatnagar-Gross-Krook (LBGK) approximation is combined together with an unstructured finite-volume lattice Boltzmann model. The method is constructed on an overlapping grid system, which allows the coexistence of a uniform lattice nodes spacing and a coordinate-free lattice structure. The natural adaptivity of the hybrid grid system makes the method particularly suitable to handle problems involving complex geometries. Moreover, the provided scheme ensures a high-accuracy solution near walls, given the capability of the unstructured submodel of achieving the desired level of refinement in a very flexible way. For these reasons, the HLBM represents a prospective tool for solving multiscale problems. The proposed method is here applied to the benchmark problem of a two-dimensional flow past a circular cylinder for a wide range of Reynolds numbers and its numerical performances are measured and compared with the standard LBGK ones.
NASA Astrophysics Data System (ADS)
Ba, Yan; Liu, Haihu; Li, Qing; Kang, Qinjun; Sun, Jinju
2016-08-01
In this paper we propose a color-gradient lattice Boltzmann (LB) model for simulating two-phase flows with high density ratio and high Reynolds number. The model applies a multirelaxation-time (MRT) collision operator to enhance the stability of the simulation. A source term, which is derived by the Chapman-Enskog analysis, is added into the MRT LB equation so that the Navier-Stokes equations can be exactly recovered. Also, a form of the equilibrium density distribution function is used to simplify the source term. To validate the proposed model, steady flows of a static droplet and the layered channel flow are first simulated with density ratios up to 1000. Small values of spurious velocities and interfacial tension errors are found in the static droplet test, and improved profiles of velocity are obtained by the present model in simulating channel flows. Then, two cases of unsteady flows, Rayleigh-Taylor instability and droplet splashing on a thin film, are simulated. In the former case, the density ratio of 3 and Reynolds numbers of 256 and 2048 are considered. The interface shapes and spike and bubble positions are in good agreement with the results of previous studies. In the latter case, the droplet spreading radius is found to obey the power law proposed in previous studies for the density ratio of 100 and Reynolds number up to 500.
NASA Astrophysics Data System (ADS)
Huang, Rongzong; Wu, Huiying
2016-06-01
A total enthalpy-based lattice Boltzmann (LB) method with adaptive mesh refinement (AMR) is developed in this paper to efficiently simulate solid-liquid phase change problem where variables vary significantly near the phase interface and thus finer grid is required. For the total enthalpy-based LB method, the velocity field is solved by an incompressible LB model with multiple-relaxation-time (MRT) collision scheme, and the temperature field is solved by a total enthalpy-based MRT LB model with the phase interface effects considered and the deviation term eliminated. With a kinetic assumption that the density distribution function for solid phase is at equilibrium state, a volumetric LB scheme is proposed to accurately realize the nonslip velocity condition on the diffusive phase interface and in the solid phase. As compared with the previous schemes, this scheme can avoid nonphysical flow in the solid phase. As for the AMR approach, it is developed based on multiblock grids. An indicator function is introduced to control the adaptive generation of multiblock grids, which can guarantee the existence of overlap area between adjacent blocks for information exchange. Since MRT collision schemes are used, the information exchange is directly carried out in the moment space. Numerical tests are firstly performed to validate the strict satisfaction of the nonslip velocity condition, and then melting problems in a square cavity with different Prandtl numbers and Rayleigh numbers are simulated, which demonstrate that the present method can handle solid-liquid phase change problem with high efficiency and accuracy.
Equilibrium sampling by reweighting nonequilibrium simulation trajectories.
Yang, Cheng; Wan, Biao; Xu, Shun; Wang, Yanting; Zhou, Xin
2016-03-01
Based on equilibrium molecular simulations, it is usually difficult to efficiently visit the whole conformational space of complex systems, which are separated into some metastable regions by high free energy barriers. Nonequilibrium simulations could enhance transitions among these metastable regions and then be applied to sample equilibrium distributions in complex systems, since the associated nonequilibrium effects can be removed by employing the Jarzynski equality (JE). Here we present such a systematical method, named reweighted nonequilibrium ensemble dynamics (RNED), to efficiently sample equilibrium conformations. The RNED is a combination of the JE and our previous reweighted ensemble dynamics (RED) method. The original JE reproduces equilibrium from lots of nonequilibrium trajectories but requires that the initial distribution of these trajectories is equilibrium. The RED reweights many equilibrium trajectories from an arbitrary initial distribution to get the equilibrium distribution, whereas the RNED has both advantages of the two methods, reproducing equilibrium from lots of nonequilibrium simulation trajectories with an arbitrary initial conformational distribution. We illustrated the application of the RNED in a toy model and in a Lennard-Jones fluid to detect its liquid-solid phase coexistence. The results indicate that the RNED sufficiently extends the application of both the original JE and the RED in equilibrium sampling of complex systems.
Equilibrium sampling by reweighting nonequilibrium simulation trajectories
NASA Astrophysics Data System (ADS)
Yang, Cheng; Wan, Biao; Xu, Shun; Wang, Yanting; Zhou, Xin
2016-03-01
Based on equilibrium molecular simulations, it is usually difficult to efficiently visit the whole conformational space of complex systems, which are separated into some metastable regions by high free energy barriers. Nonequilibrium simulations could enhance transitions among these metastable regions and then be applied to sample equilibrium distributions in complex systems, since the associated nonequilibrium effects can be removed by employing the Jarzynski equality (JE). Here we present such a systematical method, named reweighted nonequilibrium ensemble dynamics (RNED), to efficiently sample equilibrium conformations. The RNED is a combination of the JE and our previous reweighted ensemble dynamics (RED) method. The original JE reproduces equilibrium from lots of nonequilibrium trajectories but requires that the initial distribution of these trajectories is equilibrium. The RED reweights many equilibrium trajectories from an arbitrary initial distribution to get the equilibrium distribution, whereas the RNED has both advantages of the two methods, reproducing equilibrium from lots of nonequilibrium simulation trajectories with an arbitrary initial conformational distribution. We illustrated the application of the RNED in a toy model and in a Lennard-Jones fluid to detect its liquid-solid phase coexistence. The results indicate that the RNED sufficiently extends the application of both the original JE and the RED in equilibrium sampling of complex systems.
Lattice Boltzmann simulations of convection heat transfer in porous media
NASA Astrophysics Data System (ADS)
Liu, Qing; He, Ya-Ling
2017-01-01
A non-orthogonal multiple-relaxation-time (MRT) lattice Boltzmann (LB) method is developed to study convection heat transfer in porous media at the representative elementary volume scale based on the generalized non-Darcy model. In the method, two different LB models are constructed: one is constructed in the framework of the double-distribution-function approach, and the other is constructed in the framework of the hybrid approach. In particular, the transformation matrices used in the MRT-LB models are non-orthogonal matrices. The present method is applied to study mixed convection flow in a porous channel and natural convection flow in a porous cavity. It is found that the numerical results are in good agreement with the analytical solutions and/or other results reported in previous studies. Furthermore, the non-orthogonal MRT-LB method shows better numerical stability in comparison with the BGK-LB method.
Conservation laws and exact solutions of the Boltzmann equation
Mattis, D.C.; Szpilka, A.M.; Chen, H.
1989-03-10
The distribution function f which satisfies the time-dependent Boltzmann equation (BE) for a Lorentz model with perfectly elastic random scatterers is proved nonnegative, and is computed exactly when backscattering dominates. Joule heating and Ohm's law are recovered, although f has no steady-state limit, contrary to the relaxation-time approximation. (The conventional approximation to the time-independent BE also yields OHm's law but not the Joule heating and, worse, it unphysically predicts f < O.) The exact solution is compared with various effective-temperature approximations, and is shown to remain very nearly unchanged over a wide range of times even in the presence of a small amount of inelastic scattering.
Non-Equilibrium Conductivity at Quantum Critical Points
NASA Astrophysics Data System (ADS)
Berridge, Andrew; Bhaseen, M. J.; Green, A. G.
2013-03-01
The behaviour of quantum systems driven out of equilibrium is a field in which we are still searching for general principles and universal results. Quantum critical systems are useful in this search as their out of equilibrium steady states may inherit universal features from equilibrium. While this has been shown in some cases, the calculational techniques used often involve simplified models or calculational tricks, which can obscure some of the underlying physical processes. Here we use a Boltzmann transport approach to study the steady-state non-equilibrium properties - conductivity and current noise, of the Bose-Hubbard model head-on. We must explicitly consider heat-flow and rate limiting processes in the establishment of the steady-state to show that it can indeed be universal. Our analysis reveals the importance of the hydrodynamic limit and the limitations of current approaches.
An efficient annealing in Boltzmann machine in Hopfield neural network
NASA Astrophysics Data System (ADS)
Kin, Teoh Yeong; Hasan, Suzanawati Abu; Bulot, Norhisam; Ismail, Mohammad Hafiz
2012-09-01
This paper proposes and implements Boltzmann machine in Hopfield neural network doing logic programming based on the energy minimization system. The temperature scheduling in Boltzmann machine enhancing the performance of doing logic programming in Hopfield neural network. The finest temperature is determined by observing the ratio of global solution and final hamming distance using computer simulations. The study shows that Boltzmann Machine model is more stable and competent in term of representing and solving difficult combinatory problems.
Deterministic numerical solutions of the Boltzmann equation using the fast spectral method
NASA Astrophysics Data System (ADS)
Wu, Lei; White, Craig; Scanlon, Thomas J.; Reese, Jason M.; Zhang, Yonghao
2013-10-01
The Boltzmann equation describes the dynamics of rarefied gas flows, but the multidimensional nature of its collision operator poses a real challenge for its numerical solution. In this paper, the fast spectral method [36], originally developed by Mouhot and Pareschi for the numerical approximation of the collision operator, is extended to deal with other collision kernels, such as those corresponding to the soft, Lennard-Jones, and rigid attracting potentials. The accuracy of the fast spectral method is checked by comparing our numerical solutions of the space-homogeneous Boltzmann equation with the exact Bobylev-Krook-Wu solutions for a gas of Maxwell molecules. It is found that the accuracy is improved by replacing the trapezoidal rule with Gauss-Legendre quadrature in the calculation of the kernel mode, and the conservation of momentum and energy are ensured by the Lagrangian multiplier method without loss of spectral accuracy. The relax-to-equilibrium processes of different collision kernels with the same value of shear viscosity are then compared; the numerical results indicate that different forms of the collision kernels can be used as long as the shear viscosity (not only the value, but also its temperature dependence) is recovered. An iteration scheme is employed to obtain stationary solutions of the space-inhomogeneous Boltzmann equation, where the numerical errors decay exponentially. Four classical benchmarking problems are investigated: the normal shock wave, and the planar Fourier/Couette/force-driven Poiseuille flows. For normal shock waves, our numerical results are compared with a finite difference solution of the Boltzmann equation for hard sphere molecules, experimental data, and molecular dynamics simulation of argon using the realistic Lennard-Jones potential. For planar Fourier/Couette/force-driven Poiseuille flows, our results are compared with the direct simulation Monte Carlo method. Excellent agreements are observed in all test cases
Numerical Treatment of the Boltzmann Equation for Self-Propelled Particle Systems
NASA Astrophysics Data System (ADS)
Thüroff, Florian; Weber, Christoph A.; Frey, Erwin
2014-10-01
Kinetic theories constitute one of the most promising tools to decipher the characteristic spatiotemporal dynamics in systems of actively propelled particles. In this context, the Boltzmann equation plays a pivotal role, since it provides a natural translation between a particle-level description of the system's dynamics and the corresponding hydrodynamic fields. Yet, the intricate mathematical structure of the Boltzmann equation substantially limits the progress toward a full understanding of this equation by solely analytical means. Here, we propose a general framework to numerically solve the Boltzmann equation for self-propelled particle systems in two spatial dimensions and with arbitrary boundary conditions. We discuss potential applications of this numerical framework to active matter systems and use the algorithm to give a detailed analysis to a model system of self-propelled particles with polar interactions. In accordance with previous studies, we find that spatially homogeneous isotropic and broken-symmetry states populate two distinct regions in parameter space, which are separated by a narrow region of spatially inhomogeneous, density-segregated moving patterns. We find clear evidence that these three regions in parameter space are connected by first-order phase transitions and that the transition between the spatially homogeneous isotropic and polar ordered phases bears striking similarities to liquid-gas phase transitions in equilibrium systems. Within the density-segregated parameter regime, we find a novel stable limit-cycle solution of the Boltzmann equation, which consists of parallel lanes of polar clusters moving in opposite directions, so as to render the overall symmetry of the system's ordered state nematic, despite purely polar interactions on the level of single particles.
Relevance of equilibrium in multifragmentation
Furuta, Takuya; Ono, Akira
2009-01-15
The relevance of equilibrium in a multifragmentation reaction of very central {sup 40}Ca + {sup 40}Ca collisions at 35 MeV/nucleon is investigated by using simulations of antisymmetrized molecular dynamics (AMD). Two types of ensembles are compared. One is the reaction ensemble of the states at each reaction time t in collision events simulated by AMD, and the other is the equilibrium ensemble prepared by solving the AMD equation of motion for a many-nucleon system confined in a container for a long time. The comparison of the ensembles is performed for the fragment charge distribution and the excitation energies. Our calculations show that there exists an equilibrium ensemble that well reproduces the reaction ensemble at each reaction time t for the investigated period 80{<=}t{<=}300 fm/c. However, there are some other observables that show discrepancies between the reaction and equilibrium ensembles. These may be interpreted as dynamical effects in the reaction. The usual static equilibrium at each instant is not realized since any equilibrium ensemble with the same volume as that of the reaction system cannot reproduce the fragment observables.
Lattice Boltzmann equation method for multiple immiscible continuum fluids
NASA Astrophysics Data System (ADS)
Spencer, T. J.; Halliday, I.; Care, C. M.
2010-12-01
This paper generalizes the two-component algorithm of Sec. , extending it, in Sec. , to describe N>2 mutually immiscible fluids in the isothermal continuum regime. Each fluid has an independent interfacial tension. While retaining all its computational advantages, we remove entirely the empiricism associated with contact behavior in our previous multiple immiscible fluid models [M. M. Dupin , Phys. Rev. E 73, 055701(R) (2006)10.1103/PhysRevE.73.055701; Med. Eng. Phys. 28, 13 (2006)10.1016/j.medengphy.2005.04.015] while solidifying the physical foundations. Moreover, the model relies upon a fluid-fluid segregation which is simpler, computationally faster, more free of artifacts (i.e., the interfacial microcurrent), and upon an interface-inducing force distribution which is analytic. The method is completely symmetric between any numbers of immiscible fluids and stable over a wide range of directly input interfacial tension. We present data on the steady-state properties of multiple interface model, which are in good agreement with theory [R. E. Johnson and S. S. Sadhal, Annu. Rev. Fluid Mech. 17, 289 (1985)10.1146/annurev.fl.17.010185.001445], specifically on the shapes of multidrop systems. Section is an analysis of the kinetic and continuum-scale descriptions of the underlying two-component lattice Boltzmann model for immiscible fluids, extendable to more than two immiscible fluids. This extension requires (i) the use of a more local kinetic equation perturbation which is (ii) free from a reliance on measured interfacial curvature. It should be noted that viewed simply as a two-component method, the continuum algorithm is inferior to our previous methods, reported by Lishchuk [Phys. Rev. E 67, 036701 (2003)]10.1103/PhysRevE.76.036701 and Halliday [Phys. Rev. E 76, 026708 (2007)]10.1103/PhysRevE.76.026708. Greater stability and parameter range is achieved in multiple drop simulations by using the forced multi-relaxation-time lattice Boltzmann method developed
Theory of the lattice Boltzmann equation: symmetry properties of discrete velocity sets.
Rubinstein, Robert; Luo, Li-Shi
2008-03-01
The lattice Boltzmann equation replaces continuous particle velocity space by a finite set; the velocity distribution function then varies over a finite-dimensional vector space instead of over an infinite-dimensional function space. The number of linearly independent moments of the distribution function in a lattice Boltzmann model cannot exceed the number of velocities; finite dimensionality therefore necessarily induces linear dependences among the moments that do not exist in a continuous theory. Given a finite velocity set, it is important to know which moments are free of these dependences. Elementary group theory is applied to the solution of this problem. It is found that decomposing the velocity set into subsets that transform among themselves under an appropriate symmetry group makes it straightforward to uncover linear dependences among the moments. The construction of some standard two- and three-dimensional models is reviewed from this viewpoint, and procedures for constructing higher-dimensional models are suggested.
Yoshimura, Kazuyoshi; Kuwabara, Sinzi
2011-05-20
Relaxation phenomena in the binary gas-mixture with different temperature and different velocities are discussed on the basis of two Boltzmann equations. The Hermite expansion method, extended by H.Grad to multidimensional space, is applied to express distribution functions and the Galerkin method is used to solve two Boltzmann equations. Thus, a system of differential equations for the expansion coefficients is obtained. The time development of the system is calculated numerically.
NASA Astrophysics Data System (ADS)
Gao, Y.-Q.; Liu, F.-H.
2016-03-01
The transverse momentum spectra of charged particles produced in Au + Au collisions at the relativistic heavy ion collider and in Pb + Pb collisions at the large hadron collider with different centrality intervals are described by the multisource thermal model which is based on different statistic distributions for a singular source. Each source in the present work is described by the Tsallis distribution and the Boltzmann distribution, respectively. Then, the interacting system is described by the (two-component) Tsallis distribution and the (two-component) Boltzmann distribution, respectively. The results calculated by the two distributions are in agreement with the experimental data of the Solenoidal Tracker At Relativistic heavy ion collider, Pioneering High Energy Nuclear Interaction eXperiment, and A Large Ion Collider Experiment Collaborations. The effective temperature parameters extracted from the two distributions on the descriptions of heavy-ion data at the relativistic heavy ion collider and large hadron collider are obtained to show a linear correlation.
Lattice-Boltzmann-based Simulations of Diffusiophoresis
NASA Astrophysics Data System (ADS)
Castigliego, Joshua; Kreft Pearce, Jennifer
We present results from a lattice-Boltzmann-base Brownian Dynamics simulation on diffusiophoresis and the separation of particles within the system. A gradient in viscosity that simulates a concentration gradient in a dissolved polymer allows us to separate various types of particles by their deformability. As seen in previous experiments, simulated particles that have a higher deformability react differently to the polymer matrix than those with a lower deformability. Therefore, the particles can be separated from each other. This simulation, in particular, was intended to model an oceanic system where the particles of interest were zooplankton, phytoplankton and microplastics. The separation of plankton from the microplastics was achieved.
Lattice Boltzmann methods for moving boundary flows
NASA Astrophysics Data System (ADS)
Inamuro, Takaji
2012-04-01
The lattice Boltzmann methods (LBMs) for moving boundary flows are presented. The LBM for two-phase fluid flows with the same density and the LBM combined with the immersed boundary method are described. In addition, the LBM on a moving multi-block grid is explained. Three numerical examples (a droplet moving in a constricted tube, the lift generation of a flapping wing and the sedimentation of an elliptical cylinder) are shown in order to demonstrate the applicability of the LBMs to moving boundary problems.
Lattice Boltzmann approach to thermal transpiration
Sofonea, Victor
2006-11-15
Diffuse reflection boundary conditions are introduced in a thermal lattice Boltzmann model to allow for variable fluid density and temperature along the walls. The capability of this model to capture the main characteristics of the thermal transpiration phenomenon in a box at nonvanishing Knudsen numbers is demonstrated. The thermal creep velocity is found to be proportional to the temperature gradient imposed at the wall, whereas the accuracy of the simulation results are found to be of first or second order, depending on the numerical scheme.
Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method
NASA Technical Reports Server (NTRS)
Chen, Hudong; Chen, Shiyi; Matthaeus, William H.
1992-01-01
A lattice Boltzmann model is presented which gives the complete Navier-Stokes equation and may provide an efficient parallel numerical method for solving various fluid problems. The model uses the single-time relaxation approximation and a particular Maxwell-type distribution. The model eliminates exactly (1) the non-Galilean invariance caused by a density-dependent coefficient in the convection term and (2) a velocity-dependent equation of state.
Neutrino transport in type II supernovae: Boltzmann solver vs. Monte Carlo method
NASA Astrophysics Data System (ADS)
Yamada, Shoichi; Janka, Hans-Thomas; Suzuki, Hideyuki
1999-04-01
We have coded a Boltzmann solver based on a finite difference scheme (S_N method) aiming at calculations of neutrino transport in type II supernovae. Close comparison between the Boltzmann solver and a Monte Carlo transport code has been made for realistic atmospheres of post bounce core models under the assumption of a static background. We have also investigated in detail the dependence of the results on the numbers of radial, angular, and energy grid points and the way to discretize the spatial advection term which is used in the Boltzmann solver. A general relativistic calculation has been done for one of the models. We find good overall agreement between the two methods. This gives credibility to both methods which are based on completely different formulations. In particular, the number and energy fluxes and the mean energies of the neutrinos show remarkably good agreement, because these quantities are determined in a region where the angular distribution of the neutrinos is nearly isotropic and they are essentially frozen in later on. On the other hand, because of a relatively small number of angular grid points (which is inevitable due to limitations of the computation time) the Boltzmann solver tends to slightly underestimate the flux factor and the Eddington factor outside the (mean) ``neutrinosphere'' where the angular distribution of the neutrinos becomes highly anisotropic. As a result, the neutrino number (and energy) density is somewhat overestimated in this region. This fact suggests that the Boltzmann solver should be applied to calculations of the neutrino heating in the hot-bubble region with some caution because there might be a tendency to overestimate the energy deposition rate in disadvantageous situations. A comparison shows that this trend is opposite to the results obtained with a multi-group flux-limited diffusion approximation of neutrino transport. Employing three different flux limiters, we find that all of them lead to a significant
Equilibrium simulations of proteins using molecular fragment replacement and NMR chemical shifts.
Boomsma, Wouter; Tian, Pengfei; Frellsen, Jes; Ferkinghoff-Borg, Jesper; Hamelryck, Thomas; Lindorff-Larsen, Kresten; Vendruscolo, Michele
2014-09-23
Methods of protein structure determination based on NMR chemical shifts are becoming increasingly common. The most widely used approaches adopt the molecular fragment replacement strategy, in which structural fragments are repeatedly reassembled into different complete conformations in molecular simulations. Although these approaches are effective in generating individual structures consistent with the chemical shift data, they do not enable the sampling of the conformational space of proteins with correct statistical weights. Here, we present a method of molecular fragment replacement that makes it possible to perform equilibrium simulations of proteins, and hence to determine their free energy landscapes. This strategy is based on the encoding of the chemical shift information in a probabilistic model in Markov chain Monte Carlo simulations. First, we demonstrate that with this approach it is possible to fold proteins to their native states starting from extended structures. Second, we show that the method satisfies the detailed balance condition and hence it can be used to carry out an equilibrium sampling from the Boltzmann distribution corresponding to the force field used in the simulations. Third, by comparing the results of simulations carried out with and without chemical shift restraints we describe quantitatively the effects that these restraints have on the free energy landscapes of proteins. Taken together, these results demonstrate that the molecular fragment replacement strategy can be used in combination with chemical shift information to characterize not only the native structures of proteins but also their conformational fluctuations.
Boltzmann's "H"-Theorem and the Assumption of Molecular Chaos
ERIC Educational Resources Information Center
Boozer, A. D.
2011-01-01
We describe a simple dynamical model of a one-dimensional ideal gas and use computer simulations of the model to illustrate two fundamental results of kinetic theory: the Boltzmann transport equation and the Boltzmann "H"-theorem. Although the model is time-reversal invariant, both results predict that the behaviour of the gas is time-asymmetric.…
Boltzmann equations for neutrinos with flavor mixings
NASA Astrophysics Data System (ADS)
Yamada, Shoichi
2000-11-01
With a view of applications to the simulations of supernova explosions and protoneutron star cooling, we derive the Boltzmann equations for the neutrino transport with flavor mixing based on the real time formalism of the nonequilibrium field theory and the gradient expansion of the Green function. The relativistic kinematics is properly taken into account. The advection terms are derived in the mean field approximation for the neutrino self-energy while the collision terms are obtained in the Born approximation. The resulting equations take the familiar form of the Boltzmann equation with corrections due to mixing both in the advection part and in the collision part. These corrections are essentially the same as those derived by Sirera et al. for the advection terms and those by Raffelt et al. for the collision terms, respectively, though the formalism employed here is different from theirs. The derived equations will be easily implemented in numerical codes employed in the simulations of supernova explosions and protoneutron star cooling.
The Poisson-Helmholtz-Boltzmann model.
Bohinc, K; Shrestha, A; May, S
2011-10-01
We present a mean-field model of a one-component electrolyte solution where the mobile ions interact not only via Coulomb interactions but also through a repulsive non-electrostatic Yukawa potential. Our choice of the Yukawa potential represents a simple model for solvent-mediated interactions between ions. We employ a local formulation of the mean-field free energy through the use of two auxiliary potentials, an electrostatic and a non-electrostatic potential. Functional minimization of the mean-field free energy leads to two coupled local differential equations, the Poisson-Boltzmann equation and the Helmholtz-Boltzmann equation. Their boundary conditions account for the sources of both the electrostatic and non-electrostatic interactions on the surface of all macroions that reside in the solution. We analyze a specific example, two like-charged planar surfaces with their mobile counterions forming the electrolyte solution. For this system we calculate the pressure between the two surfaces, and we analyze its dependence on the strength of the Yukawa potential and on the non-electrostatic interactions of the mobile ions with the planar macroion surfaces. In addition, we demonstrate that our mean-field model is consistent with the contact theorem, and we outline its generalization to arbitrary interaction potentials through the use of a Laplace transformation.
Convolution Inequalities for the Boltzmann Collision Operator
NASA Astrophysics Data System (ADS)
Alonso, Ricardo J.; Carneiro, Emanuel; Gamba, Irene M.
2010-09-01
We study integrability properties of a general version of the Boltzmann collision operator for hard and soft potentials in n-dimensions. A reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution, where the weight is given by an operator invariant under rotations. Using a symmetrization technique in L p we prove a Young’s inequality for hard potentials, which is sharp for Maxwell molecules in the L 2 case. Further, we find a new Hardy-Littlewood-Sobolev type of inequality for Boltzmann collision integrals with soft potentials. The same method extends to radially symmetric, non-increasing potentials that lie in some {Ls_{weak}} or L s . The method we use resembles a Brascamp, Lieb and Luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting. Consequently, it is closely connected to the classical analysis of Young and Hardy-Littlewood-Sobolev inequalities. In all cases, the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section (in the spirit of Grad cut-off). As an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions.
Entropic Lattice Boltzmann Methods for Fluid Mechanics
NASA Astrophysics Data System (ADS)
Chikatamarla, Shyam; Boesch, Fabian; Sichau, David; Karlin, Ilya
2013-11-01
With its roots in statistical mechanics and kinetic theory, the lattice Boltzmann method (LBM) is a paradigm-changing innovation, offering for the first time an intrinsically parallel CFD algorithm. Over the past two decades, LBM has achieved numerous results in the field of CFD and is now in a position to challenge state-of-the art CFD techniques. Our major restyling of LBM resulted in an unconditionally stable entropic LBM which restored Second Law (Boltzmann H theorem) in the LBM kinetics and thus enabled affordable direct simulations of fluid turbulence. We review here recent advances in ELBM as a practical, modeling-free tool for simulation of turbulent flows in complex geometries. We shall present recent simulations including turbulent channel flow, flow past a circular cylinder, knotted vortex tubes, and flow past a surface mounted cube. ELBM listed all admissible lattices supporting a discrete entropy function and has classified them in hierarchically increasing order of accuracy. Applications of these higher-order lattices to simulations of turbulence and thermal flows shall also be presented. This work was supported CSCS grant s437.
Magnetospheric equilibrium with anisotropic pressure
Cheng, C.Z.
1991-07-01
Self-consistent magnetospheric equilibrium with anisotropic pressure is obtained by employing an iterative metric method for solving the inverse equilibrium equation in an optimal flux coordinate system. A method of determining plasma parallel and perpendicular pressures from either analytic particle distribution or particle distribution measured along the satellite's path is presented. The numerical results of axisymmetric magnetospheric equilibrium including the effects of finite beta, pressure anisotropy, and boundary conditions are presented for a bi-Maxwellian particle distribution. For the isotropic pressure cases, the finite beta effect produces an outward expansion of the constant magnetic flux surfaces in relation to the dipole field lines, and along the magnetic field the toroidal ring current is maximum at the magnetic equator. The effect of pressure anisotropy is found to further expand the flux surfaces outward. Along the magnetic field lines the westward ring current can be peak away from the equator due to an eastward current contribution resulting from pressure anisotropy. As pressure anisotropy increases, the peak westward current can become more singular. The outer boundary flux surface has significant effect on the magnetospheric equilibrium. For the outer flux boundary resembling dayside compressed flux surface due to solar wind pressure, the deformation of the magnetic field can be quite different from that for the outer flux boundary resembling the tail-like surface. 23 refs., 17 figs.
Monte Carlo variance reduction approaches for non-Boltzmann tallies
Booth, T.E.
1992-12-01
Quantities that depend on the collective effects of groups of particles cannot be obtained from the standard Boltzmann transport equation. Monte Carlo estimates of these quantities are called non-Boltzmann tallies and have become increasingly important recently. Standard Monte Carlo variance reduction techniques were designed for tallies based on individual particles rather than groups of particles. Experience with non-Boltzmann tallies and analog Monte Carlo has demonstrated the severe limitations of analog Monte Carlo for many non-Boltzmann tallies. In fact, many calculations absolutely require variance reduction methods to achieve practical computation times. Three different approaches to variance reduction for non-Boltzmann tallies are described and shown to be unbiased. The advantages and disadvantages of each of the approaches are discussed.
Quantum-statistical equilibrium and the ``law'' of constant Fermi potential
NASA Astrophysics Data System (ADS)
Le Coz, Yannick L.
2003-02-01
We apply the general quantum-statistical density-matrix formalism to an independent-electron gas within a space-dependent external electric potential, under equilibrium conditions. This problem is analogous to an ideal semiconductor homojunction diode. We solve the resulting equilibrium density-matrix equation using a perturbation theory. Next, we derive a first-order quantum correction to the classical Maxwell-Boltzmann density-potential formula. The correction appears as an added curvature term in external potential. It represents expected quantum-mechanical scattering against a spatially varying potential. Our results indicate that the commonly encountered thermodynamic or statistical-mechanical "law" of constant, equilibrium Fermi potential—with Fermi potential a parameter in the Maxwell-Boltzmann density-potential formula—is not fundamentally exact. In a general space-dependent potential, this "law," we prove, is simply a classical approximation.
Munafò, A; Panesi, M; Magin, T E
2014-02-01
A Boltzmann rovibrational collisional coarse-grained model is proposed to reduce a detailed kinetic mechanism database developed at NASA Ames Research Center for internal energy transfer and dissociation in N(2)-N interactions. The coarse-grained model is constructed by lumping the rovibrational energy levels of the N(2) molecule into energy bins. The population of the levels within each bin is assumed to follow a Boltzmann distribution at the local translational temperature. Excitation and dissociation rate coefficients for the energy bins are obtained by averaging the elementary rate coefficients. The energy bins are treated as separate species, thus allowing for non-Boltzmann distributions of their populations. The proposed coarse-grained model is applied to the study of nonequilibrium flows behind normal shock waves and within converging-diverging nozzles. In both cases, the flow is assumed inviscid and steady. Computational results are compared with those obtained by direct solution of the master equation for the rovibrational collisional model and a more conventional multitemperature model. It is found that the proposed coarse-grained model is able to accurately resolve the nonequilibrium dynamics of internal energy excitation and dissociation-recombination processes with only 20 energy bins. Furthermore, the proposed coarse-grained model provides a superior description of the nonequilibrium phenomena occurring in shock heated and nozzle flows when compared with the conventional multitemperature models.
Horsten, N. Baelmans, M.; Dekeyser, W.; Samaey, G.
2016-01-15
We derive fluid neutral approximations for a simplified 1D edge plasma model, suitable to study the neutral behavior close to the target of a nuclear fusion divertor, and compare its solutions to the solution of the corresponding kinetic Boltzmann equation. The plasma is considered as a fixed background extracted from a detached 2D simulation. We show that the Maxwellian equilibrium distribution is already obtained very close to the target, justifying the use of a fluid approximation. We compare three fluid neutral models: (i) a diffusion model; (ii) a pressure-diffusion model (i.e., a combination of a continuity and momentum equation) assuming equal neutral and ion temperatures; and (iii) the pressure-diffusion model coupled to a neutral energy equation taking into account temperature differences between neutrals and ions. Partial reflection of neutrals reaching the boundaries is included in both the kinetic and fluid models. We propose two methods to obtain an incident neutral flux boundary condition for the fluid models: one based on a diffusion approximation and the other assuming a truncated Chapman-Enskog distribution. The pressure-diffusion model predicts the plasma sources very well. The diffusion boundary condition gives slightly better results overall. Although including an energy equation still improves the results, the assumption of equal ion and neutral temperature already gives a very good approximation.
NASA Astrophysics Data System (ADS)
Mishra, Subhash C.; Vernekar, Rohan Ranganath
2012-11-01
Application of the lattice Boltzmann method (LBM) recently proposed by Asinari et al. [Asinari P, Mishra SC, Borchiellini R. A lattice Boltzmann formulation to the analysis of radiative heat transfer problems in a participating medium. Numer Heat Transfer B 2010; 57:126-146] is extended to the analysis of transport of collimated radiation in a planar participating medium. To deal with azimuthally symmetric radiation in planar medium, a new lattice structure for the LBM is used. The transport of the collimated component in the medium is analysed by two different, viz., flux splitting and direct approaches. For different angles of incidence of the collimated radiation, the LBM formulation is tested for the effects of the extinction coefficient, the anisotropy factor, and the boundary emissivities on heat flux and emissive power distributions. Results are compared with the benchmark results obtained using the finite volume method. Both the approaches in LBM provide accurate results.
PB-AM: An open-source, fully analytical linear poisson-boltzmann solver.
Felberg, Lisa E; Brookes, David H; Yap, Eng-Hui; Jurrus, Elizabeth; Baker, Nathan A; Head-Gordon, Teresa
2016-11-02
We present the open source distributed software package Poisson-Boltzmann Analytical Method (PB-AM), a fully analytical solution to the linearized PB equation, for molecules represented as non-overlapping spherical cavities. The PB-AM software package includes the generation of outputs files appropriate for visualization using visual molecular dynamics, a Brownian dynamics scheme that uses periodic boundary conditions to simulate dynamics, the ability to specify docking criteria, and offers two different kinetics schemes to evaluate biomolecular association rate constants. Given that PB-AM defines mutual polarization completely and accurately, it can be refactored as a many-body expansion to explore 2- and 3-body polarization. Additionally, the software has been integrated into the Adaptive Poisson-Boltzmann Solver (APBS) software package to make it more accessible to a larger group of scientists, educators, and students that are more familiar with the APBS framework. © 2016 Wiley Periodicals, Inc.
Kinetic theory of correlated fluids: from dynamic density functional to Lattice Boltzmann methods.
Marconi, Umberto Marini Bettolo; Melchionna, Simone
2009-07-07
Using methods of kinetic theory and liquid state theory we propose a description of the nonequilibrium behavior of molecular fluids, which takes into account their microscopic structure and thermodynamic properties. The present work represents an alternative to the recent dynamic density functional theory, which can only deal with colloidal fluids and is not apt to describe the hydrodynamic behavior of a molecular fluid. The method is based on a suitable modification of the Boltzmann transport equation for the phase space distribution and provides a detailed description of the local structure of the fluid and its transport coefficients. Finally, we propose a practical scheme to solve numerically and efficiently the resulting kinetic equation by employing a discretization procedure analogous to the one used in the Lattice Boltzmann method.
Cost of s-fold decisions in exact Maxwell Boltzmann, Bose Einstein and Fermi Dirac statistics
NASA Astrophysics Data System (ADS)
Niven, Robert K.
2006-06-01
The exact forms of the degenerate Maxwell-Boltzmann (MB), Bose-Einstein (BE) and Fermi-Dirac (FD) entropy functions, derived by Boltzmann's principle without the Stirling approximation [R.K. Niven, Physics Letters A, 342(4) (2005) 286], are further examined. Firstly, an apparent paradox in quantization effects is resolved using the Laplace-Jaynes interpretation of probability. The energy cost of learning that a system, distributed over s equiprobable states, is in one such state (an “ s-fold decision”) is then calculated for each statistic. The analysis confirms that the cost depends on one's knowledge of the number of entities N and (for BE and FD statistics) the degeneracy, extending the findings of Niven (2005).
Poisson-Boltzmann versus Size-Modified Poisson-Boltzmann Electrostatics Applied to Lipid Bilayers.
Wang, Nuo; Zhou, Shenggao; Kekenes-Huskey, Peter M; Li, Bo; McCammon, J Andrew
2014-12-26
Mean-field methods, such as the Poisson-Boltzmann equation (PBE), are often used to calculate the electrostatic properties of molecular systems. In the past two decades, an enhancement of the PBE, the size-modified Poisson-Boltzmann equation (SMPBE), has been reported. Here, the PBE and the SMPBE are reevaluated for realistic molecular systems, namely, lipid bilayers, under eight different sets of input parameters. The SMPBE appears to reproduce the molecular dynamics simulation results better than the PBE only under specific parameter sets, but in general, it performs no better than the Stern layer correction of the PBE. These results emphasize the need for careful discussions of the accuracy of mean-field calculations on realistic systems with respect to the choice of parameters and call for reconsideration of the cost-efficiency and the significance of the current SMPBE formulation.
The stationary non-equilibrium plasma of cosmic-ray electrons and positrons
NASA Astrophysics Data System (ADS)
Tomaschitz, Roman
2016-06-01
The statistical properties of the two-component plasma of cosmic-ray electrons and positrons measured by the AMS-02 experiment on the International Space Station and the HESS array of imaging atmospheric Cherenkov telescopes are analyzed. Stationary non-equilibrium distributions defining the relativistic electron-positron plasma are derived semi-empirically by performing spectral fits to the flux data and reconstructing the spectral number densities of the electronic and positronic components in phase space. These distributions are relativistic power-law densities with exponential cutoff, admitting an extensive entropy variable and converging to the Maxwell-Boltzmann or Fermi-Dirac distributions in the non-relativistic limit. Cosmic-ray electrons and positrons constitute a classical (low-density high-temperature) plasma due to the low fugacity in the quantized partition function. The positron fraction is assembled from the flux densities inferred from least-squares fits to the electron and positron spectra and is subjected to test by comparing with the AMS-02 flux ratio measured in the GeV interval. The calculated positron fraction extends to TeV energies, predicting a broad spectral peak at about 1 TeV followed by exponential decay.
Thermal equilibrium properties of surface hopping with an implicit Langevin bath
Sherman, M. C.; Corcelli, S. A.
2015-01-14
The ability of fewest switches surface hopping (FSSH) approach, where the classical degrees of freedom are coupled to an implicit Langevin bath, to establish and maintain an appropriate thermal equilibrium was evaluated in the context of a three site model for electron transfer. The electron transfer model consisted of three coupled diabatic states that each depends harmonically on the collective bath coordinate. This results in three states with increasing energy in the adiabatic representation. The adiabatic populations and distributions of the collective solvent coordinate were monitored during the course of 250 ns FSSH-Langevin (FSSH-L) simulations performed at a broad range of temperatures and for three different nonadiabatic coupling strengths. The agreement between the FSSH-L simulations and numerically exact results for the adiabatic population ratios and solvent coordinate distributions was generally favorable. The FSSH-L method produces a correct Boltzmann distribution of the solvent coordinate on each of the adiabats, but the integrated populations are slightly incorrect because FSSH does not rigorously obey detailed balance. The overall agreement is better at high temperatures and for high nonadiabatic coupling, which agrees with a previously reported analytical and simulation analysis [J. R. Schmidt, P. V. Parandekar, and J. C. Tully, J. Chem. Phys. 129, 044104 (2008)] on a two-level system coupled to a classical bath.
Thermodynamic theory of equilibrium fluctuations
Mishin, Y.
2015-12-15
The postulational basis of classical thermodynamics has been expanded to incorporate equilibrium fluctuations. The main additional elements of the proposed thermodynamic theory are the concept of quasi-equilibrium states, a definition of non-equilibrium entropy, a fundamental equation of state in the entropy representation, and a fluctuation postulate describing the probability distribution of macroscopic parameters of an isolated system. Although these elements introduce a statistical component that does not exist in classical thermodynamics, the logical structure of the theory is different from that of statistical mechanics and represents an expanded version of thermodynamics. Based on this theory, we present a regular procedure for calculations of equilibrium fluctuations of extensive parameters, intensive parameters and densities in systems with any number of fluctuating parameters. The proposed fluctuation formalism is demonstrated by four applications: (1) derivation of the complete set of fluctuation relations for a simple fluid in three different ensembles; (2) fluctuations in finite-reservoir systems interpolating between the canonical and micro-canonical ensembles; (3) derivation of fluctuation relations for excess properties of grain boundaries in binary solid solutions, and (4) derivation of the grain boundary width distribution for pre-melted grain boundaries in alloys. The last two applications offer an efficient fluctuation-based approach to calculations of interface excess properties and extraction of the disjoining potential in pre-melted grain boundaries. Possible future extensions of the theory are outlined.
Lattice Boltzmann model for numerical relativity
NASA Astrophysics Data System (ADS)
Ilseven, E.; Mendoza, M.
2016-02-01
In the Z4 formulation, Einstein equations are written as a set of flux conservative first-order hyperbolic equations that resemble fluid dynamics equations. Based on this formulation, we construct a lattice Boltzmann model for numerical relativity and validate it with well-established tests, also known as "apples with apples." Furthermore, we find that by increasing the relaxation time, we gain stability at the cost of losing accuracy, and by decreasing the lattice spacings while keeping a constant numerical diffusivity, the accuracy and stability of our simulations improve. Finally, in order to show the potential of our approach, a linear scaling law for parallelization with respect to number of CPU cores is demonstrated. Our model represents the first step in using lattice kinetic theory to solve gravitational problems.
Entropic Lattice Boltzmann Algorithms for Turbulence
NASA Astrophysics Data System (ADS)
Vahala, George; Yepez, Jeffrey; Soe, Min; Vahala, Linda; Keating, Brian; Carter, Jonathan
2007-11-01
For turbulent flows in non-trivial geometry, the scaling of CFD codes (now necessarily non-pseudo spectral) quickly saturate with the number of PEs. By projecting into a lattice kinetic phase space, the turbulent dynamics are simpler and much easier to solve since the underlying kinetic equation has only local algebraic nonlinearities in the macroscopic variables with simple linear kinetic advection. To achieve arbitrary high Reynolds number, a discrete H-theorem constraint is imposed on the collision operator resulting in an entropic lattice Boltzmann (ELB) algorithm that is unconditionally stable and scales almost perfectly with PE's on any supercomputer architecture. At this mesoscopic level, there are various kinetic lattices (ELB-27, ELB-19, ELB-15) which will recover the Navier-Stokes equation to leading order in the Chapman-Enskog asymptotics. We comment on the morphology of turbulence and its correlation to the rate of change of enstrophy as well as simulations on 1600^3 grids.
Boltzmann equation with double-well potentials
NASA Astrophysics Data System (ADS)
Chiacchiera, Silvia; Macrı, Tommaso; Trombettoni, Andrea
2016-10-01
We study the dynamics of an interacting classical gas trapped in a double-well potential at finite temperature. Two model potentials are considered: a cubic box with a square barrier in the middle, and a harmonic trap with a Gaussian barrier along one direction. The study is performed using the Boltzmann equation, solved numerically via the test-particle method. We introduce and discuss a simple analytical model that allows one to provide estimates of the relaxation time, which are compared with numerical results. Finally, we use our findings to make numerical and analytical predictions for the case of a fermionic mixture in the normal-fluid phase in a realistic double-well potential relevant for experiments with cold atoms.
Lattice Boltzmann simulations of lymphatic pumping
NASA Astrophysics Data System (ADS)
Kunert, Christian; Padera, Tim P.; Munn, Lance L.
2012-02-01
Lymphatic flow plays an important role in the progress of many diseases, including lymphedema and metastasis. However lymphatic pumping and flow is poorly understood. Here, we present a computer model that is based on biological observations of lymphatic pumping. Fluid flow is simulated by a D2Q9 lattice Boltzmann model. The boundary of the vessels moves according to shear-induced nitric oxide production, and wall motion transfers momentum to the fluid to induce flow. Because the model only includes local properties, it can be highly parallelized. In our case we utilize graphic processors (GPU) to achieve high performance computation. We show that the model provides stable pumping over a wide range of parameter values, with optimum flow achieved in the biological ranges. Furthermore, we investigate the efficiency by analyzing the flow rate and pumping frequency in order to compare the behavior of the model with existing in vivo data.
Adaptive filtering for the lattice Boltzmann method
NASA Astrophysics Data System (ADS)
Marié, Simon; Gloerfelt, Xavier
2017-03-01
In this study, a new selective filtering technique is proposed for the Lattice Boltzmann Method. This technique is based on an adaptive implementation of the selective filter coefficient σ. The proposed model makes the latter coefficient dependent on the shear stress in order to restrict the use of the spatial filtering technique in sheared stress region where numerical instabilities may occur. Different parameters are tested on 2D test-cases sensitive to numerical stability and on a 3D decaying Taylor-Green vortex. The results are compared to the classical static filtering technique and to the use of a standard subgrid-scale model and give significant improvements in particular for low-order filter consistent with the LBM stencil.
Enhanced gradient for training restricted Boltzmann machines.
Cho, Kyunghyun; Raiko, Tapani; Ilin, Alexander
2013-03-01
Restricted Boltzmann machines (RBMs) are often used as building blocks in greedy learning of deep networks. However, training this simple model can be laborious. Traditional learning algorithms often converge only with the right choice of metaparameters that specify, for example, learning rate scheduling and the scale of the initial weights. They are also sensitive to specific data representation. An equivalent RBM can be obtained by flipping some bits and changing the weights and biases accordingly, but traditional learning rules are not invariant to such transformations. Without careful tuning of these training settings, traditional algorithms can easily get stuck or even diverge. In this letter, we present an enhanced gradient that is derived to be invariant to bit-flipping transformations. We experimentally show that the enhanced gradient yields more stable training of RBMs both when used with a fixed learning rate and an adaptive one.
Dissipative Boltzmann-Robertson-Walker cosmologies
Hiscock, W.A.; Salmonson, J. )
1991-05-15
The equations governing a flat Robertson-Walker cosmological model containing a dissipative Boltzmann gas are integrated numerically. The bulk viscous stress is modeled using the Eckart and Israel-Stewart theories of dissipative relativistic fluids; the resulting cosmologies are compared and contrasted. The Eckart models are shown to always differ in a significant quantitative way from the Israel-Stewart models. It thus appears inappropriate to use the pathological (nonhyperbolic) Eckart theory for cosmological applications. For large bulk viscosities, both cosmological models approach asymptotic nonequilibrium states; in the Eckart model the total pressure is negative, while in the Israel-Stewart model the total pressure is asymptotically zero. The Eckart model also expands more rapidly than the Israel-Stewart models. These results suggest that bulk-viscous'' inflation may be an artifact of using a pathological fluid theory such as the Eckart theory.
Lattice Boltzmann model for numerical relativity.
Ilseven, E; Mendoza, M
2016-02-01
In the Z4 formulation, Einstein equations are written as a set of flux conservative first-order hyperbolic equations that resemble fluid dynamics equations. Based on this formulation, we construct a lattice Boltzmann model for numerical relativity and validate it with well-established tests, also known as "apples with apples." Furthermore, we find that by increasing the relaxation time, we gain stability at the cost of losing accuracy, and by decreasing the lattice spacings while keeping a constant numerical diffusivity, the accuracy and stability of our simulations improve. Finally, in order to show the potential of our approach, a linear scaling law for parallelization with respect to number of CPU cores is demonstrated. Our model represents the first step in using lattice kinetic theory to solve gravitational problems.
Lattice Boltzmann simulations of multiple-droplet interaction dynamics
NASA Astrophysics Data System (ADS)
Zhou, Wenchao; Loney, Drew; Fedorov, Andrei G.; Degertekin, F. Levent; Rosen, David W.
2014-03-01
A lattice Boltzmann (LB) formulation, which is consistent with the phase-field model for two-phase incompressible fluid, is proposed to model the interface dynamics of droplet impingement. The interparticle force is derived by comparing the macroscopic transport equations recovered from LB equations with the governing equations of the continuous phase-field model. The inconsistency between the existing LB implementations and the phase-field model in calculating the relaxation time at the phase interface is identified and an approximation is proposed to ensure the consistency with the phase-field model. It is also shown that the commonly used equilibrium velocity boundary for the binary fluid LB scheme does not conserve momentum at the wall boundary and a modified scheme is developed to ensure the momentum conservation at the boundary. In addition, a geometric formulation of the wetting boundary condition is proposed to replace the popular surface energy formulation and results show that the geometric approach enforces the prescribed contact angle better than the surface energy formulation in both static and dynamic wetting. The proposed LB formulation is applied to simulating droplet impingement dynamics in three dimensions and results are compared to those obtained with the continuous phase-field model, the LB simulations reported in the literature, and experimental data from the literature. The results show that the proposed LB simulation approach yields not only a significant speed improvement over the phase-field model in simulating droplet impingement dynamics on a submillimeter length scale, but also better accuracy than both the phase-field model and the previously reported LB techniques when compared to experimental data. Upon validation, the proposed LB modeling methodology is applied to the study of multiple-droplet impingement and interactions in three dimensions, which demonstrates its powerful capability of simulating extremely complex interface
Real-time keypoint recognition using restricted Boltzmann machine.
Yuan, Miaolong; Tang, Huajin; Li, Haizhou
2014-11-01
Feature point recognition is a key component in many vision-based applications, such as vision-based robot navigation, object recognition and classification, image-based modeling, and augmented reality. Real-time performance and high recognition rates are of crucial importance to these applications. In this brief, we propose a novel method for real-time keypoint recognition using restricted Boltzmann machine (RBM). RBMs are generative models that can learn probability distributions of many different types of data including labeled and unlabeled data sets. Due to the inherent noise of the training data sets, we use an RBM to model statistical distributions of the training data. Furthermore, the learned RBM can be used as a competitive classifier to recognize the keypoints in real-time during the tracking stage, thus making it advantageous to be employed in applications that require real-time performance. Experiments have been conducted under a variety of conditions to demonstrate the effectiveness and generalization of the proposed approach.
Volumetric lattice Boltzmann simulation for blood flow in aorta arteries
NASA Astrophysics Data System (ADS)
Deep, Debanjan; Yu, Huidan (Whitney); Teague, Shawn
2012-11-01
Complicated moving boundaries pose a major challenge in computational fluid dynamics for complex flows, especially in the biomechanics of both blood flow in the cardiovascular system and air flow in the respiratory system where the compliant nature of the vessels can have significant effects on the flow rate and wall shear stress. We develop a computation approach to treat arbitrarily moving boundaries using a volumetric representation of lattice Boltzmann method, which distributes fluid particles inside lattice cells. A volumetric bounce-back procedure is applied in the streaming step while momentum exchange between the fluid and moving solid boundary are accounted for in the collision sub-step. Additional boundary-induced migration is introduced to conserve fluid mass as the boundary moves across fluid cells. The volumetric LBM (VLBM) is used to simulate blood flow in both normal and dilated aorta arteries. We first compare flow structure and pressure distribution in steady state with results from Navier-Stokes based solver and good agreements are achieved. Then we focus on wall stress within the aorta for different heart pumping condition and present quantitative measurement of wall shear and normal stress.
Flow visualisation of downhill skiers using the lattice Boltzmann method
NASA Astrophysics Data System (ADS)
Asai, Takeshi; Hong, Sungchan; Ijuin, Koichi
2017-03-01
In downhill alpine skiing, skiers often exceed speeds of 120 km h-1, with air resistance substantially affecting the overall race times. To date, studies on air resistance in alpine skiing have used wind tunnels and actual skiers to examine the relationship between the gliding posture and magnitude of drag and for the design of skiing equipment. However, these studies have not revealed the flow velocity distribution and vortex structure around the skier. In the present study, computational fluid dynamics are employed with the lattice Boltzmann method to derive the relationship between total drag and the flow velocity around a downhill skier in the full-tuck position. Furthermore, the flow around the downhill skier is visualised, and its vortex structure is examined. The results show that the total drag force in the downhill skier model is 27.0 N at a flow velocity of 15 m s-1, increasing to 185.8 N at 40 m s-1. From analysis of the drag distribution and the flow profile, the head, upper arms, lower legs, and thighs (including buttocks) are identified as the major sources of drag on a downhill skier. Based on these results, the design of suits and equipment for reducing the drag from each location should be the focus of research and development in ski equipment. This paper describes a pilot study that introduces undergraduate students of physics or engineering into this research field. The results of this study are easy to understand for undergraduate students.
High-performance reconfigurable hardware architecture for restricted Boltzmann machines.
Ly, Daniel Le; Chow, Paul
2010-11-01
Despite the popularity and success of neural networks in research, the number of resulting commercial or industrial applications has been limited. A primary cause for this lack of adoption is that neural networks are usually implemented as software running on general-purpose processors. Hence, a hardware implementation that can exploit the inherent parallelism in neural networks is desired. This paper investigates how the restricted Boltzmann machine (RBM), which is a popular type of neural network, can be mapped to a high-performance hardware architecture on field-programmable gate array (FPGA) platforms. The proposed modular framework is designed to reduce the time complexity of the computations through heavily customized hardware engines. A method to partition large RBMs into smaller congruent components is also presented, allowing the distribution of one RBM across multiple FPGA resources. The framework is tested on a platform of four Xilinx Virtex II-Pro XC2VP70 FPGAs running at 100 MHz through a variety of different configurations. The maximum performance was obtained by instantiating an RBM of 256 × 256 nodes distributed across four FPGAs, which resulted in a computational speed of 3.13 billion connection-updates-per-second and a speedup of 145-fold over an optimized C program running on a 2.8-GHz Intel processor.
Lattice Boltzmann Method for Two-Dimensional Unsteady Incompressible Flow
NASA Astrophysics Data System (ADS)
Mužík, Juraj
2016-12-01
A Lattice Boltzmann method is used to analyse incompressible fluid flow in a two-dimensional cavity and flow in the channel past cylindrical obstacle. The method solves the Boltzmann's transport equation using simple computational grid - lattice. With the proper choice of the collision operator, the Boltzmann's equation can be converted into incompressible Navier-Stokes equation. Lid-driven cavity benchmark case for various Reynolds numbers and flow past cylinder is presented in the article. The method produces stable solutions with results comparable to those in literature and is very easy to implement.
Mean-Field Inference in Gaussian Restricted Boltzmann Machine
NASA Astrophysics Data System (ADS)
Takahashi, Chako; Yasuda, Muneki
2016-03-01
A Gaussian restricted Boltzmann machine (GRBM) is a Boltzmann machine defined on a bipartite graph and is an extension of usual restricted Boltzmann machines. A GRBM consists of two different layers: a visible layer composed of continuous visible variables and a hidden layer composed of discrete hidden variables. In this paper, we derive two different inference algorithms for GRBMs based on the naïve mean-field approximation (NMFA). One is an inference algorithm for whole variables in a GRBM, and the other is an inference algorithm for partial variables in a GBRBM. We compare the two methods analytically and numerically and show that the latter method is better.
Quantum statistical theory of semiconductor junctions in thermal equilibrium
NASA Technical Reports Server (NTRS)
Von Roos, O.
1977-01-01
Free carrier and electric field distributions of one-dimensional semiconductor junctions are evaluated using a quantum mechanical phase-space distribution and its corresponding Boltzmann equation. Attention is given to quantum and exchange corrections in cases of high doping concentrations when carrier densities become degenerate. Quantitative differences between degenerate and classical junction characteristics, e.g., maximum electric field and built-in voltage and carrier concentration within the transition region, are evaluated numerically.
NASA Astrophysics Data System (ADS)
Guo, Xiaohui
Fluid and thermal problems are widely encountered in micro/nano-scale devices, the characteristic lengths of which are from hundreds of microns down to tens of nanometers. A great number of such devices involve fundamental components like microchannels, capillaries, membranes and cantilever beams. Continuum assumptions that lead to classical governing equations such as Navier-Stokes equations and Fourier Laws break down when the characteristic size shrinks by an order of millions. In addition, conventional sensors, actuators and controllers turn to be insufficient to depict the flow, thermal, or electrical fields in micro-devices without impacting the original conditions greatly. Therefore, the development of numerical methods becomes indispensable in design and performance analysis of micro-electro-mechanical systems (MEMS). The main goal of this PhD research is the development, implementation and application of comprehensive deterministic Boltzmann-ESBGK modeling framework to micro-scale fluid-thermal phenomena. Investigation of gas flows in short rectangular microchannels has been carried out to understand the rarefaction effects on the reduced mass-flow-rate as well as the non-equilibrium effects on the temperature components. At high Knudsen numbers, the reduced mass-flow-rate only depends on the pressure ratio and the temperature components deviate at the channel exit. For gas flows in long microchannels with and without constrictions, the Navier-Stokes equations with first-order slip boundary conditions are solved. Numerical results accurately predict the entrance pressure drop comparing to high-resolution experimental data using pressure-sensitive-paint (PSP). Simultions show clearly that the compressibility effects become less important than the rarefaction effects at low pressures. The coupled gas-phonon Boltzmann solver has been developed. The reduced distribution functions are used in the two-dimensional code to reduce the computational cost. The
NASA Astrophysics Data System (ADS)
Chvala, Frantisek
Subjected to an external electromagnetic field, a rare two-component spatially homogeneous gas consisting of charged and neutral particles is considered. The velocity distribution of the neutral particles being assumed known, the mixture is characterized by the velocity distribution of the charged particles, which is determined as a mild solution of the Boltzmann kinetic equation. Relying upon functional-analytic properties of the collision term, existence and uniqueness of the mild solution are established in some Lebesgue-type function spaces involving exponential weights.
Dewar, K.M.; Montreuil, B.; Grondin, L.; Reader, T.A. )
1989-08-01
The binding properties of the substituted benzamide raclopride to dopamine D2 receptors were studied with membrane preparations from rat and rabbit neostriatum. An analysis of the association kinetics suggested a single binding site but the data from the dissociation experiments were better described by a two-site model. Examination of saturation curves at equilibrium revealed a single class of binding sites in the neostriatum from both species (rat: maximum binding capacity (Bmax) = 247 fmol/mg of protein; rabbit: Bmax = 337 fmol/mg of protein). In cortical regions known to possess a distinct dopaminergic innervation (piriform-entorhinal areas and cingulate cortex) the Bmax values ranged between 9 and 22 fmol/mg of protein. ({sup 3}H)Raclopride binding sites (less than 12 fmol/mg of protein) were also detectable in the dorsal and ventral hippocampus as well as in the somatosensory and visual cortices. The selectivity in the neostriatum was examined by competition experiments with dopaminergic drugs. The rank of potency of agonists and antagonists to displace ({sup 3}H)raclopride binding revealed its selectivity for the dopamine D2 receptor and was essentially the same for both species. Antagonist competition curves could be fitted to a single site but inhibition by agonists was better described assuming a two-site model. The stereospecificity of binding was demonstrated by the use of the enantiomer pairs. These results validate the utilization of the novel benzamide ({sup 3}H)raclopride as a selective marker of dopamine D2 receptors.
How accurate is Poisson-Boltzmann theory for monovalent ions near highly charged interfaces?
Bu, Wei; Vaknin, David; Travesset, Alex
2006-06-20
Surface sensitive synchrotron X-ray scattering studies were performed to obtain the distribution of monovalent ions next to a highly charged interface. A lipid phosphate (dihexadecyl hydrogen-phosphate) was spread as a monolayer at the air-water interface to control surface charge density. Using anomalous reflectivity off and at the L3 Cs+ resonance, we provide spatial counterion (Cs+) distributions next to the negatively charged interfaces. Five decades in bulk concentrations are investigated, demonstrating that the interfacial distribution is strongly dependent on bulk concentration. We show that this is due to the strong binding constant of hydronium H3O+ to the phosphate group, leading to proton-transfer back to the phosphate group and to a reduced surface charge. The increase of Cs+ concentration modifies the contact value potential, thereby causing proton release. This process effectively modifies surface charge density and enables exploration of ion distributions as a function of effective surface charge-density. The experimentally obtained ion distributions are compared to distributions calculated by Poisson-Boltzmann theory accounting for the variation of surface charge density due to proton release and binding. We also discuss the accuracy of our experimental results in discriminating possible deviations from Poisson-Boltzmann theory.
Niu, Xiao-Dong; Hyodo, Shi-Aki; Munekata, Toshihisa; Suga, Kazuhiko
2007-09-01
It is well known that the Navier-Stokes equations cannot adequately describe gas flows in the transition and free-molecular regimes. In these regimes, the Boltzmann equation (BE) of kinetic theory is invoked to govern the flows. However, this equation cannot be solved easily, either by analytical techniques or by numerical methods. Hence, in order to efficiently maneuver around this equation for modeling microscale gas flows, a kinetic lattice Boltzmann method (LBM) has been introduced in recent years. This method is regarded as a numerical approach for solving the BE in discrete velocity space with Gauss-Hermite quadrature. In this paper, a systematic description of the kinetic LBM, including the lattice Boltzmann equation, the diffuse-scattering boundary condition for gas-surface interactions, and definition of the relaxation time, is provided. To capture the nonlinear effects due to the high-order moments and wall boundaries, an effective relaxation time and a modified regularization procedure of the nonequilibrium part of the distribution function are further presented based on previous work [Guo et al., J. Appl. Phys. 99, 074903 (2006); Shan et al., J. Fluid Mech. 550, 413 (2006)]. The capability of the kinetic LBM of simulating microscale gas flows is illustrated based on the numerical investigations of micro Couette and force-driven Poiseuille flows.
Boltzmann-type control of opinion consensus through leaders
Albi, G.; Pareschi, L.; Zanella, M.
2014-01-01
The study of formations and dynamics of opinions leading to the so-called opinion consensus is one of the most important areas in mathematical modelling of social sciences. Following the Boltzmann-type control approach recently introduced by the first two authors, we consider a group of opinion leaders who modify their strategy accordingly to an objective functional with the aim of achieving opinion consensus. The main feature of the Boltzmann-type control is that, owing to an instantaneous binary control formulation, it permits the minimization of the cost functional to be embedded into the microscopic leaders’ interactions of the corresponding Boltzmann equation. The related Fokker–Planck asymptotic limits are also derived, which allow one to give explicit expressions of stationary solutions. The results demonstrate the validity of the Boltzmann-type control approach and the capability of the leaders’ control to strategically lead the followers’ opinion. PMID:25288820
Boltzmann-type control of opinion consensus through leaders.
Albi, G; Pareschi, L; Zanella, M
2014-11-13
The study of formations and dynamics of opinions leading to the so-called opinion consensus is one of the most important areas in mathematical modelling of social sciences. Following the Boltzmann-type control approach recently introduced by the first two authors, we consider a group of opinion leaders who modify their strategy accordingly to an objective functional with the aim of achieving opinion consensus. The main feature of the Boltzmann-type control is that, owing to an instantaneous binary control formulation, it permits the minimization of the cost functional to be embedded into the microscopic leaders' interactions of the corresponding Boltzmann equation. The related Fokker-Planck asymptotic limits are also derived, which allow one to give explicit expressions of stationary solutions. The results demonstrate the validity of the Boltzmann-type control approach and the capability of the leaders' control to strategically lead the followers' opinion.
Conservative form of Boltzmann's equation in general relativity
NASA Astrophysics Data System (ADS)
Shibata, Masaru; Nagakura, Hiroki; Sekiguchi, Yuichiro; Yamada, Shoichi
2014-04-01
We derive a conservative form of Boltzmann's equation in general relativity, which is concisely written. Several explicit forms of this equation are written for black-hole spacetime with several coordinate conditions in real spacetime and momentum-space coordinates.
Quantum linear Boltzmann equation with finite intercollision time
Diosi, Lajos
2009-12-15
Inconsistencies are pointed out in the usual quantum versions of the classical linear Boltzmann equation constructed for a quantized test particle in a gas. These are related to the incorrect formal treatment of momentum decoherence. We prove that ideal collisions with the molecules would result in complete momentum decoherence, the persistence of coherence is only due to the finite intercollision time. A corresponding quantum linear Boltzmann equation is proposed.
Multiple-Relaxation-Time Lattice Boltzmann Models in 3D
NASA Technical Reports Server (NTRS)
dHumieres, Dominique; Ginzburg, Irina; Krafczyk, Manfred; Lallemand, Pierre; Luo, Li-Shi; Bushnell, Dennis M. (Technical Monitor)
2002-01-01
This article provides a concise exposition of the multiple-relaxation-time lattice Boltzmann equation, with examples of fifteen-velocity and nineteen-velocity models in three dimensions. Simulation of a diagonally lid-driven cavity flow in three dimensions at Re=500 and 2000 is performed. The results clearly demonstrate the superior numerical stability of the multiple-relaxation-time lattice Boltzmann equation over the popular lattice Bhatnagar-Gross-Krook equation.
Pseudo-Boltzmann model for modeling the junctionless transistors
NASA Astrophysics Data System (ADS)
Avila-Herrera, F.; Cerdeira, A.; Roldan, J. B.; Sánchez-Moreno, P.; Tienda-Luna, I. M.; Iñiguez, B.
2014-05-01
Calculation of the carrier concentrations in semiconductors using the Fermi-Dirac integral requires complex numerical calculations; in this context, practically all analytical device models are based on Boltzmann statistics, even though it is known that it leads to an over-estimation of carriers densities for high doping concentrations. In this paper, a new approximation to Fermi-Dirac integral, called Pseudo-Boltzmann model, is presented for modeling junctionless transistors with high doping concentrations.
Consistency of pseudolikelihood estimation of fully visible Boltzmann machines.
Hyvärinen, Aapo
2006-10-01
A Boltzmann machine is a classic model of neural computation, and a number of methods have been proposed for its estimation. Most methods are plagued by either very slow convergence or asymptotic bias in the resulting estimates. Here we consider estimation in the basic case of fully visible Boltzmann machines. We show that the old principle of pseudolikelihood estimation provides an estimator that is computationally very simple yet statistically consistent.
Fluctuations around equilibrium laws in ergodic continuous-time random walks.
Schulz, Johannes H P; Barkai, Eli
2015-06-01
We study occupation time statistics in ergodic continuous-time random walks. Under thermal detailed balance conditions, the average occupation time is given by the Boltzmann-Gibbs canonical law. But close to the nonergodic phase, the finite-time fluctuations around this mean are large and nontrivial. They exhibit dual time scaling and distribution laws: the infinite density of large fluctuations complements the Lévy-stable density of bulk fluctuations. Neither of the two should be interpreted as a stand-alone limiting law, as each has its own deficiency: the infinite density has an infinite norm (despite particle conservation), while the stable distribution has an infinite variance (although occupation times are bounded). These unphysical divergences are remedied by consistent use and interpretation of both formulas. Interestingly, while the system's canonical equilibrium laws naturally determine the mean occupation time of the ergodic motion, they also control the infinite and Lévy-stable densities of fluctuations. The duality of stable and infinite densities is in fact ubiquitous for these dynamics, as it concerns the time averages of general physical observables.
NASA Astrophysics Data System (ADS)
Wang, Y.; Shu, C.; Yang, L. M.
2015-12-01
An improved multiphase lattice Boltzmann flux solver (MLBFS) is proposed in this work for effective simulation of three-dimensional (3D) multiphase flows with large density ratio and high Reynolds number. As a finite volume scheme, the MLBFS originally proposed in [27] applies the finite volume method to solve for macroscopic flow variables directly. The fluxes are reconstructed locally at each cell interface by using the standard LBM solutions. Due to the modeling error of the standard LBM, the reconstructed fluxes deviate from those in the Navier-Stokes equations; and to compensate this error, a complex tensor is introduced in the original MLBFS. However, the computation of the tensor introduces additional complexity and usually needs a relatively thicker interface thickness to maintain numerical stability, which makes the solver be complex and inefficient in the 3D case. To remove this drawback, in this work, a theoretical analysis to the formulations obtained from the Chapman-Enskog expansion is conducted. It is shown that the modeling error can be effectively removed by modifying the computation of the equilibrium density distribution function. With this improvement, the proposed 3D MLBFS not only avoids the calculation of the compensation tensor but also is able to maintain numerical stability with very thin interface thickness. Several benchmark cases, including the challenging droplet impacting on a dry surface, head-on collisions of binary droplets and droplet splashing on a thin film with density ratio 1000 and Reynolds number up to 3000, are studied to validate the proposed solver. The obtained results agree well with the published data.
Relaxation of hot and massive tracers using numerical solutions of the Boltzmann equation
NASA Astrophysics Data System (ADS)
Khurana, Saheba; Thachuk, Mark
2016-03-01
A numerical method using B-splines is used to solve the linear Boltzmann equation describing the energy relaxation of massive tracer particles moving through a dilute bath gas. The smooth and rough hard sphere and Maxwell molecule models are used with a variety of mass ratios and initial energies to test the capability of the numerical method. Massive tracers are initialized with energies typically found in energy loss experiments in mass spectrometry using biomolecules. The method is also used to examine the applicability of known expressions for the kinetic energy decay from the Fokker-Planck equation for the Rayleigh gas, where we find that results are generally good provided that the initial energy is properly bounded. Otherwise, the energy decay is not constant and a more complex behaviour occurs. The validity of analytical expressions for drag coefficients for spherical particles under specular and diffuse scattering is also tested. We find such expressions are generally good for hard spheres but cannot account, as expected, for the softer repulsive walls of the Maxwell (and real) molecules. Overall, the numerical method performed well even when tracers more than 400 times as massive as the bath were initialized with energies very far from equilibrium. This is a range of applicability beyond many of the standard methods for solving the Boltzmann equation.
Relaxation of hot and massive tracers using numerical solutions of the Boltzmann equation.
Khurana, Saheba; Thachuk, Mark
2016-03-14
A numerical method using B-splines is used to solve the linear Boltzmann equation describing the energy relaxation of massive tracer particles moving through a dilute bath gas. The smooth and rough hard sphere and Maxwell molecule models are used with a variety of mass ratios and initial energies to test the capability of the numerical method. Massive tracers are initialized with energies typically found in energy loss experiments in mass spectrometry using biomolecules. The method is also used to examine the applicability of known expressions for the kinetic energy decay from the Fokker-Planck equation for the Rayleigh gas, where we find that results are generally good provided that the initial energy is properly bounded. Otherwise, the energy decay is not constant and a more complex behaviour occurs. The validity of analytical expressions for drag coefficients for spherical particles under specular and diffuse scattering is also tested. We find such expressions are generally good for hard spheres but cannot account, as expected, for the softer repulsive walls of the Maxwell (and real) molecules. Overall, the numerical method performed well even when tracers more than 400 times as massive as the bath were initialized with energies very far from equilibrium. This is a range of applicability beyond many of the standard methods for solving the Boltzmann equation.
Lattice Boltzmann Simulations of Evaporating Droplets with Nanoparticles
NASA Astrophysics Data System (ADS)
Zhao, Mingfei; Yong, Xin
2016-11-01
Elucidating the nanoparticle dynamics in drying droplets provides fundamental hydrodynamic insight into the evaporation-induced self-assembly, which is of great importance to materials printing and thin film processing. We develop a free-energy-based multiphase lattice Boltzmann model coupled with Lagrangian particle tracking to simulate evaporating particle-laden droplets on a solid substrate with specified wetting behavior. This work focuses on the interplay between the evaporation-driven advection and the self-organization of nanoparticles inside the droplet and at the droplet surface. For static droplets, the different parameters, fluid-particle interaction strength and particle number, governing the nanoparticle-droplet dynamics are systematically investigated, such as particle radial and circumferential distribution. We clarify the effect of nanoparticle presence on the droplet surface tension and wetting behavior. For evaporating droplets, we observe how droplet evaporation modulates the self-assembly of nanoparticles when the droplet has different static contact angles and hysteresis windows. We also confirm that the number of nanoparticles at the liquid-vapor interface influences the evaporation flux at the liquid-vapor interface.
Computer programs for the Boltzmann collision matrix elements
NASA Astrophysics Data System (ADS)
Das, P.
1989-09-01
When the distribution function in the kinetic theory of gases is expanded in a basis of orthogonal functions, the Boltzmann collision operators can be evaluated in terms of appropriate matrix elements. These matrix elements are usually given in terms of highly complex algebraic expressions. When Burnett functions, which consist of Sonine polynomials and spherical harmonics, are used as the basis, the irreducible tensor formalism provides expressions for the matrix elements that are algebraically simple, possess high symmetry, and are computationally more economical than in any other basis. The package reported here consists of routines to compute such matrix elements in a Burnett function basis for a mixture of hard sphere gases, as also the loss integral of a Burnett mode and the functions themselves. The matrix elements involve the Clebsch-Gordan and Brody-Moshinsky coefficients, both of which are used here for unusually high values of their arguments. For the purpose of validation both coefficients are computed using two different methods. Though written for hard sphere molecules the package can, with only slight modification, be adapted to more general molecular models as well.
Force Evaluation in the Lattice Boltzmann Method Involving Curved Geometry
NASA Technical Reports Server (NTRS)
Mei, Renwei; Yu, Dazhi; Shyy, Wei; Luo, Li-Shi; Bushnell, Dennis M. (Technical Monitor)
2002-01-01
The present work investigates two approaches for force evaluation in the lattice Boltzmann equation: the momentum- exchange method and the stress-integration method on the surface of a body. The boundary condition for the particle distribution functions on curved geometries is handled with second order accuracy based on our recent works. The stress-integration method is computationally laborious for two-dimensional flows and in general difficult to implement for three-dimensional flows, while the momentum-exchange method is reliable, accurate, and easy to implement for both two-dimensional and three-dimensional flows. Several test cases are selected to evaluate the present methods, including: (i) two-dimensional pressure-driven channel flow; (ii) two-dimensional uniform flow past a column of cylinders; (iii) two-dimensional flow past a cylinder asymmetrically placed in a channel (with vortex shedding); (iv) three-dimensional pressure-driven flow in a circular pipe; and (v) three-dimensional flow past a sphere. The drag evaluated by using the momentum-exchange method agrees well with the exact or other published results.
An axisymmetric multiple-relaxation-time lattice Boltzmann scheme
NASA Astrophysics Data System (ADS)
Xie, Wenjun
2015-01-01
A multiple-relaxation-time (MRT) lattice Boltzmann (LB) scheme developed for axisymmetric flows recovers the complete continuity and Navier-Stokes equations. This scheme follows the strategy of the standard D2Q9 model by using a single particle distribution function and a simple "collision-streaming" updating rule. The extra terms related to axisymmetry in the macroscopic equations are recovered by adding source terms into the LB equation, which are simple and involve no gradients. The compressible effect retained in the Navier-Stokes equations is recovered by introducing a term related to the reversed transformation matrix for MRT collision operator, so as to produce a correct bulk viscosity, making it suitable for compressible flows with high frequency and low Mach number. The validity of the scheme is demonstrated by testing the Hagen-Poiseuille flow and 3D Womersley flow, as well as the standing acoustic waves in a closed cylindrical chamber. The numerical experiments show desirable stability at low viscosities, enabling to simulate a standing ultrasound field in centimeters space.
Large-scale lattice-Boltzmann simulations over lambda networks
NASA Astrophysics Data System (ADS)
Saksena, R.; Coveney, P. V.; Pinning, R.; Booth, S.
Amphiphilic molecules are of immense industrial importance, mainly due to their tendency to align at interfaces in a solution of immiscible species, e.g., oil and water, thereby reducing surface tension. Depending on the concentration of amphiphiles in the solution, they may assemble into a variety of morphologies, such as lamellae, micelles, sponge and cubic bicontinuous structures exhibiting non-trivial rheological properties. The main objective of this work is to study the rheological properties of very large, defect-containing gyroidal systems (of up to 10243 lattice sites) using the lattice-Boltzmann method. Memory requirements for the simulation of such large lattices exceed that available to us on most supercomputers and so we use MPICH-G2/MPIg to investigate geographically distributed domain decomposition simulations across HPCx in the UK and TeraGrid in the US. Use of MPICH-G2/MPIg requires the port-forwarder to work with the grid middleware on HPCx. Data from the simulations is streamed to a high performance visualisation resource at UCL (London) for rendering and visualisation. Lighting the Blue Touchpaper for UK e-Science - Closing Conference of ESLEA Project March 26-28 2007 The George Hotel, Edinburgh, UK
Meshless lattice Boltzmann method for the simulation of fluid flows.
Musavi, S Hossein; Ashrafizaadeh, Mahmud
2015-02-01
A meshless lattice Boltzmann numerical method is proposed. The collision and streaming operators of the lattice Boltzmann equation are separated, as in the usual lattice Boltzmann models. While the purely local collision equation remains the same, we rewrite the streaming equation as a pure advection equation and discretize the resulting partial differential equation using the Lax-Wendroff scheme in time and the meshless local Petrov-Galerkin scheme based on augmented radial basis functions in space. The meshless feature of the proposed method makes it a more powerful lattice Boltzmann solver, especially for cases in which using meshes introduces significant numerical errors into the solution, or when improving the mesh quality is a complex and time-consuming process. Three well-known benchmark fluid flow problems, namely the plane Couette flow, the circular Couette flow, and the impulsively started cylinder flow, are simulated for the validation of the proposed method. Excellent agreement with analytical solutions or with previous experimental and numerical results in the literature is observed in all the simulations. Although the computational resources required for the meshless method per node are higher compared to that of the standard lattice Boltzmann method, it is shown that for cases in which the total number of nodes is significantly reduced, the present method actually outperforms the standard lattice Boltzmann method.
Getting Freshman in Equilibrium.
ERIC Educational Resources Information Center
Journal of Chemical Education, 1983
1983-01-01
Various aspects of chemical equilibrium were discussed in six papers presented at the Seventh Biennial Conference on Chemical Education (Stillwater, Oklahoma 1982). These include student problems in understanding hydrolysis, helping students discover/uncover topics, equilibrium demonstrations, instructional strategies, and flaws to kinetic…
A SIMPLE METHOD FOR MODELING COLLISION PROCESSES IN PLASMAS WITH A KAPPA ENERGY DISTRIBUTION
Hahn, M.; Savin, D. W.
2015-08-20
We demonstrate that a nonthermal distribution of particles described by a kappa distribution can be accurately approximated by a weighted sum of Maxwell–Boltzmann distributions. We apply this method to modeling collision processes in kappa-distribution plasmas, with a particular focus on atomic processes important for solar physics. The relevant collision process rate coefficients are generated by summing appropriately weighted Maxwellian rate coefficients. This method reproduces the rate coefficients for a kappa distribution to an estimated accuracy of better than 3%. This is equal to or better than the accuracy of rate coefficients generated using “reverse-engineering” methods, which attempt to extract the needed cross sections from the published Maxwellian rate coefficient data and then reconvolve the extracted cross sections with the desired kappa distribution. Our approach of summing Maxwellian rate coefficients is easy to implement using existing spectral analysis software. Moreover, the weights in the sum of the Maxwell–Boltzmann distribution rate coefficients can be found for any value of the parameter κ, thereby enabling one to model plasmas with a time-varying κ. Tabulated Maxwellian fitting parameters are given for specific values of κ from 1.7 to 100. We also provide polynomial fits to these parameters over this entire range. Several applications of our technique are presented, including the plasma equilibrium charge state distribution (CSD), predicting line ratios, modeling the influence of electron impact multiple ionization on the equilibrium CSD of kappa-distribution plasmas, and calculating the time-varying CSD of plasmas during a solar flare.
A Simple Method for Modeling Collision Processes in Plasmas with a Kappa Energy Distribution
NASA Astrophysics Data System (ADS)
Hahn, M.; Savin, D. W.
2015-08-01
We demonstrate that a nonthermal distribution of particles described by a kappa distribution can be accurately approximated by a weighted sum of Maxwell-Boltzmann distributions. We apply this method to modeling collision processes in kappa-distribution plasmas, with a particular focus on atomic processes important for solar physics. The relevant collision process rate coefficients are generated by summing appropriately weighted Maxwellian rate coefficients. This method reproduces the rate coefficients for a kappa distribution to an estimated accuracy of better than 3%. This is equal to or better than the accuracy of rate coefficients generated using “reverse-engineering” methods, which attempt to extract the needed cross sections from the published Maxwellian rate coefficient data and then reconvolve the extracted cross sections with the desired kappa distribution. Our approach of summing Maxwellian rate coefficients is easy to implement using existing spectral analysis software. Moreover, the weights in the sum of the Maxwell-Boltzmann distribution rate coefficients can be found for any value of the parameter κ, thereby enabling one to model plasmas with a time-varying κ. Tabulated Maxwellian fitting parameters are given for specific values of κ from 1.7 to 100. We also provide polynomial fits to these parameters over this entire range. Several applications of our technique are presented, including the plasma equilibrium charge state distribution (CSD), predicting line ratios, modeling the influence of electron impact multiple ionization on the equilibrium CSD of kappa-distribution plasmas, and calculating the time-varying CSD of plasmas during a solar flare.
Lattice Boltzmann model for resistive relativistic magnetohydrodynamics.
Mohseni, F; Mendoza, M; Succi, S; Herrmann, H J
2015-08-01
In this paper, we develop a lattice Boltzmann model for relativistic magnetohydrodynamics (MHD). Even though the model is derived for resistive MHD, it is shown that it is numerically robust even in the high conductivity (ideal MHD) limit. In order to validate the numerical method, test simulations are carried out for both ideal and resistive limits, namely the propagation of Alfvén waves in the ideal MHD and the evolution of current sheets in the resistive regime, where very good agreement is observed comparing to the analytical results. Additionally, two-dimensional magnetic reconnection driven by Kelvin-Helmholtz instability is studied and the effects of different parameters on the reconnection rate are investigated. It is shown that the density ratio has a negligible effect on the magnetic reconnection rate, while an increase in shear velocity decreases the reconnection rate. Additionally, it is found that the reconnection rate is proportional to σ-1/2, σ being the conductivity, which is in agreement with the scaling law of the Sweet-Parker model. Finally, the numerical model is used to study the magnetic reconnection in a stellar flare. Three-dimensional simulation suggests that the reconnection between the background and flux rope magnetic lines in a stellar flare can take place as a result of a shear velocity in the photosphere.
Modeling adsorption with lattice Boltzmann equation
Guo, Long; Xiao, Lizhi; Shan, Xiaowen; Zhang, Xiaoling
2016-01-01
The research of adsorption theory has recently gained renewed attention due to its critical relevance to a number of trending industrial applications, hydrogen storage and shale gas exploration for instance. The existing theoretical foundation, laid mostly in the early twentieth century, was largely based on simple heuristic molecular interaction models and static interaction potential which, although being insightful in illuminating the fundamental mechanisms, are insufficient for computations with realistic adsorbent structure and adsorbate hydrodynamics, both critical for real-life applications. Here we present and validate a novel lattice Boltzmann model incorporating both adsorbate-adsorbate and adsorbate-adsorbent interactions with hydrodynamics which, for the first time, allows adsorption to be computed with real-life details. Connection with the classic Ono-Kondo lattice theory is established and various adsorption isotherms, both within and beyond the IUPAC classification are observed as a pseudo-potential is varied. This new approach not only enables an important physical to be simulated for real-life applications, but also provides an enabling theoretical framework within which the fundamentals of adsorption can be studied. PMID:27256325
Accurate lineshape spectroscopy and the Boltzmann constant
Truong, G.-W.; Anstie, J. D.; May, E. F.; Stace, T. M.; Luiten, A. N.
2015-01-01
Spectroscopy has an illustrious history delivering serendipitous discoveries and providing a stringent testbed for new physical predictions, including applications from trace materials detection, to understanding the atmospheres of stars and planets, and even constraining cosmological models. Reaching fundamental-noise limits permits optimal extraction of spectroscopic information from an absorption measurement. Here, we demonstrate a quantum-limited spectrometer that delivers high-precision measurements of the absorption lineshape. These measurements yield a very accurate measurement of the excited-state (6P1/2) hyperfine splitting in Cs, and reveals a breakdown in the well-known Voigt spectral profile. We develop a theoretical model that accounts for this breakdown, explaining the observations to within the shot-noise limit. Our model enables us to infer the thermal velocity dispersion of the Cs vapour with an uncertainty of 35 p.p.m. within an hour. This allows us to determine a value for Boltzmann's constant with a precision of 6 p.p.m., and an uncertainty of 71 p.p.m. PMID:26465085
Lattice Boltzmann model for resistive relativistic magnetohydrodynamics
NASA Astrophysics Data System (ADS)
Mohseni, F.; Mendoza, M.; Succi, S.; Herrmann, H. J.
2015-08-01
In this paper, we develop a lattice Boltzmann model for relativistic magnetohydrodynamics (MHD). Even though the model is derived for resistive MHD, it is shown that it is numerically robust even in the high conductivity (ideal MHD) limit. In order to validate the numerical method, test simulations are carried out for both ideal and resistive limits, namely the propagation of Alfvén waves in the ideal MHD and the evolution of current sheets in the resistive regime, where very good agreement is observed comparing to the analytical results. Additionally, two-dimensional magnetic reconnection driven by Kelvin-Helmholtz instability is studied and the effects of different parameters on the reconnection rate are investigated. It is shown that the density ratio has a negligible effect on the magnetic reconnection rate, while an increase in shear velocity decreases the reconnection rate. Additionally, it is found that the reconnection rate is proportional to σ-1 / 2, σ being the conductivity, which is in agreement with the scaling law of the Sweet-Parker model. Finally, the numerical model is used to study the magnetic reconnection in a stellar flare. Three-dimensional simulation suggests that the reconnection between the background and flux rope magnetic lines in a stellar flare can take place as a result of a shear velocity in the photosphere.
Modeling adsorption with lattice Boltzmann equation.
Guo, Long; Xiao, Lizhi; Shan, Xiaowen; Zhang, Xiaoling
2016-06-03
The research of adsorption theory has recently gained renewed attention due to its critical relevance to a number of trending industrial applications, hydrogen storage and shale gas exploration for instance. The existing theoretical foundation, laid mostly in the early twentieth century, was largely based on simple heuristic molecular interaction models and static interaction potential which, although being insightful in illuminating the fundamental mechanisms, are insufficient for computations with realistic adsorbent structure and adsorbate hydrodynamics, both critical for real-life applications. Here we present and validate a novel lattice Boltzmann model incorporating both adsorbate-adsorbate and adsorbate-adsorbent interactions with hydrodynamics which, for the first time, allows adsorption to be computed with real-life details. Connection with the classic Ono-Kondo lattice theory is established and various adsorption isotherms, both within and beyond the IUPAC classification are observed as a pseudo-potential is varied. This new approach not only enables an important physical to be simulated for real-life applications, but also provides an enabling theoretical framework within which the fundamentals of adsorption can be studied.
NASA Astrophysics Data System (ADS)
You, K.-I.; Lee, D.-K.; Lee, S. G.; Bak, J. G.; Hahn, S. H.; Lao, L.; Kstar Team
2011-10-01
We have installed the EFIT code on our computing system and made some modification to reconstruct the plasma equilibrium of KSTAR (Korea Superconducting Tokamak Advanced Research). KSTAR PF and TF coil systems use a CICC (Cable-In-Conduit Conductor) type superconductor. The CICC jacket material for most PF and all TF coils is Incoloy 908, which is a magnetic material with relative magnetic permeability greater than 10 in low external field. We newly introduced Diamagnetic Loop and variational Motion Stark Effect signals to equilibrium reconstruction. In this paper, we present some results of equilibrium reconstruction with the EFIT code, assess the effects of newly introduced diagnsotics signal on the equilibrium reconstruction and compare the EFIT results with the various diagnostics data in various plasma conditions including H- and L- modes. In addition, we will show the Incoloy908 effects on the plasma equilibrium.
On Weak Solutions to the Linear Boltzmann Equation with Inelastic Coulomb Collisions
Pettersson, Rolf
2011-05-20
This paper considers the time- and space-dependent linear Boltzmann equation with general boundary conditions in the case of inelastic (granular) collisions. First, in the (angular) cut-off case, mild L{sup 1}-solutions are constructed as limits of the iterate functions and boundedness of higher velocity moments are discussed in the case of inverse power collisions forces. Then the problem of the weak solutions, as weak limit of a sequence of mild solutions, is studied for a bounded body, in the case of very soft interactions (including the Coulomb case). Furthermore, strong convergence of weak solutions to the equilibrium, when time goes to infinity, is discussed, using a generalized H-theorem, together with a translation continuity property.
Lattice Boltzmann accelerated direct simulation Monte Carlo for dilute gas flow simulations
NASA Astrophysics Data System (ADS)
Di Staso, G.; Clercx, H. J. H.; Succi, S.; Toschi, F.
2016-11-01
Hybrid particle-continuum computational frameworks permit the simulation of gas flows by locally adjusting the resolution to the degree of non-equilibrium displayed by the flow in different regions of space and time. In this work, we present a new scheme that couples the direct simulation Monte Carlo (DSMC) with the lattice Boltzmann (LB) method in the limit of isothermal flows. The former handles strong non-equilibrium effects, as they typically occur in the vicinity of solid boundaries, whereas the latter is in charge of the bulk flow, where non-equilibrium can be dealt with perturbatively, i.e. according to Navier-Stokes hydrodynamics. The proposed concurrent multiscale method is applied to the dilute gas Couette flow, showing major computational gains when compared with the full DSMC scenarios. In addition, it is shown that the coupling with LB in the bulk flow can speed up the DSMC treatment of the Knudsen layer with respect to the full DSMC case. In other words, LB acts as a DSMC accelerator. This article is part of the themed issue 'Multiscale modelling at the physics-chemistry-biology interface'.
Influence of state-to-state vibrational distributions on transport coefficients of a single gas
NASA Astrophysics Data System (ADS)
Kustova, Elena V.; Kremer, Gilberto M.
2016-11-01
In this work the influence of the size of vibrationally and rotationally excited molecules on the collision integrals required for the calculation of state-to-state transport coefficients is discussed. Several diatomic molecules are considered: N2, O2, NO, H2, Cl2. It is shown that whereas the molecular size is not affected by rotational excitation, it strongly depends on the vibrational state. Particular emphasis is given to the shear viscosity and thermal conductivity coefficients calculated in the temperature range 2 500-20 000 K for equilibrium Boltzmann vibrational distributions. It is shown that under conditions of local thermal equilibrium, the effect of vibrational excitation on the shear viscosity and thermal conductivity coefficients are found to be negligible for temperatures below 5 000 K, except for the case of Cl2 molecule where at 5 000 K the effect is about 10%. For T > 10 000 K, the contribution of excited states becomes important and reaches 10-25%.
Alemani, Davide; Pappalardo, Francesco; Pennisi, Marzio; Motta, Santo; Brusic, Vladimir
2012-02-28
In the last decades the Lattice Boltzmann method (LB) has been successfully used to simulate a variety of processes. The LB model describes the microscopic processes occurring at the cellular level and the macroscopic processes occurring at the continuum level with a unique function, the probability distribution function. Recently, it has been tried to couple deterministic approaches with probabilistic cellular automata (probabilistic CA) methods with the aim to model temporal evolution of tumor growths and three dimensional spatial evolution, obtaining hybrid methodologies. Despite the good results attained by CA-PDE methods, there is one important issue which has not been completely solved: the intrinsic stochastic nature of the interactions at the interface between cellular (microscopic) and continuum (macroscopic) level. CA methods are able to cope with the stochastic phenomena because of their probabilistic nature, while PDE methods are fully deterministic. Even if the coupling is mathematically correct, there could be important statistical effects that could be missed by the PDE approach. For such a reason, to be able to develop and manage a model that takes into account all these three level of complexity (cellular, molecular and continuum), we believe that PDE should be replaced with a statistic and stochastic model based on the numerical discretization of the Boltzmann equation: The Lattice Boltzmann (LB) method. In this work we introduce a new hybrid method to simulate tumor growth and immune system, by applying Cellular Automata Lattice Boltzmann (CA-LB) approach.
A unified gas-kinetic scheme for continuum and rarefied flows IV: Full Boltzmann and model equations
Liu, Chang; Xu, Kun; Sun, Quanhua; Cai, Qingdong
2016-06-01
where it is needed. The central ingredient of the UGKS is the coupled treatment of particle transport and collision in the flux evaluation across a cell interface, where a continuous flow dynamics from kinetic to hydrodynamic scales is modeled. The newly developed UGKS has the asymptotic preserving (AP) property of recovering the NS solutions in the continuum flow regime, and the full Boltzmann solution in the rarefied regime. In the mostly unexplored transition regime, the UGKS itself provides a valuable tool for the non-equilibrium flow study. The mathematical properties of the scheme, such as stability, accuracy, and the asymptotic preserving, will be analyzed in this paper as well.
NASA Astrophysics Data System (ADS)
Chen, Xin
2014-04-01
Understanding the roles of the temporary and spatial structures of quantum functional noise in open multilevel quantum molecular systems attracts a lot of theoretical interests. I want to establish a rigorous and general framework for functional quantum noises from the constructive and computational perspectives, i.e., how to generate the random trajectories to reproduce the kernel and path ordering of the influence functional with effective Monte Carlo methods for arbitrary spectral densities. This construction approach aims to unify the existing stochastic models to rigorously describe the temporary and spatial structure of Gaussian quantum noises. In this paper, I review the Euclidean imaginary time influence functional and propose the stochastic matrix multiplication scheme to calculate reduced equilibrium density matrices (REDM). In addition, I review and discuss the Feynman-Vernon influence functional according to the Gaussian quadratic integral, particularly its imaginary part which is critical to the rigorous description of the quantum detailed balance. As a result, I establish the conditions under which the influence functional can be interpreted as the average of exponential functional operator over real-valued Gaussian processes for open multilevel quantum systems. I also show the difference between the local and nonlocal phonons within this framework. With the stochastic matrix multiplication scheme, I compare the normalized REDM with the Boltzmann equilibrium distribution for open multilevel quantum systems.
Chen, Xin
2014-04-21
Understanding the roles of the temporary and spatial structures of quantum functional noise in open multilevel quantum molecular systems attracts a lot of theoretical interests. I want to establish a rigorous and general framework for functional quantum noises from the constructive and computational perspectives, i.e., how to generate the random trajectories to reproduce the kernel and path ordering of the influence functional with effective Monte Carlo methods for arbitrary spectral densities. This construction approach aims to unify the existing stochastic models to rigorously describe the temporary and spatial structure of Gaussian quantum noises. In this paper, I review the Euclidean imaginary time influence functional and propose the stochastic matrix multiplication scheme to calculate reduced equilibrium density matrices (REDM). In addition, I review and discuss the Feynman-Vernon influence functional according to the Gaussian quadratic integral, particularly its imaginary part which is critical to the rigorous description of the quantum detailed balance. As a result, I establish the conditions under which the influence functional can be interpreted as the average of exponential functional operator over real-valued Gaussian processes for open multilevel quantum systems. I also show the difference between the local and nonlocal phonons within this framework. With the stochastic matrix multiplication scheme, I compare the normalized REDM with the Boltzmann equilibrium distribution for open multilevel quantum systems.
Chemical Principles Revisited: Chemical Equilibrium.
ERIC Educational Resources Information Center
Mickey, Charles D.
1980-01-01
Describes: (1) Law of Mass Action; (2) equilibrium constant and ideal behavior; (3) general form of the equilibrium constant; (4) forward and reverse reactions; (5) factors influencing equilibrium; (6) Le Chatelier's principle; (7) effects of temperature, changing concentration, and pressure on equilibrium; and (8) catalysts and equilibrium. (JN)
Kinetic Effects of Non-Equilibrium Plasma on Partially Premixed Flame Extinction
2011-01-01
dissociative attachment processes. The Boltzmann equation calculates the rate coefficients of the electron impact elementary reactions by averaging the...ion-ion neutralization processes, ion-molecule reactions, and electron attachment and detachment processes. Note that the present model does not solve...partially premixed methane flames was studied at 60 Torr by blending 2% CH4 into the oxidizer stream. The non-equilibrium discharge accelerated
Geva, A B; Shtram, L; Policker, S
2000-07-01
One of the most fascinating aspects of brain research is the subject of language. As in many other cases, the malfunctions that occur in different persons for various reasons give us insight on the mechanisms that support our ability to talk, read and listen. Following the work of Plaut and associates, we deal with the dyslexia disorder, which is the overall name for a large number of reading disorders. A Boltzmann machine neural network scheme was trained to implement the nonlinear mapping task of graphic representation into semantic representation, which may model the brain sections responsible for the translation of a written word into meanings and syllables. After training, various types of lesions were applied and the performance of the network was tested in order to measure the effect of each lesion on the error rate and type distribution that were detected. The system's errors were classified into several categories and the distribution of errors between the categories was studied. Using the simulations, it is demonstrated that a finite scheduling process in the Boltzmann machine causes the distribution of the network's errors to be unique and different from its expected error distribution. The phenomenon is given a mathematical explanation rooted in the statistical mechanics basics of the Boltzmann machine. Test results suggest the localization of certain reading functions within the network. Comparison is made to relevant types of dyslexia and shows resemblance in major symptoms as well as in certain known side effects.
Taruya, Atsushi; Sakagami, Masa-aki
2003-05-09
With particular attention to the recently postulated introduction of a nonextensive generalization of Boltzmann-Gibbs statistics, we study the long-term stellar dynamical evolution of self-gravitating systems on time scales much longer than the two-body relaxation time. In a self-gravitating N-body system confined in an adiabatic wall, we show that the quasiequilibrium sequence arising from the Tsallis entropy, so-called stellar polytropes, plays an important role in characterizing the transient states away from the Boltzmann-Gibbs equilibrium state.
Response reactions: equilibrium coupling.
Hoffmann, Eufrozina A; Nagypal, Istvan
2006-06-01
It is pointed out and illustrated in the present paper that if a homogeneous multiple equilibrium system containing k components and q species is composed of the reactants actually taken and their reactions contain only k + 1 species, then we have a unique representation with (q - k) stoichiometrically independent reactions (SIRs). We define these as coupling reactions. All the other possible combinations with k + 1 species are the coupled reactions that are in equilibrium when the (q - k) SIRs are in equilibrium. The response of the equilibrium state for perturbation is determined by the coupling and coupled equilibria. Depending on the circumstances and the actual thermodynamic data, the effect of coupled equilibria may overtake the effect of the coupling ones, leading to phenomena that are in apparent contradiction with Le Chatelier's principle.
Electrostatic interaction of two charged macroparticles in an equilibrium plasma
NASA Astrophysics Data System (ADS)
Filippov, A. V.; Pal', A. F.; Starostin, A. N.
2015-11-01
This article is a critical review of publications devoted to studying the electrostatic interaction of two charged macroparticles in an equilibrium plasma. It is shown from an analysis of the force of interaction based on the Maxwell stress tensor that two macroparticles with identical charges in the Poisson-Boltzmann model always repel each other both in isothermal and nonisothermal plasmas. At distances between macroparticles for which the Boltzmann exponents can be linearized, the interaction between macroparticles is completely described by the Debye-Hückel model. The correction to free energy due to the electrostatic interaction in the system of two macroparticles is determined by integrating the correction to the internal energy and by direct calculation of the correction for entropy. It is shown that the free energy coincides with the Yukawa potential. The coincidence of the interaction energy obtained by integrating the force of interaction with the free energy leads to the conclusion about the potential nature of the force of interaction between two macroparticles in an equilibrium plasma. The effect of the outer boundary on the electrostatic interaction force is analyzed; it is shown that the type of interaction depends on the choice of the boundary conditions at the outer boundary. It is also shown that the accumulation of space charge near the outer boundary can lead to the attraction of similarly charged particles at distances comparable with the radius of the outer boundary.
Electrostatic interaction of two charged macroparticles in an equilibrium plasma
Filippov, A. V. Pal’, A. F.; Starostin, A. N.
2015-11-15
This article is a critical review of publications devoted to studying the electrostatic interaction of two charged macroparticles in an equilibrium plasma. It is shown from an analysis of the force of interaction based on the Maxwell stress tensor that two macroparticles with identical charges in the Poisson–Boltzmann model always repel each other both in isothermal and nonisothermal plasmas. At distances between macroparticles for which the Boltzmann exponents can be linearized, the interaction between macroparticles is completely described by the Debye–Hückel model. The correction to free energy due to the electrostatic interaction in the system of two macroparticles is determined by integrating the correction to the internal energy and by direct calculation of the correction for entropy. It is shown that the free energy coincides with the Yukawa potential. The coincidence of the interaction energy obtained by integrating the force of interaction with the free energy leads to the conclusion about the potential nature of the force of interaction between two macroparticles in an equilibrium plasma. The effect of the outer boundary on the electrostatic interaction force is analyzed; it is shown that the type of interaction depends on the choice of the boundary conditions at the outer boundary. It is also shown that the accumulation of space charge near the outer boundary can lead to the attraction of similarly charged particles at distances comparable with the radius of the outer boundary.
Ripszam, Matyas; Haglund, Peter
2015-02-01
Dissolved organic carbon (DOC) plays a key role in determining the environmental fate of semivolatile organic environmental contaminants. The goal of the present study was to develop a method using commercially available hardware to rapidly characterize the sorption properties of DOC in water samples. The resulting method uses negligible-depletion direct immersion solid-phase microextraction (SPME) and gas chromatography-mass spectrometry. Its performance was evaluated using Nordic reference fulvic acid and 40 priority environmental contaminants that cover a wide range of physicochemical properties. Two SPME fibers had to be used to cope with the span of properties, 1 coated with polydimethylsiloxane and 1 coated with polystyrene divinylbenzene polydimethylsiloxane, for nonpolar and semipolar contaminants, respectively. The measured DOC-water distribution constants showed reasonably good reproducibility (standard deviation ≤ 0.32) and good correlation (R(2) = 0.80) with log octanol-water partition coefficients for nonpolar persistent organic pollutants. The sample pretreatment is limited to filtration, and the method is easy to adjust to different DOC concentrations. These experiments also utilized the latest SPME automation that largely decreases total cycle time (to 20 min or shorter) and increases sample throughput, which is advantageous in cases when many samples of DOC must be characterized or when the determinations must be performed quickly, for example, to avoid precipitation, aggregation, and other changes of DOC structure and properties. The data generated by this method are valuable as a basis for transport and fate modeling studies.
Computing Equilibrium Chemical Compositions
NASA Technical Reports Server (NTRS)
Mcbride, Bonnie J.; Gordon, Sanford
1995-01-01
Chemical Equilibrium With Transport Properties, 1993 (CET93) computer program provides data on chemical-equilibrium compositions. Aids calculation of thermodynamic properties of chemical systems. Information essential in design and analysis of such equipment as compressors, turbines, nozzles, engines, shock tubes, heat exchangers, and chemical-processing equipment. CET93/PC is version of CET93 specifically designed to run within 640K memory limit of MS-DOS operating system. CET93/PC written in FORTRAN.
White, R D; Ness, K F; Robson, R E; Li, B
1999-08-01
A multiterm solution of the Boltzmann equation has been developed and used to calculate transport coefficients of charged-particle swarms in gases under the influence of electric and magnetic fields crossed at arbitrary angles psi. The hierarchy resulting from a spherical harmonic decomposition of the Boltzmann equation in the hydrodynamic regime [Ness, Phys. Rev. A 47, 327 (1993)] is solved numerically by representing the speed dependence of the phase-space distribution function in terms of an expansion in Sonine polynomials about a weighted sum of Maxwellian distributions at different temperatures. Results are given for charged-particle swarms in certain model gases over a range of psi and field strengths. The variation of the transport coefficients with psi is addressed using physical arguments. The errors associated with the two-term approximation and inadequacies of Legendre polynomial expansions are highlighted.
Wu, Zhen; Zhang, Xian; Zhou, Chunjiao; Pang, Jing-Lin; Zhang, Panyue
2017-02-22
Single-molecule aluminum salt AlCl3, medium polymerized polyaluminum chloride (PAC), and high polymerized polyaluminum chloride (HPAC) were prepared in a laboratory. The characteristics and coagulation properties of these prepared aluminum salts were investigated. The Langmuir, Freundlich, and Sips adsorption isotherms were first used to describe the adsorption neutralization process in coagulation, and the Boltzmann equation was used to fit the reaction kinetics of floc growth in flocculation. It was novel to find that the experimental data fitted well with the Sips and Boltzmann equation, and the significance of parameters in the equations was discussed simultaneously. Through the Sips equation, the adsorption neutralization reaction was proved to be spontaneous and the adsorption neutralization capacity was HPAC > PAC > AlCl3. Sips equation also indicated that the zeta potential of water samples would reach a limit with the increase of coagulant dosage, and the equilibrium zeta potential values were 30.25, 30.23, and 27.25 mV for AlCl3, PAC, and HPAC, respectively. The lower equilibrium zeta potential value of HPAC might be the reason why the water sample was not easy to achieve restabilization at a high coagulant dosage. Through the Boltzmann equation modeling, the maximum average floc size formed by AlCl3, PAC, and HPAC were 196.0, 188.0, and 203.6 μm, respectively, and the halfway time of reactions were 31.23, 17.08, and 9.55 min, respectively. The HPAC showed the strongest floc formation ability and the fastest floc growth rate in the flocculation process, which might be caused by the stronger adsorption and bridging functions of Alb and Alc contained in HPAC.
Lattice Boltzmann Methods to Address Fundamental Boiling and Two-Phase Problems
Uddin, Rizwan
2012-01-01
This report presents the progress made during the fourth (no cost extension) year of this three-year grant aimed at the development of a consistent Lattice Boltzmann formulation for boiling and two-phase flows. During the first year, a consistent LBM formulation for the simulation of a two-phase water-steam system was developed. Results of initial model validation in a range of thermo-dynamic conditions typical for Boiling Water Reactors (BWRs) were shown. Progress was made on several fronts during the second year. Most important of these included the simulation of the coalescence of two bubbles including the surface tension effects. Work during the third year focused on the development of a new lattice Boltzmann model, called the artificial interface lattice Boltzmann model (AILB model) for the 3 simulation of two-phase dynamics. The model is based on the principle of free energy minimization and invokes the Gibbs-Duhem equation in the formulation of non-ideal forcing function. This was reported in detail in the last progress report. Part of the efforts during the last (no-cost extension) year were focused on developing a parallel capability for the 2D as well as for the 3D codes developed in this project. This will be reported in the final report. Here we report the work carried out on testing the AILB model for conditions including the thermal effects. A simplified thermal LB model, based on the thermal energy distribution approach, was developed. The simplifications are made after neglecting the viscous heat dissipation and the work done by pressure in the original thermal energy distribution model. Details of the model are presented here, followed by a discussion of the boundary conditions, and then results for some two-phase thermal problems.
Physical scales in the Wigner–Boltzmann equation
Nedjalkov, M.; Selberherr, S.; Ferry, D.K.; Vasileska, D.; Dollfus, P.; Querlioz, D.; Dimov, I.; Schwaha, P.
2013-01-01
The Wigner–Boltzmann equation provides the Wigner single particle theory with interactions with bosonic degrees of freedom associated with harmonic oscillators, such as phonons in solids. Quantum evolution is an interplay of two transport modes, corresponding to the common coherent particle-potential processes, or to the decoherence causing scattering due to the oscillators. Which evolution mode will dominate depends on the scales of the involved physical quantities. A dimensionless formulation of the Wigner–Boltzmann equation is obtained, where these scales appear as dimensionless strength parameters. A notion called scaling theorem is derived, linking the strength parameters to the coupling with the oscillators. It is shown that an increase of this coupling is equivalent to a reduction of both the strength of the electric potential, and the coherence length. Secondly, the existence of classes of physically different, but mathematically equivalent setups of the Wigner–Boltzmann evolution is demonstrated. PMID:23504194
Properties of the Boltzmann equation in the classical approximation
Epelbaum, Thomas; Gelis, François; Tanji, Naoto; ...
2014-12-30
We examine the Boltzmann equation with elastic point-like scalar interactions in two different versions of the the classical approximation. Although solving numerically the Boltzmann equation with the unapproximated collision term poses no problem, this allows one to study the effect of the ultraviolet cutoff in these approximations. This cutoff dependence in the classical approximations of the Boltzmann equation is closely related to the non-renormalizability of the classical statistical approximation of the underlying quantum field theory. The kinetic theory setup that we consider here allows one to study in a much simpler way the dependence on the ultraviolet cutoff, since onemore » has also access to the non-approximated result for comparison.« less
George Hartley Bryan, Ludwig Boltzmann, and the Stability of Flight
NASA Astrophysics Data System (ADS)
Boyd, T. James M.
2012-03-01
A century ago, George Hartley Bryan (1864-1928) published his classic book, Stability in Aviation. I draw together some strands from events that awakened his interest in the nascent science of aviation, in particular the stability of flight. Prominent among those who influenced him was Ludwig Boltzmann (1844-1906), who held Bryan in high esteem for his contributions to thermodynamics and kinetic theory. I argue that the seeds of Bryan's interest in aviation were sown at the British Association meeting at Oxford in the summer of 1894, at which Boltzmann was guest of honor. A joint discussion between Section A (Mathematical and Physical Science) and Section G (Mechanical Science) was devoted to the problems of flight, during the course of which Boltzmann revealed a hitherto unsuspected enthusiasm for flying.
Properties of the Boltzmann equation in the classical approximation
Epelbaum, Thomas; Gelis, François; Tanji, Naoto; Wu, Bin
2014-12-30
We examine the Boltzmann equation with elastic point-like scalar interactions in two different versions of the the classical approximation. Although solving numerically the Boltzmann equation with the unapproximated collision term poses no problem, this allows one to study the effect of the ultraviolet cutoff in these approximations. This cutoff dependence in the classical approximations of the Boltzmann equation is closely related to the non-renormalizability of the classical statistical approximation of the underlying quantum field theory. The kinetic theory setup that we consider here allows one to study in a much simpler way the dependence on the ultraviolet cutoff, since one has also access to the non-approximated result for comparison.
Effect of q-nonextensive distribution of electrons on the sheath in dusty plasma
NASA Astrophysics Data System (ADS)
Driouch, Ismael; Chatei, Hassan
2017-01-01
In this paper, a sheath model has been developed to investigate the characteristics of a magnetised dusty plasma sheath in the presence of a q-nonextensive distribution of electrons. For this, we have established a one-dimensional fluid model. The electrons are considered following the q-nonextensive distribution (i.e. the deflection of the electrons for their Maxwell-Boltzmann distribution); however the ions and dust grains are described by fluid equations. According to multi-fluid equations and some dimensionless variables, the dimensionless equations are obtained and solved numerically. The effect of the nonextensivity q-parameter on the plasma sheath parameters is examined. A significant change is observed in the quantities characterising the sheath when the electrons evolve far away from their thermodynamic equilibrium Maxwellian ( q = 1) assumption.
Equations of state in a lattice Boltzmann model
NASA Astrophysics Data System (ADS)
Yuan, Peng; Schaefer, Laura
2006-04-01
In this paper we consider the incorporation of various equations of state into the single-component multiphase lattice Boltzmann model. Several cubic equations of state, including the van der Waals, Redlich-Kwong, and Peng-Robinson, as well as a noncubic equation of state (Carnahan-Starling), are incorporated into the lattice Boltzmann model. The details of phase separation in these nonideal single-component systems are presented by comparing the numerical simulation results in terms of density ratios, spurious currents, and temperature ranges. A comparison with a real fluid system, i.e., the properties of saturated water and steam, is also presented.
Evaluation of Two Lattice Boltzmann Models for Multiphase Flows
NASA Astrophysics Data System (ADS)
Hou, Shuling; Shan, Xiaowen; Zou, Qisu; Doolen, Gary D.; Soll, Wendy E.
1997-12-01
Two lattice Boltzmann models for multiphase flows, the immiscible fluid model proposed by Rothman and Keller (R-K) and the multicomponent nonideal gas lattice Boltzmann model by Shan and Chen (S-C), are studied numerically to compare their abilities to simulate the physics of multiphase flows. The test problem is the simulation of a static bubble. Isotropy, strength of surface tension, thickness of the interface, spurious currents, Laplace's law, and steadiness of the bubble are examined. The results show that the S-C model is a major improvement over the R-K model.
Advanced mean-field theory of the restricted Boltzmann machine
NASA Astrophysics Data System (ADS)
Huang, Haiping; Toyoizumi, Taro
2015-05-01
Learning in restricted Boltzmann machine is typically hard due to the computation of gradients of log-likelihood function. To describe the network state statistics of the restricted Boltzmann machine, we develop an advanced mean-field theory based on the Bethe approximation. Our theory provides an efficient message-passing-based method that evaluates not only the partition function (free energy) but also its gradients without requiring statistical sampling. The results are compared with those obtained by the computationally expensive sampling-based method.
Martín-Hernández, G; Mastinu, P F; Praena, J; Dzysiuk, N; Capote Noy, R; Pignatari, M
2012-08-01
The need of neutron capture cross section measurements for astrophysics motivates present work, where calculations to generate stellar neutron spectra at different temperatures are performed. The accelerator-based (7)Li(p,n)(7)Be reaction is used. Shaping the proton beam energy and the sample covering a specific solid angle, neutron activation for measuring stellar-averaged capture cross section can be done. High-quality Maxwell-Boltzmann neutron spectra are predicted. Assuming a general behavior of the neutron capture cross section a weighted fit of the spectrum to Maxwell-Boltzmann distributions is successfully introduced.
NASA Astrophysics Data System (ADS)
Shizgal, Bernie
2016-03-01
The paper by Burini et al. [7] presents an interesting use of the Boltzmann equation of kinetic theory to model real learning processes. The authors provide a comprehensive discussion of the basic concepts involved in their modelling work. The Boltzmann equation as used by physicists and chemists to model a variety of transport processes in many diverse fields is based on the notion of the binary collisions between identifiable particles in the defined system [9]. The particles exchange energy on collision and the distribution function, which depends on the three velocity components and the three spatial coordinates, varies with time. The classical or quantum collision dynamics between particles play a central role in the definition of the kernels in the integral operators that define the Boltzmann equation [8].
NASA Astrophysics Data System (ADS)
Azmir, O. Shahrul; Azwadi, C. S. Nor
2010-06-01
This paper presents numerical study of flow behavior from a heated concentric annulus cylinder at various Rayleigh number Ra, Prandtl number Pr while the aspect ratio is fixed to 5.0 of the outer and inner cylinders. The Finite Different Lattice Boltzmann Method (FDLBM) numerical scheme is proposed to improve the computational efficiency and numerical stability of the conventional method. The proposed FELBM applied UTOPIA approach (third order accuracy in space) to study the temperature distribution and the vortex formation in the annulus cylinder. The comparison of the flow pattern and temperature distribution for every case via streamline, vortices and temperature distribution contour with published paper in literature were carried out for the validation purposes. Current investigation concluded that the UTOPIA FDLBM is an efficient approach for the current problem in hand and good agreement with the benchmark solution.
A novel scheme for curved moving boundaries in the lattice Boltzmann method
NASA Astrophysics Data System (ADS)
Xu, Lina; Rao, Parthib; Schaefer, Laura
2016-06-01
We propose a novel scheme to handle moving boundaries, both straight and curved, within the lattice Boltzmann method (LBM). In this scheme, which is broadly based on the Chapman-Enskog expansion, the fictitious distributions are constructed exactly on the moving boundary. This is in contrast to existing methods where such distributions are constructed on neighboring nodes which may not lie on the moving boundary. The post-collisional distributions on the fluid nodes near the moving boundary are then constructed using first- or second-order interpolations. The proposed scheme also overcomes the requirement to have separate interpolation formulations for different values of the intersection parameter. Several validation tests presented here indicate improved accuracy and numerical stability, compliance with Galilean invariance principle, an ability to preserve the geometric fidelity of curved surfaces.
Boltzmann-conserving classical dynamics in quantum time-correlation functions: “Matsubara dynamics”
Hele, Timothy J. H.; Willatt, Michael J.; Muolo, Andrea; Althorpe, Stuart C.
2015-04-07
We show that a single change in the derivation of the linearized semiclassical-initial value representation (LSC-IVR or “classical Wigner approximation”) results in a classical dynamics which conserves the quantum Boltzmann distribution. We rederive the (standard) LSC-IVR approach by writing the (exact) quantum time-correlation function in terms of the normal modes of a free ring-polymer (i.e., a discrete imaginary-time Feynman path), taking the limit that the number of polymer beads N → ∞, such that the lowest normal-mode frequencies take their “Matsubara” values. The change we propose is to truncate the quantum Liouvillian, not explicitly in powers of ħ{sup 2} at ħ{sup 0} (which gives back the standard LSC-IVR approximation), but in the normal-mode derivatives corresponding to the lowest Matsubara frequencies. The resulting “Matsubara” dynamics is inherently classical (since all terms O(ħ{sup 2}) disappear from the Matsubara Liouvillian in the limit N → ∞) and conserves the quantum Boltzmann distribution because the Matsubara Hamiltonian is symmetric with respect to imaginary-time translation. Numerical tests show that the Matsubara approximation to the quantum time-correlation function converges with respect to the number of modes and gives better agreement than LSC-IVR with the exact quantum result. Matsubara dynamics is too computationally expensive to be applied to complex systems, but its further approximation may lead to practical methods.
A hydrodynamically-consistent MRT lattice Boltzmann model on a 2D rectangular grid
NASA Astrophysics Data System (ADS)
Peng, Cheng; Min, Haoda; Guo, Zhaoli; Wang, Lian-Ping
2016-12-01
A multiple-relaxation time (MRT) lattice Boltzmann (LB) model on a D2Q9 rectangular grid is designed theoretically and validated numerically in the present work. By introducing stress components into the equilibrium moments, this MRT-LB model restores the isotropy of diffusive momentum transport at the macroscopic level (or in the continuum limit), leading to moment equations that are fully consistent with the Navier-Stokes equations. The model is derived by an inverse design process which is described in detail. Except one moment associated with the energy square, all other eight equilibrium moments can be theoretically and uniquely determined. The model is then carefully validated using both the two-dimensional decaying Taylor-Green vortex flow and lid-driven cavity flow, with different grid aspect ratios. The corresponding results from an earlier model (Bouzidi et al. (2001) [28]) are also presented for comparison. The results of Bouzidi et al.'s model show problems associated with anisotropy of viscosity coefficients, while the present model exhibits full isotropy and is accurate and stable.
Determination of nonaxisymmetric equilibrium
Elkin, D.
1980-01-01
The Princeton Equilibrium Code is modified to determine the equilibrium surfaces for a large aspect ratio toroidal system with helical magnetic fields. The code may easily be made to include any variety of modes. Verification of the code is made by comparison with an analytic solution for l = 3. Previously observed shifting of the magnetic axis with increasing pressure or with a changed externally applied vertical field is obtained. The case l = 0, a bumpy torus, gives convergence only for the lenient convergence tolerance of epsilon/sub b/ = 1.0 x 10-/sup 2/.
Beyond Equilibrium Thermodynamics
NASA Astrophysics Data System (ADS)
Öttinger, Hans Christian
2005-01-01
Beyond Equilibrium Thermodynamics fills a niche in the market by providing a comprehensive introduction to a new, emerging topic in the field. The importance of non-equilibrium thermodynamics is addressed in order to fully understand how a system works, whether it is in a biological system like the brain or a system that develops plastic. In order to fully grasp the subject, the book clearly explains the physical concepts and mathematics involved, as well as presenting problems and solutions; over 200 exercises and answers are included. Engineers, scientists, and applied mathematicians can all use the book to address their problems in modelling, calculating, and understanding dynamic responses of materials.
NASA Technical Reports Server (NTRS)
Mei, Ren-Wei; Shyy, Wei; Yu, Da-Zhi; Luo, Li-Shi; Rudy, David (Technical Monitor)
2001-01-01
The lattice Boltzmann equation (LBE) is a kinetic formulation which offers an alternative computational method capable of solving fluid dynamics for various systems. Major advantages of the method are owing to the fact that the solution for the particle distribution functions is explicit, easy to implement, and the algorithm is natural to parallelize. In this final report, we summarize the works accomplished in the past three years. Since most works have been published, the technical details can be found in the literature. Brief summary will be provided in this report. In this project, a second-order accurate treatment of boundary condition in the LBE method is developed for a curved boundary and tested successfully in various 2-D and 3-D configurations. To evaluate the aerodynamic force on a body in the context of LBE method, several force evaluation schemes have been investigated. A simple momentum exchange method is shown to give reliable and accurate values for the force on a body in both 2-D and 3-D cases. Various 3-D LBE models have been assessed in terms of efficiency, accuracy, and robustness. In general, accurate 3-D results can be obtained using LBE methods. The 3-D 19-bit model is found to be the best one among the 15-bit, 19-bit, and 27-bit LBE models. To achieve desired grid resolution and to accommodate the far field boundary conditions in aerodynamics computations, a multi-block LBE method is developed by dividing the flow field into various blocks each having constant lattice spacing. Substantial contribution to the LBE method is also made through the development of a new, generalized lattice Boltzmann equation constructed in the moment space in order to improve the computational stability, detailed theoretical analysis on the stability, dispersion, and dissipation characteristics of the LBE method, and computational studies of high Reynolds number flows with singular gradients. Finally, a finite difference-based lattice Boltzmann method is
Lattice Boltzmann method for linear oscillatory noncontinuum flows
NASA Astrophysics Data System (ADS)
Shi, Yong; Yap, Ying Wan; Sader, John E.
2014-03-01
Oscillatory gas flows are commonly generated by micro- and nanoelectromechanical systems. Due to their small size and high operating frequencies, these devices often produce noncontinuum gas flows. Theoretical analysis of such flows requires solution of the unsteady Boltzmann equation, which can present a formidable challenge. In this article, we explore the applicability of the lattice Boltzmann (LB) method to such linear oscillatory noncontinuum flows; this method is derived from the linearized Boltzmann Bhatnagar-Gross-Krook (BGK) equation. We formulate four linearized LB models in the frequency domain, based on Gaussian-Hermite quadratures of different algebraic precision (AP). The performance of each model is assessed by comparison to high-accuracy numerical solutions to the linearized Boltzmann-BGK equation for oscillatory Couette flow. The numerical results demonstrate that high even-order LB models provide superior performance over the greatest noncontinuum range. Our results also highlight intrinsic deficiencies in the current LB framework, which is incapable of capturing noncontinuum behavior at high oscillation frequencies, regardless of quadrature AP and the Knudsen number.
Measuring Boltzmann's Constant with Carbon Dioxide
ERIC Educational Resources Information Center
Ivanov, Dragia; Nikolov, Stefan
2013-01-01
In this paper we present two experiments to measure Boltzmann's constant--one of the fundamental constants of modern-day physics, which lies at the base of statistical mechanics and thermodynamics. The experiments use very basic theory, simple equipment and cheap and safe materials yet provide very precise results. They are very easy and…
Lattice-Boltzmann methods for suspensions of solid particles
NASA Astrophysics Data System (ADS)
Ladd, Anthony J. C.
2015-09-01
In this paper, dedicated to Prof. Jean-Pierre Hansen, I will summarise the development of lattice-Boltzmann methods for simulating the dynamics of colloidal suspensions. I will describe the main ideas and subsequent improvements, and place them in the wider context of particle-based methods for fluid dynamics.
Lattice Boltzmann beyond Navier-Stokes: Where do we stand?
NASA Astrophysics Data System (ADS)
Succi, Sauro
2016-11-01
The main steps taking the Lattice Boltzmann (LB) method beyond the realm of continuum hydrodynamics are discussed, along with an appraisal of future prospects for coupling LB with other computational kinetic methods, such as Bird's Direct Simulation Monte Carlo and/or Discrete Velocity Models.
Determining Planetary Temperatures with the Stefan-Boltzmann Law
ERIC Educational Resources Information Center
LoPresto, Michael C.; Hagoort, Nichole
2011-01-01
What follows is a description of several activities involving the Stefan-Boltzmann radiation law that can provide laboratory experience beyond what is normally found in traditional introductory thermodynamics experiments on thermal expansion, specific heat, and heats of transformation. The activities also provide more extensive coverage of and…
Multiscale lattice Boltzmann schemes for low Mach number flows.
Filippova, Olga; Schwade, Bettina; Hänel, Dieter
2002-03-15
A low Mach number approximation (LMNA) of the Navier-Stokes equations is widely used in numerical methods for the simulation of low-speed thermal and athermal flows. The advanced lattice Boltzmann approach (Bhatnagar-Gross-Krook) for the solution of the LMNA equations is discussed and its performance is compared with the performance of the commercial CFD code FLUENT 5.
A Parallel Lattice Boltzmann Model of a Carotid Artery
NASA Astrophysics Data System (ADS)
Boyd, J.; Ryan, S. J.; Buick, J. M.
2008-11-01
A parallel implementation of the lattice Boltzmann model is considered for a three dimensional model of the carotid artery. The computational method and its parallel implementation are described. The performance of the parallel implementation on a Beowulf cluster is presented, as are preliminary hemodynamic results.
An efficient learning procedure for deep Boltzmann machines.
Salakhutdinov, Ruslan; Hinton, Geoffrey
2012-08-01
We present a new learning algorithm for Boltzmann machines that contain many layers of hidden variables. Data-dependent statistics are estimated using a variational approximation that tends to focus on a single mode, and data-independent statistics are estimated using persistent Markov chains. The use of two quite different techniques for estimating the two types of statistic that enter into the gradient of the log likelihood makes it practical to learn Boltzmann machines with multiple hidden layers and millions of parameters. The learning can be made more efficient by using a layer-by-layer pretraining phase that initializes the weights sensibly. The pretraining also allows the variational inference to be initialized sensibly with a single bottom-up pass. We present results on the MNIST and NORB data sets showing that deep Boltzmann machines learn very good generative models of handwritten digits and 3D objects. We also show that the features discovered by deep Boltzmann machines are a very effective way to initialize the hidden layers of feedforward neural nets, which are then discriminatively fine-tuned.
An Updated Equilibrium Machine
ERIC Educational Resources Information Center
Schultz, Emeric
2008-01-01
A device that can demonstrate equilibrium, kinetic, and thermodynamic concepts is described. The device consists of a leaf blower attached to a plastic container divided into two chambers by a barrier of variable size and form. Styrofoam balls can be exchanged across the barrier when the leaf blower is turned on and various air pressures are…
Shi, Xing; Lin, Guang; Zou, Jianfeng; Fedosov, Dmitry A.
2013-07-20
To model red blood cell (RBC) deformation in flow, the recently developed LBM-DLM/FD method ([Shi and Lim, 2007)29], derived from the lattice Boltzmann method and the distributed Lagrange multiplier/fictitious domain methodthe fictitious domain method, is extended to employ the mesoscopic network model for simulations of red blood cell deformation. The flow is simulated by the lattice Boltzmann method with an external force, while the network model is used for modeling red blood cell deformation and the fluid-RBC interaction is enforced by the Lagrange multiplier. To validate parameters of the RBC network model, sThe stretching numerical tests on both coarse and fine meshes are performed and compared with the corresponding experimental data to validate the parameters of the RBC network model. In addition, RBC deformation in pipe flow and in shear flow is simulated, revealing the capacity of the current method for modeling RBC deformation in various flows.
A new splitting scheme to the discrete Boltzmann equation for non-ideal gases on non-uniform meshes
NASA Astrophysics Data System (ADS)
Patel, Saumil; Lee, Taehun
2016-12-01
We present a novel numerical procedure for solving the discrete Boltzmann equations (DBE) on non-uniform meshes. Our scheme is based on the Strang splitting method where we seek to investigate two-phase flow applications. In this note, we investigate the onset of parasitic currents which arise in many computational two-phase algorithms. To the best of our knowledge, the results presented in this work show, for the first time, a spectral element discontinuous Galerkin (SEDG) discretization of a discrete Boltzmann equation which successfully eliminates parasitic currents on non-uniform meshes. With the hope that this technique can be used for applications in complex geometries, calculations are performed on non-uniform mesh distributions by using high-order (spectral), body-fitting quadrilateral elements. Validation and verification of our work is carried out by comparing results against the classical 2D Young-Laplace law problem for a static drop.
Third-order analysis of pseudopotential lattice Boltzmann model for multiphase flow
NASA Astrophysics Data System (ADS)
Huang, Rongzong; Wu, Huiying
2016-12-01
In this work, a third-order Chapman-Enskog analysis of the multiple-relaxation-time (MRT) pseudopotential lattice Boltzmann (LB) model for multiphase flow is performed for the first time. The leading terms on the interaction force, consisting of an anisotropic and an isotropic term, are successfully identified in the third-order macroscopic equation recovered by the lattice Boltzmann equation (LBE), and then new mathematical insights into the pseudopotential LB model are provided. For the third-order anisotropic term, numerical tests show that it can cause the stationary droplet to become out-of-round, which suggests the isotropic property of the LBE needs to be seriously considered in the pseudopotential LB model. By adopting the classical equilibrium moment or setting the so-called "magic" parameter to 1/12, the anisotropic term can be eliminated, which is found from the present third-order analysis and also validated numerically. As for the third-order isotropic term, when and only when it is considered, accurate continuum form pressure tensor can be definitely obtained, by which the predicted coexistence densities always agree well with the numerical results. Compared with this continuum form pressure tensor, the classical discrete form pressure tensor is accurate only when the isotropic term is a specific one. At last, in the framework of the present third-order analysis, a consistent scheme for third-order additional term is proposed, which can be used to independently adjust the coexistence densities and surface tension. Numerical tests are subsequently carried out to validate the present scheme.
Lattice Boltzmann Methods for Fluid Structure Interaction
2012-09-01
benchmarks are published in [38]-[44]. The trailing vortex region for this benchmark was measured visually following flow simulation. Numeric results...matches well with experimentally measured values for vortex shedding for flow of a cyinder with Reynolds number equal to 200. F. HETEROGENEOUS PARALLEL...The mean field theory was introduced in [91] for non-ideal gas flow . In this method two distribution functions are used. The first distribution
On removal of charge singularity in Poisson-Boltzmann equation.
Cai, Qin; Wang, Jun; Zhao, Hong-Kai; Luo, Ray
2009-04-14
The Poisson-Boltzmann theory has become widely accepted in modeling electrostatic solvation interactions in biomolecular calculations. However the standard practice of atomic point charges in molecular mechanics force fields introduces singularity into the Poisson-Boltzmann equation. The finite-difference/finite-volume discretization approach to the Poisson-Boltzmann equation alleviates the numerical difficulty associated with the charge singularity but introduces discretization error into the electrostatic potential. Decomposition of the electrostatic potential has been explored to remove the charge singularity explicitly to achieve higher numerical accuracy in the solution of the electrostatic potential. In this study, we propose an efficient method to overcome the charge singularity problem. In our framework, two separate equations for two different potentials in two different regions are solved simultaneously, i.e., the reaction field potential in the solute region and the total potential in the solvent region. The proposed method can be readily implemented with typical finite-difference Poisson-Boltzmann solvers and return the singularity-free reaction field potential with a single run. Test runs on 42 small molecules and 4 large proteins show a very high agreement between the reaction field energies computed by the proposed method and those by the classical finite-difference Poisson-Boltzmann method. It is also interesting to note that the proposed method converges faster than the classical method, though additional time is needed to compute Coulombic potential on the dielectric boundary. The higher precision, accuracy, and efficiency of the proposed method will allow for more robust electrostatic calculations in molecular mechanics simulations of complex biomolecular systems.
Catt, B; Snyder, M
2014-06-15
Purpose: To investigate the use of the linear Boltzmann transport equation as a dose calculation tool which can account for interface effects, while still having faster computation times than Monte Carlo methods. In particular, we introduce a forward scattering approximation, in hopes of improving calculation time without a significant hindrance to accuracy. Methods: Two coupled Boltzmann transport equations were constructed, one representing the fluence of photons within the medium, and the other, the fluence of electrons. We neglect the scattering term within the electron transport equation, resulting in an extreme forward scattering approximation to reduce computational complexity. These equations were then solved using a numerical technique for solving partial differential equations, known as a finite difference scheme, where the fluence at each discrete point in space is calculated based on the fluence at the previous point in the particle's path. Using this scheme, it is possible to develop a solution to the Boltzmann transport equations by beginning with boundary conditions and iterating across the entire medium. The fluence of electrons can then be used to find the dose at any point within the medium. Results: Comparisons with Monte Carlo simulations indicate that even simplistic techniques for solving the linear Boltzmann transport equation yield expected interface effects, which many popular dose calculation algorithms are not capable of predicting. Implementation of a forward scattering approximation does not appear to drastically reduce the accuracy of this algorithm. Conclusion: Optimized implementations of this algorithm have been shown to be very accurate when compared with Monte Carlo simulations, even in build up regions where many models fail. Use of a forward scattering approximation could potentially give a reasonably accurate dose distribution in a shorter amount of time for situations where a completely accurate dose distribution is not
ON THE EQUILIBRIUM STRUCTURE OF SIMPLE LIQUIDS
It is shown that the repulsive (not merely the positive) portion of the Lennard - Jones potential quantitatively dominates the equilibrium structure of...the Lennard - Jones liquid. A simple and accurate approximation for the radial distribution function at high densities is presented.
Equilibrium Molecular Thermodynamics from Kirkwood Sampling
2015-01-01
We present two methods for barrierless equilibrium sampling of molecular systems based on the recently proposed Kirkwood method (J. Chem. Phys.2009, 130, 134102). Kirkwood sampling employs low-order correlations among internal coordinates of a molecule for random (or non-Markovian) sampling of the high dimensional conformational space. This is a geometrical sampling method independent of the potential energy surface. The first method is a variant of biased Monte Carlo, where Kirkwood sampling is used for generating trial Monte Carlo moves. Using this method, equilibrium distributions corresponding to different temperatures and potential energy functions can be generated from a given set of low-order correlations. Since Kirkwood samples are generated independently, this method is ideally suited for massively parallel distributed computing. The second approach is a variant of reservoir replica exchange, where Kirkwood sampling is used to construct a reservoir of conformations, which exchanges conformations with the replicas performing equilibrium sampling corresponding to different thermodynamic states. Coupling with the Kirkwood reservoir enhances sampling by facilitating global jumps in the conformational space. The efficiency of both methods depends on the overlap of the Kirkwood distribution with the target equilibrium distribution. We present proof-of-concept results for a model nine-atom linear molecule and alanine dipeptide. PMID:25915525
Equilibrium molecular thermodynamics from Kirkwood sampling.
Somani, Sandeep; Okamoto, Yuko; Ballard, Andrew J; Wales, David J
2015-05-21
We present two methods for barrierless equilibrium sampling of molecular systems based on the recently proposed Kirkwood method (J. Chem. Phys. 2009, 130, 134102). Kirkwood sampling employs low-order correlations among internal coordinates of a molecule for random (or non-Markovian) sampling of the high dimensional conformational space. This is a geometrical sampling method independent of the potential energy surface. The first method is a variant of biased Monte Carlo, where Kirkwood sampling is used for generating trial Monte Carlo moves. Using this method, equilibrium distributions corresponding to different temperatures and potential energy functions can be generated from a given set of low-order correlations. Since Kirkwood samples are generated independently, this method is ideally suited for massively parallel distributed computing. The second approach is a variant of reservoir replica exchange, where Kirkwood sampling is used to construct a reservoir of conformations, which exchanges conformations with the replicas performing equilibrium sampling corresponding to different thermodynamic states. Coupling with the Kirkwood reservoir enhances sampling by facilitating global jumps in the conformational space. The efficiency of both methods depends on the overlap of the Kirkwood distribution with the target equilibrium distribution. We present proof-of-concept results for a model nine-atom linear molecule and alanine dipeptide.
NASA Astrophysics Data System (ADS)
Bernhoff, N.
2012-11-01
Half-space problems for the Boltzmann equation are of great importance in the study of the asymptotic behavior of the solutions of boundary value problems of the Boltzmann equation for small Knudsen numbers. Half-space problems provide the boundary conditions for the fluid-dynamic-type equations and Knudsen-layer corrections to the solution of the fluid-dynamic-type equations in a neighborhood of the boundary. Here we consider a half-space problem of condensation for a pure vapor in the presence of a non-condensable gas by using discrete velocity models (DVMs) of the Boltzmann equation. The Boltzmann equation can be approximated by DVMs up to any order, and these DVMs can be applied for numerical methods, but also for mathematical studies to bring deeper understanding and new ideas. For one-dimensional half-space problems, the discrete Boltzmann equation (the general DVM) reduces to a system of ODEs. We obtain that the number of parameters to be specified in the boundary conditions depends on whether the condensing vapor flow is subsonic or supersonic. This behavior has earlier been found numerically. We want to stress that our results are valid for any finite number of velocities. This is an extension of known results for single-component gases (and for binary mixtures of two vapors) to the case when a non-condensable gas is present. The vapor is assumed to tend to an assigned Maxwellian, with a flow velocity towards the condensed phase, at infinity, while the non-condensable gas tends to zero at infinity. Steady condensation of the vapor takes place at the condensed phase, which is held at a constant temperature. We assume that the vapor is completely absorbed, that the non-condensable gas is diffusively reflected at the condensed phase, and that vapor molecules leaving the condensed phase are distributed according to a given distribution. The conditions, on the given distribution at the condensed phase, needed for the existence of a unique solution of the
Two-Class Structure of Income Distribution in the Usa:. Exponential Bulk and Power-Law Tail
NASA Astrophysics Data System (ADS)
Yakovenko, V. M.; Silva, A. Christian
2007-07-01
Personal income distribution in the USA has a well-defined two-class structure. The majority of population (97-99 %) belongs to the lower class characterized by the exponential Boltzmann-Gibbs ("thermal") distribution, whereas the upper class (1-3 % of population) has a Pareto power-law ("superthermal") distribution. By analyzing income data for 1983-2001, we show that the "thermal" part is stationary in time, save for a gradual increase of the effective temperature, whereas the "superthermal" tail swells and shrinks following the stock market. We discuss the concept of equilibrium inequality in a society, based on the principle of maximal entropy, and quantitatively show that it applies to the majority of population.
Non-Equilibrium Molecular Dynamics
NASA Astrophysics Data System (ADS)
Ciccotti, Giovanni; Kapral, Raymond; Sergi, Alessandro
Statistical mechanics provides a well-established link between microscopic equilibrium states and thermodynamics. If one considers systems out of equilibrium, the link between microscopic dynamical properties and non-equilibrium macroscopic states is more difficult to establish [1,2]. For systems lying near equilibrium, linear response theory provides a route to derive linear macroscopic laws and the microscopic expressions for the transport properties that enter the constitutive relations. If the system is displaced far from equilibrium, no fully general theory exists to treat such systems. By restricting consideration to a class of non-equilibrium states which arise from perturbations (linear or non-linear) of an equilibrium state, methods can be developed to treat non-equilibrium states. Furthermore, non-equilibrium molecular dynamics (NEMD) simulation methods can be devised to provide estimates for the transport properties of these systems.
The electron Boltzmann equation in a plasma generated by fission fragments
NASA Technical Reports Server (NTRS)
Hassan, H. A.; Deese, J. E.
1976-01-01
A Boltzmann equation formulation is presented for the determination of the electron distribution function in a plasma generated by fission fragments. The formulation takes into consideration ambipolar diffusion, elastic and inelastic collisions, recombination and ionization, and allows for the fact that the primary electrons are not monoenergetic. Calculations for He in a tube coated with fissionable material show that, over a wide pressure and neutron flux range, the distribution function is non-Maxwellian, but the electrons are essentially thermal. Moreover, about a third of the energy of the primary electrons is transferred into the inelastic levels of He. This fraction of energy transfer is almost independent of pressure and neutron flux but increases sharply in the presence of a sustainer electric field.
NASA Astrophysics Data System (ADS)
Kang, XiuYing
2015-01-01
By using the lattice Boltzmann method (LBM) pulsatile blood flows were simulated in three-dimensional moderate stenosed and recanalized carotid bifurcations to understand local hemodynamics and its relevance in arterial atherosclerosis formation and progression. The helical flow patterns, secondary flow and wall dynamical pressure spatiotemporal distributions were investigated, which leads to the disturbed shear forces in the carotid artery bifurcations. The wall shear stress distributions indicated by time-averaged wall shear stress (TAWSS), oscillatory shear index (OSI), and the relative residence time (RRT) in a cardiac cycle revealed the regions where atherosclerotic plaques are prone to form, extend or rupture. This study also illustrates the point that locally disturbed flow may be considered as an indicator for early atherosclerosis diagnosis. Additionally the present work demonstrates the robust and highly efficient advantages of the LBM for the hemodynamics study of the human blood vessel system.
Lattice Boltzmann technique for heat transport phenomena coupled with melting process
NASA Astrophysics Data System (ADS)
Ibrahem, A. M.; El-Amin, M. F.; Mohammadein, A. A.; Gorla, Rama Subba Reddy
2017-01-01
In this work, the heat transport phenomena coupled with melting process are studied by using the enthalpy-based lattice Boltzmann method (LBM). The proposed model is a modified version of thermal LB model, where could avoid iteration steps and ensures high accuracy. The Bhatnagar-Gross-Krook (BGK) approximation with a D1Q2 lattice was used to determine the temperature field for one-dimensional melting by conduction and multi-distribution functions (MDF) with D2Q9 lattice was used to determine the density, velocity and temperature fields for two-dimensional melting by natural convection. Different boundary conditions including Dirichlet, adiabatic and bounce-back boundary conditions were used. The influence of increasing Rayleigh number (from 103 to 105) on temperature distribution and melting process is studied. The obtained results show that a good agreement with the analytical solution for melting by conduction case and with the benchmark solution for melting by convection.
Montufar, Guido; Ay, Nihat
2011-05-01
We improve recently published results about resources of restricted Boltzmann machines (RBM) and deep belief networks (DBN)required to make them universal approximators. We show that any distribution pon the set {0,1}(n) of binary vectors of length n can be arbitrarily well approximated by an RBM with k-1 hidden units, where k is the minimal number of pairs of binary vectors differing in only one entry such that their union contains the support set of p. In important cases this number is half the cardinality of the support set of p (given in Le Roux & Bengio, 2008). We construct a DBN with 2n/ 2(n-b) , b ∼ log n, hidden layers of width n that is capable of approximating any distribution on {0,1}(n) arbitrarily well. This confirms a conjecture presented in Le Roux and Bengio (2010).
Molecular velocity distributions and generalized scale invariance in the turbulent atmosphere.
Tuck, Adrian F; Hovde, Susan J; Richard, Erik C; Gao, Ru-Shan; Bui, T Paul; Swartz, William H; Lloyd, Steven A
2005-01-01
Airborne observations of ozone, temperature and the spectral actinic photon flux for ozone in the Arctic lower stratosphere April-September 1997 and January-March 2000 allow a connection to be made between the rate of production of translationally hot atoms and molecules via ozone photodissociation and the intermittency of temperature. Seen in the context of non-equilibrium statistical mechanics literature results from molecular dynamics simulations, the observed correlation between the molecular scale production of translationally hot atoms and molecules and the macroscopic fluid mechanical intermittency of temperature may imply a departure from Maxwell-Boltzmann distributions of molecular velocities, with consequences for chemistry, radiative line shapes and turbulence in the atmosphere, arising from overpopulated high velocity tails of the probability distribution functions (PDFs).
NASA Technical Reports Server (NTRS)
Metzger, Philip T.
2006-01-01
Ergodicity is proved for granular contact forces. To obtain this proof from first principles, this paper generalizes Boltzmann's stosszahlansatz (molecular chaos) so that it maintains the necessary correlations and symmetries of granular packing ensembles. Then it formally counts granular contact force states and thereby defines the proper analog of Boltzmann's H functional. This functional is used to prove that (essentially) all static granular packings must exist at maximum entropy with respect to their contact forces. Therefore, the propagation of granular contact forces through a packing is a truly ergodic process in the Boltzmannian sense, or better, it is self-ergodic. Self-ergodicity refers to the non-dynamic, internal relationships that exist between the layer-by-layer and column-by-column subspaces contained within the phase space locus of any particular granular packing microstate. The generalized H Theorem also produces a recursion equation that may be solved numerically to obtain the density of single particle states and hence the distribution of granular contact forces corresponding to the condition of self-ergodicity. The predictions of the theory are overwhelmingly validated by comparison to empirical data from discrete element modeling.
An improved lattice Boltzmann method for simulating advective-diffusive processes in fluids
NASA Astrophysics Data System (ADS)
Aursjø, Olav; Jettestuen, Espen; Vinningland, Jan Ludvig; Hiorth, Aksel
2017-03-01
Lattice Boltzmann methods are widely used to simulate advective-diffusive processes in fluids. Lattice Bhatnagar-Gross-Krook methods presented in the literature mostly just exhibit first order spatial accuracy and introduce errors proportional to the velocity squared. Formulations proposed to alleviate this have only been partly successful and are valid only in certain specific situations. We present and demonstrate here a formulation that produces no such second order errors. This formulation suggests that a subtle, but important, adjustment is all it takes to improve the accuracy of the method. The key to the improved accuracy of this new model is the non-standard definition of the concentration that relates to the distribution function describing the advection-diffusion in lattice Boltzmann. The main advantage of the algorithm comes to view when simulating situations where fluid density variations appear. The present formulation of the advection-diffusion algorithm will, by taking into account these fluid density variations, drastically reduce the errors produced compared to the standard formulations. We also show how a source term is included in this new formulation without it losing its second order spatial accuracy.
NASA Technical Reports Server (NTRS)
Shebalin, John V.
1997-01-01
The entropy associated with absolute equilibrium ensemble theories of ideal, homogeneous, fluid and magneto-fluid turbulence is discussed and the three-dimensional fluid case is examined in detail. A sigma-function is defined, whose minimum value with respect to global parameters is the entropy. A comparison is made between the use of global functions sigma and phase functions H (associated with the development of various H-theorems of ideal turbulence). It is shown that the two approaches are complimentary though conceptually different: H-theorems show that an isolated system tends to equilibrium while sigma-functions allow the demonstration that entropy never decreases when two previously isolated systems are combined. This provides a more complete picture of entropy in the statistical mechanics of ideal fluids.
NASA Astrophysics Data System (ADS)
Godrèche, C.
2011-03-01
Preface; 1. Shape and growth of crystals P. Nozières; 2. Instabilities of planar solidification fronts B. Caroli, C. Caroli and B. Roulet; 3. An introduction to the kinetics of first-order phase transition J. S. Langer; 4. Dendritic growth and related topics Y. Pomeau and M. Ben Amar; 5. Growth and aggregation far from equilibrium L. M. Sander; 6. Kinetic roughening of growing surfaces J. Krug and H. Spohn; Acknowledgements; References; Index.
Molecular equilibrium with condensation
NASA Astrophysics Data System (ADS)
Sharp, C. M.; Huebner, W. F.
1990-02-01
Minimization of the Gibbs energy of formation for species of chemical elements and compounds in their gas and condensed phases determines their relative abundances in a mixture in chemical equilibrium. The procedure is more general and more powerful than previous abundance determinations in multiphase astrophysical mixtures. Some results for astrophysical equations of state are presented, and the effects of condensation on opacity are briefly indicated.
Nicholls, David C.; Dopita, Michael A.; Sutherland, Ralph S.
2012-06-20
The measurement of electron temperatures and metallicities in H II regions and planetary nebulae (PNe) has-for several decades-presented a problem: results obtained using different techniques disagree. What is worse, they disagree consistently. There have been numerous attempts to explain these discrepancies, but none has provided a satisfactory solution to the problem. In this paper, we explore the possibility that electrons in H II regions and PNe depart from a Maxwell-Boltzmann equilibrium energy distribution. We adopt a '{kappa}-distribution' for the electron energies. Such distributions are widely found in solar system plasmas, where they can be directly measured. This simple assumption is able to explain the temperature and metallicity discrepancies in H II regions and PNe arising from the different measurement techniques. We find that the energy distribution does not need to depart dramatically from an equilibrium distribution. From an examination of data from H II regions and PNe, it appears that {kappa} {approx}> 10 is sufficient to encompass nearly all objects. We argue that the kappa-distribution offers an important new insight into the physics of gaseous nebulae, both in the Milky Way and elsewhere, and one that promises significantly more accurate estimates of temperature and metallicity in these regions.
Equilibrium Electroconvective Instability
NASA Astrophysics Data System (ADS)
Rubinstein, I.; Zaltzman, B.
2015-03-01
Since its prediction 15 years ago, hydrodynamic instability in concentration polarization at a charge-selective interface has been attributed to nonequilibrium electro-osmosis related to the extended space charge which develops at the limiting current. This attribution had a double basis. On the one hand, it has been recognized that neither equilibrium electro-osmosis nor bulk electroconvection can yield instability for a perfectly charge-selective solid. On the other hand, it has been shown that nonequilibrium electro-osmosis can. The first theoretical studies in which electro-osmotic instability was predicted and analyzed employed the assumption of perfect charge selectivity for the sake of simplicity and so did the subsequent studies of various time-dependent and nonlinear features of electro-osmotic instability. In this Letter, we show that relaxing the assumption of perfect charge selectivity (tantamount to fixing the electrochemical potential of counterions in the solid) allows for the equilibrium electroconvective instability. In addition, we suggest a simple experimental test for determining the true, either equilibrium or nonequilibrium, origin of instability in concentration polarization.
NASA Astrophysics Data System (ADS)
Bhakta, Subrata; Ghosh, Uttam; Sarkar, Susmita
2017-02-01
In this paper, we have investigated the effect of secondary electron emission on nonlinear propagation of dust acoustic waves in a complex plasma where equilibrium dust charge is negative. The primary electrons, secondary electrons, and ions are Boltzmann distributed, and only dust grains are inertial. Electron-neutral and ion-neutral collisions have been neglected with the assumption that electron and ion mean free paths are very large compared to the plasma Debye length. Both adiabatic and nonadiabatic dust charge variations have been separately taken into account. In the case of adiabatic dust charge variation, nonlinear propagation of dust acoustic waves is governed by the KdV (Korteweg-de Vries) equation, whereas for nonadiabatic dust charge variation, it is governed by the KdV-Burger equation. The solution of the KdV equation gives a dust acoustic soliton, whose amplitude and width depend on the secondary electron yield. Similarly, the KdV-Burger equation provides a dust acoustic shock wave. This dust acoustic shock wave may be monotonic or oscillatory in nature depending on the fact that whether it is dissipation dominated or dispersion dominated. Our analysis shows that secondary electron emission increases nonadiabaticity induced dissipation and consequently increases the monotonicity of the dust acoustic shock wave. Such a dust acoustic shock wave may accelerate charge particles and cause bremsstrahlung radiation in space plasmas whose physical process may be affected by secondary electron emission from dust grains. The effect of the secondary electron emission on the stability of the equilibrium points of the KdV-Burger equation has also been investigated. This equation has two equilibrium points. The trivial equilibrium point with zero potential is a saddle and hence unstable in nature. The nontrivial equilibrium point with constant nonzero potential is a stable node up to a critical value of the wave velocity and a stable focus above it. This critical
Permeability of Partially Molten Rocks from Lattice-Boltzmann Modeling
NASA Astrophysics Data System (ADS)
Garapic, G.; Faul, U.
2013-12-01
Timescales of melt transport at mid-ocean ridges from mantle source to the surface depend on permeability of the partially molten mantle. The permeability is usually predicted indirectly from experimental observations based on porosities that are much higher than the porosities inferred for the partially molten mantle. Low porosities are for example predicted by geochemical models from the onset of melt migration. Since melting starts at the grain scale, permeability of the partially molten mantle will depend on the grain-scale melt distribution. We reconstructed a 3-D view of melt geometry of two experimentally produced samples of partially molten olivine which demonstrates that melt exists in thin layers on two-grain boundaries (Garapić et al.,G3, 2013). The wetted two-grain boundaries have a width about 100 times smaller than the average grain size. Additionally, the pore space consists of a network of triple-junction tubules at all porosities, and large 'melt pools'. Due to the relative size of the wetted two-grain boundaries as well as the size of the triple-junction network compared to the grain size imagining and numerical analyses of partially molten samples require high resolution. Since no direct experimental permeability measurements are possible on partially molten aggregates, we investigate numerically the permeability as a function of porosity for this system. We simulate porous flow through an artificial pore volume using the lattice-Boltzmann method (LBM) and Palabos LB code. Flow simulations were done on a computer cluster on three or four 125 GB nodes with 16 processors per node. With the available memory and allowed run time the maximum size of our pore structure was 1100 voxels per edge. In its simplest form the pore structure consists of a network of cylinders within a matrix of cubic grains. To approximate the observed 3-D melt geometry we added randomly distributed sheets on cube faces ('wetted two-grain boundaries') as well as randomly
NASA Astrophysics Data System (ADS)
Dellar, Paul
2016-11-01
We present discrete kinetic and lattice Boltzmann formulations for reaction cross-diffusion systems, as commonly used to model microbiological chemotaxis and macroscopic predator-prey interactions, and their hyperbolic extensions with fluid-like persistence terms. For example, the canonical Patlak-Keller-Segal model for chemotaxis involves a flux of cells up the gradient of a chemical secreted by the cells, in addition to the usual down-gradient diffusive fluxes. Existing lattice Boltzmann approaches for such systems use finite difference approximations to compute the flux of cells due to the chemical gradient. The resulting coupling between, and necessary synchronisation of the evolution of, adjacent grid points greatly complicates boundary conditions, and efficient implementation on graphical processing units (GPUs). We present a kinetic formulation using cross-collisions between bases of moments for the two sets of distribution functions to couple the fluxes of the two species, from which we construct lattice Boltzmann algorithms using second-order Strang splitting. We demonstrate an efficient GPU implementation, and verify second-order spatial convergence towards spectral solutions for benchmark problems such as the finite-time blow-up in the Patlak-Keller-Segal model.
NASA Astrophysics Data System (ADS)
Fukuda, Ikuo
2010-03-01
A brief discussion of the ergodic description of constant temperature molecular dynamics (MD) is provided; the discussion is based on the analysis of criticisms raised in a recent paper [B. Cooke and S. C. Schmidler, J. Chem. Phys.129, 164112 (2008)]. In the paper, the following criticisms relating to the basic concepts of constant temperature MD are made in mathematical manners: (I) the Nosé-Hoover (NH) equation is not measure-preserving; (II) NH system and NH chain system are not ergodic under the Boltzmann measure; and (III) the Nosé Hamiltonian system as well as the Nosé-Poincaré Hamiltonian system is not ergodic. In this comment, I show the necessity for the reconsideration of these criticisms. The NH equation is measure-preserving, where the measure carries the Boltzmann-Gibbs density; this fact provides the compatibility between MD equation and the Boltzmann-Gibbs distribution. The arguments advanced in support of the above criticisms are unsound; ergodicities of those systems are still not theoretically judged. I discuss exact ergodic-theoretical expressions appropriate for constant temperature MD, and explain the reason behind the incorrect recognitions.
Lattice Boltzmann Representations of MHD Turbulence
NASA Astrophysics Data System (ADS)
Vahala, George; Vahala, Linda; Soe, Min; Flint, Christopher
2013-10-01
Lattice Botlzmann algorithms are an ideally parallelized method for the solutions of macroscopic nonlinear equations of physics - like resistive MHD. In its simplest LB representation one introduces a scalar distribution for the density-velocity fields and a vector distribution for the magnetic field. An important feature is that gradients of certain macroscopic fields can be represented by local moments of the mesoscopic distribution functions. In particular, div B = 0 can be exactly enforced to machine accuracy, without any divergence cleaning. One of the problems facing the explicit LB code is numerical instabilities. Methods to permit strong turbulence simulations include: (a) moving from a single BGK to multiple collisional relaxation, (b) quasi-equilibria and central moment enhanced LB representations. The LB turbulence modeling of Ansumali et al. to Navier-Stokes turbulence will be extended to MHD in which in its noted that filtering and Chapman-Enskog limits do not commute. In the NS-case, it leads to unique Samgorinsky closure scheme, with definite filter width.
On the Boltzmann relation in a cold magnetized plasma
Nasi, L.; Raimbault, J.-L.
2010-11-15
A systematic and exact comparison between the forces acting on magnetized electrons in a current-free plasma is considered within a fluid model. We show that the Boltzmann relation is fulfilled in the drift-diffusion approximation when (h{sub i}/h{sub e})(1+h{sub e}{sup 2})/(1+h{sub i}{sup 2})<<1 where h{sub e} (or h{sub i}) is the ratio of the electron (or ion) cyclotron to the collision frequency. When the nonlinear inertia terms are taken into account, the previous criterion is too rough and must be modified. In particular it is proved that the Boltzmann relation is not uniformly valid in the plasma. The case of bounded plasmas where the electron temperature must be determined self-consistently is discussed in detail.
Shock-wave structure using nonlinear model Boltzmann equations.
NASA Technical Reports Server (NTRS)
Segal, B. M.; Ferziger, J. H.
1972-01-01
The structure of strong plane shock waves in a perfect monatomic gas was studied using four nonlinear models of the Boltzmann equation. The models involved the use of a simplified collision operator with velocity-independent collision frequency, in place of the complicated Boltzmann collision operator. The models employed were the BGK and ellipsoidal models developed by earlier authors, and the polynomial and trimodal gain function models developed during the work. An exact set of moment equations was derived for the density, velocity, temperature, viscous stress, and heat flux within the shock. This set was reduced to a pair of coupled nonlinear integral equations and solved using specially adapted numerical techniques. A new and simple Gauss-Seidel iteration was developed during the work and found to be as efficient as the best earlier iteration methods.
A Boltzmann machine for the organization of intelligent machines
NASA Technical Reports Server (NTRS)
Moed, Michael C.; Saridis, George N.
1990-01-01
A three-tier structure consisting of organization, coordination, and execution levels forms the architecture of an intelligent machine using the principle of increasing precision with decreasing intelligence from a hierarchically intelligent control. This system has been formulated as a probabilistic model, where uncertainty and imprecision can be expressed in terms of entropies. The optimal strategy for decision planning and task execution can be found by minimizing the total entropy in the system. The focus is on the design of the organization level as a Boltzmann machine. Since this level is responsible for planning the actions of the machine, the Boltzmann machine is reformulated to use entropy as the cost function to be minimized. Simulated annealing, expanding subinterval random search, and the genetic algorithm are presented as search techniques to efficiently find the desired action sequence and illustrated with numerical examples.
The Lattice Boltzmann Method applied to neutron transport
Erasmus, B.; Van Heerden, F. A.
2013-07-01
In this paper the applicability of the Lattice Boltzmann Method to neutron transport is investigated. One of the main features of the Lattice Boltzmann method is the simultaneous discretization of the phase space of the problem, whereby particles are restricted to move on a lattice. An iterative solution of the operator form of the neutron transport equation is presented here, with the first collision source as the starting point of the iteration scheme. A full description of the discretization scheme is given, along with the quadrature set used for the angular discretization. An angular refinement scheme is introduced to increase the angular coverage of the problem phase space and to mitigate lattice ray effects. The method is applied to a model problem to investigate its applicability to neutron transport and the results are compared to a reference solution calculated, using MCNP. (authors)
Pointwise Description for the Linearized Fokker-Planck-Boltzmann Model
NASA Astrophysics Data System (ADS)
Wu, Kung-Chien
2015-09-01
In this paper, we study the pointwise (in the space variable) behavior of the linearized Fokker-Planck-Boltzmann model for nonsmooth initial perturbations. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as the long-wave expansion in the spectrum of the Fourier modes for the space variable, and it has polynomial time decay rate. We design a Picard-type iteration for constructing the increasingly regular kinetic-like waves, which are carried by the transport equations and have exponential time decay rate. The Mixture Lemma plays an important role in constructing the kinetic-like waves, this lemma was originally introduced by Liu-Yu (Commun Pure Appl Math 57:1543-1608, 2004) for Boltzmann equation, but the Fokker-Planck term in this paper creates some technical difficulties.
Accelerated Monte Carlo simulations with restricted Boltzmann machines
NASA Astrophysics Data System (ADS)
Huang, Li; Wang, Lei
2017-01-01
Despite their exceptional flexibility and popularity, Monte Carlo methods often suffer from slow mixing times for challenging statistical physics problems. We present a general strategy to overcome this difficulty by adopting ideas and techniques from the machine learning community. We fit the unnormalized probability of the physical model to a feed-forward neural network and reinterpret the architecture as a restricted Boltzmann machine. Then, exploiting its feature detection ability, we utilize the restricted Boltzmann machine to propose efficient Monte Carlo updates to speed up the simulation of the original physical system. We implement these ideas for the Falicov-Kimball model and demonstrate an improved acceptance ratio and autocorrelation time near the phase transition point.
Structural design using equilibrium programming
NASA Technical Reports Server (NTRS)
Scotti, Stephen J.
1992-01-01
Multiple nonlinear programming methods are combined in the method of equilibrium programming. Equilibrium programming theory has been appied to problems in operations research, and in the present study it is investigated as a framework to solve structural design problems. Several existing formal methods for structural optimization are shown to actually be equilibrium programming methods. Additionally, the equilibrium programming framework is utilized to develop a new structural design method. Selected computational results are presented to demonstrate the methods.
Local thermodynamic equilibrium in rapidly heated high energy density plasmas
Aslanyan, V.; Tallents, G. J.
2014-06-15
Emission spectra and the dynamics of high energy density plasmas created by optical and Free Electron Lasers (FELs) depend on the populations of atomic levels. Calculations of plasma emission and ionization may be simplified by assuming Local Thermodynamic Equilibrium (LTE), where populations are given by the Saha-Boltzmann equation. LTE can be achieved at high densities when collisional processes are much more significant than radiative processes, but may not be valid if plasma conditions change rapidly. A collisional-radiative model has been used to calculate the times taken by carbon and iron plasmas to reach LTE at varying densities and heating rates. The effect of different energy deposition methods, as well as Ionization Potential Depression are explored. This work shows regimes in rapidly changing plasmas, such as those created by optical lasers and FELs, where the use of LTE is justified, because timescales for plasma changes are significantly longer than the times needed to achieve an LTE ionization balance.
Pomeau, Yves; Louët, Sabine
2016-06-01
During the StatPhys Conference on 20th July 2016 in Lyon, France, Yves Pomeau and Daan Frenkel will be awarded the most important prize in the field of Statistical Mechanics: the 2016 Boltzmann Medal, named after the Austrian physicist and philosopher Ludwig Boltzmann. The award recognises Pomeau's key contributions to the Statistical Physics of non-equilibrium phenomena in general. And, in particular, for developing our modern understanding of fluid mechanics, instabilities, pattern formation and chaos. He is recognised as an outstanding theorist bridging disciplines from applied mathematics to statistical physics with a profound impact on the neighbouring fields of turbulence and mechanics. In the article Sabine Louët interviews Pomeau, who is an Editor for the European Physical Journal Special Topics. He shares his views and tells how he experienced the rise of Statistical Mechanics in the past few decades. He also touches upon the need to provide funding to people who have the rare ability to discover new things and ideas, and not just those who are good at filling in grant application forms.
Lattice Boltzmann Simulation of Particle Laden Flows in Microfluidic Systems
2003-12-01
wide application and will enable the study of colloidal/macromolecular transport in physiological systems, such as, blood filtration in the kidney... MICROFLUIDIC SYSTEMS DOE/Lawrence Livermore National Laboratory Sponsored by Defense Advanced Research Projects Agency DARPA Order No. E117...Jun 00 – Aug 02 4. TITLE AND SUBTITLE LATTICE BOLTZMANN SIMULATION OF PARTICLE LADEN FLOWS IN MICROFLUIDIC SYSTEMS 6. AUTHOR(S) David S
Accounting for adsorption and desorption in lattice Boltzmann simulations
NASA Astrophysics Data System (ADS)
Levesque, Maximilien; Duvail, Magali; Pagonabarraga, Ignacio; Frenkel, Daan; Rotenberg, Benjamin
2013-07-01
We report a Lattice-Boltzmann scheme that accounts for adsorption and desorption in the calculation of mesoscale dynamical properties of tracers in media of arbitrary complexity. Lattice Boltzmann simulations made it possible to solve numerically the coupled Navier-Stokes equations of fluid dynamics and Nernst-Planck equations of electrokinetics in complex, heterogeneous media. With the moment propagation scheme, it became possible to extract the effective diffusion and dispersion coefficients of tracers, or solutes, of any charge, e.g., in porous media. Nevertheless, the dynamical properties of tracers depend on the tracer-surface affinity, which is not purely electrostatic and also includes a species-specific contribution. In order to capture this important feature, we introduce specific adsorption and desorption processes in a lattice Boltzmann scheme through a modified moment propagation algorithm, in which tracers may adsorb and desorb from surfaces through kinetic reaction rates. The method is validated on exact results for pure diffusion and diffusion-advection in Poiseuille flows in a simple geometry. We finally illustrate the importance of taking such processes into account in the time-dependent diffusion coefficient in a more complex porous medium.
A Poisson-Boltzmann dynamics method with nonperiodic boundary condition
NASA Astrophysics Data System (ADS)
Lu, Qiang; Luo, Ray
2003-12-01
We have developed a well-behaved and efficient finite difference Poisson-Boltzmann dynamics method with a nonperiodic boundary condition. This is made possible, in part, by a rather fine grid spacing used for the finite difference treatment of the reaction field interaction. The stability is also made possible by a new dielectric model that is smooth both over time and over space, an important issue in the application of implicit solvents. In addition, the electrostatic focusing technique facilitates the use of an accurate yet efficient nonperiodic boundary condition: boundary grid potentials computed by the sum of potentials from individual grid charges. Finally, the particle-particle particle-mesh technique is adopted in the computation of the Coulombic interaction to balance accuracy and efficiency in simulations of large biomolecules. Preliminary testing shows that the nonperiodic Poisson-Boltzmann dynamics method is numerically stable in trajectories at least 4 ns long. The new model is also fairly efficient: it is comparable to that of the pairwise generalized Born solvent model, making it a strong candidate for dynamics simulations of biomolecules in dilute aqueous solutions. Note that the current treatment of total electrostatic interactions is with no cutoff, which is important for simulations of biomolecules. Rigorous treatment of the Debye-Hückel screening is also possible within the Poisson-Boltzmann framework: its importance is demonstrated by a simulation of a highly charged protein.
Assessment of Linear Finite-Difference Poisson-Boltzmann Solvers
Wang, Jun; Luo, Ray
2009-01-01
CPU time and memory usage are two vital issues that any numerical solvers for the Poisson-Boltzmann equation have to face in biomolecular applications. In this study we systematically analyzed the CPU time and memory usage of five commonly used finite-difference solvers with a large and diversified set of biomolecular structures. Our comparative analysis shows that modified incomplete Cholesky conjugate gradient and geometric multigrid are the most efficient in the diversified test set. For the two efficient solvers, our test shows that their CPU times increase approximately linearly with the numbers of grids. Their CPU times also increase almost linearly with the negative logarithm of the convergence criterion at very similar rate. Our comparison further shows that geometric multigrid performs better in the large set of tested biomolecules. However, modified incomplete Cholesky conjugate gradient is superior to geometric multigrid in molecular dynamics simulations of tested molecules. We also investigated other significant components in numerical solutions of the Poisson-Boltzmann equation. It turns out that the time-limiting step is the free boundary condition setup for the linear systems for the selected proteins if the electrostatic focusing is not used. Thus, development of future numerical solvers for the Poisson-Boltzmann equation should balance all aspects of the numerical procedures in realistic biomolecular applications. PMID:20063271
Entropic multirelaxation lattice Boltzmann models for turbulent flows
NASA Astrophysics Data System (ADS)
Bösch, Fabian; Chikatamarla, Shyam S.; Karlin, Ilya V.
2015-10-01
We present three-dimensional realizations of a class of lattice Boltzmann models introduced recently by the authors [I. V. Karlin, F. Bösch, and S. S. Chikatamarla, Phys. Rev. E 90, 031302(R) (2014), 10.1103/PhysRevE.90.031302] and review the role of the entropic stabilizer. Both coarse- and fine-grid simulations are addressed for the Kida vortex flow benchmark. We show that the outstanding numerical stability and performance is independent of a particular choice of the moment representation for high-Reynolds-number flows. We report accurate results for low-order moments for homogeneous isotropic decaying turbulence and second-order grid convergence for most assessed statistical quantities. It is demonstrated that all the three-dimensional lattice Boltzmann realizations considered herein converge to the familiar lattice Bhatnagar-Gross-Krook model when the resolution is increased. Moreover, thanks to the dynamic nature of the entropic stabilizer, the present model features less compressibility effects and maintains correct energy and enstrophy dissipation. The explicit and efficient nature of the present lattice Boltzmann method renders it a promising candidate for both engineering and scientific purposes for highly turbulent flows.
Low uncertainty Boltzmann constant determinations and the kelvin redefinition.
Fischer, J
2016-03-28
At its 25th meeting, the General Conference on Weights and Measures (CGPM) approved Resolution 1 'On the future revision of the International System of Units, the SI', which sets the path towards redefinition of four base units at the next CGPM in 2018. This constitutes a decisive advance towards the formal adoption of the new SI and its implementation. Kilogram, ampere, kelvin and mole will be defined in terms of fixed numerical values of the Planck constant, elementary charge, Boltzmann constant and Avogadro constant, respectively. The effect of the new definition of the kelvin referenced to the value of the Boltzmann constant k is that the kelvin is equal to the change of thermodynamic temperature T that results in a change of thermal energy kT by 1.380 65×10(-23) J. A value of the Boltzmann constant suitable for defining the kelvin is determined by fundamentally different primary thermometers such as acoustic gas thermometers, dielectric constant gas thermometers, noise thermometers and the Doppler broadening technique. Progress to date of the measurements and further perspectives are reported. Necessary conditions to be met before proceeding with changing the definition are given. The consequences of the new definition of the kelvin on temperature measurement are briefly outlined.
Entropic multirelaxation lattice Boltzmann models for turbulent flows.
Bösch, Fabian; Chikatamarla, Shyam S; Karlin, Ilya V
2015-10-01
We present three-dimensional realizations of a class of lattice Boltzmann models introduced recently by the authors [I. V. Karlin, F. Bösch, and S. S. Chikatamarla, Phys. Rev. E 90, 031302(R) (2014)] and review the role of the entropic stabilizer. Both coarse- and fine-grid simulations are addressed for the Kida vortex flow benchmark. We show that the outstanding numerical stability and performance is independent of a particular choice of the moment representation for high-Reynolds-number flows. We report accurate results for low-order moments for homogeneous isotropic decaying turbulence and second-order grid convergence for most assessed statistical quantities. It is demonstrated that all the three-dimensional lattice Boltzmann realizations considered herein converge to the familiar lattice Bhatnagar-Gross-Krook model when the resolution is increased. Moreover, thanks to the dynamic nature of the entropic stabilizer, the present model features less compressibility effects and maintains correct energy and enstrophy dissipation. The explicit and efficient nature of the present lattice Boltzmann method renders it a promising candidate for both engineering and scientific purposes for highly turbulent flows.
NASA Astrophysics Data System (ADS)
Bian, Xin; Li, Zhen; Deng, Mingge; Karniadakis, George Em
2015-11-01
truncated side of the SDPD simulation. In the EBC buffer, the velocity of particles is drawn from a known Gaussian distribution, that is, the Maxwell-Boltzmann distribution. Due to the finite range of spatial correlation, the density of particles in the EBC buffer must be drawn from a conditional Gaussian distribution, which takes into account the available density distribution of neighboring interior particles. We introduce a Kriging method to provide such a conditional distribution and hence preserve the spatial correlation of density. Spatial and temporal correlations of SDPD simulations in the truncated domain are compared to that in a single complete domain. We find that a gap region between the buffer and interior is important to reduce the extra dissipation generated by the artificial buffer at equilibrium, rendering more investigations necessary for thermal fluctuations in the multiscale coupling of nonequilibrium flows.
Maia, Alex S C; Nascimento, Sheila T; Nascimento, Carolina C N; Gebremedhin, Kifle G
2016-05-01
The effects of air temperature and relative humidity on thermal equilibrium of goats in a tropical region was evaluated. Nine non-pregnant Anglo Nubian nanny goats were used in the study. An indirect calorimeter was designed and developed to measure oxygen consumption, carbon dioxide production, methane production and water vapour pressure of the air exhaled from goats. Physiological parameters: rectal temperature, skin temperature, hair-coat temperature, expired air temperature and respiratory rate and volume as well as environmental parameters: air temperature, relative humidity and mean radiant temperature were measured. The results show that respiratory and volume rates and latent heat loss did not change significantly for air temperature between 22 and 26°C. In this temperature range, metabolic heat was lost mainly by convection and long-wave radiation. For temperature greater than 30°C, the goats maintained thermal equilibrium mainly by evaporative heat loss. At the higher air temperature, the respiratory and ventilation rates as well as body temperatures were significantly elevated. It can be concluded that for Anglo Nubian goats, the upper limit of air temperature for comfort is around 26°C when the goats are protected from direct solar radiation.
Global well-posedness for the Fokker-Planck-Boltzmann equation in Besov-Chemin-Lerner type spaces
NASA Astrophysics Data System (ADS)
Liu, Zhengrong; Tang, Hao
2016-06-01
In this paper, motivated by [16], we use the Littlewood-Paley theory to establish some estimates on the nonlinear collision term, which enable us to investigate the Cauchy problem of the Fokker-Planck-Boltzmann equation. When the initial data is a small perturbation of the Maxwellian equilibrium state, under the Grad's angular cutoff assumption, the unique global solution for the hard potential case is obtained in the Besov-Chemin-Lerner type spaces C ([ 0 , ∞) ; L˜ξ 2 (B2,rs)) with 1 ≤ r ≤ 2 and s > 3 / 2 or s = 3 / 2 and r = 1. Besides, we also obtain the uniform stability of the dependence on the initial data.
Yu, Huidan; Chen, Xi; Wang, Zhiqiang; Deep, Debanjan; Lima, Everton; Zhao, Ye; Teague, Shawn D
2014-06-01
In this paper, we develop a mass-conserved volumetric lattice Boltzmann method (MCVLBM) for numerically solving fluid dynamics with willfully moving arbitrary boundaries. In MCVLBM, fluid particles are uniformly distributed in lattice cells and the lattice Boltzmann equations deal with the time evolution of the particle distribution function. By introducing a volumetric parameter P(x,y,z,t) defined as the occupation of solid volume in the cell, we distinguish three types of lattice cells in the simulation domain: solid cell (pure solid occupation, P=1), fluid cell (pure fluid occupation, P=0), and boundary cell (partial solid and partial fluid, 0
Boltzmann equations are self-regularized through P and consist of three parts: (1) collision taking into account the momentum exchange between the willfully moving boundary and the flow; (2) streaming accompanying a volumetric bounce-back procedure in boundary cells; and (3) boundary-induced volumetric fluid migration moving the residual fluid particles into the flow domain when the boundary swipes over a boundary cell toward a solid cell. The MCVLBM strictly satisfies mass conservation and can handle irregular boundary orientation and motion with respect to the mesh. Validation studies are carried out in four cases. The first is to simulate fluid dynamics in syringes focusing on how MCVLBM captures the underlying physics of flow driven by a willfully moving piston. The second and third cases are two-dimensional (2D) peristaltic flow and three-dimensional (3D) pipe flow, respectively. In each case, we compare the MCVLBM simulation result with the analytical solution and achieve quantitatively good agreements. The fourth case is to simulate blood flow in human aortic arteries with a very complicated irregular boundary. We study steady flow in two dimensions and unsteady flow via the pulsation of the cardiac cycle in three dimensions. In the 2D case, both vector (velocity) and
NASA Astrophysics Data System (ADS)
Yu, Huidan; Chen, Xi; Wang, Zhiqiang; Deep, Debanjan; Lima, Everton; Zhao, Ye; Teague, Shawn D.
2014-06-01
In this paper, we develop a mass-conserved volumetric lattice Boltzmann method (MCVLBM) for numerically solving fluid dynamics with willfully moving arbitrary boundaries. In MCVLBM, fluid particles are uniformly distributed in lattice cells and the lattice Boltzmann equations deal with the time evolution of the particle distribution function. By introducing a volumetric parameter P (x,y,z,t) defined as the occupation of solid volume in the cell, we distinguish three types of lattice cells in the simulation domain: solid cell (pure solid occupation, P =1), fluid cell (pure fluid occupation, P =0), and boundary cell (partial solid and partial fluid, 0
Boltzmann equations are self-regularized through P and consist of three parts: (1) collision taking into account the momentum exchange between the willfully moving boundary and the flow; (2) streaming accompanying a volumetric bounce-back procedure in boundary cells; and (3) boundary-induced volumetric fluid migration moving the residual fluid particles into the flow domain when the boundary swipes over a boundary cell toward a solid cell. The MCVLBM strictly satisfies mass conservation and can handle irregular boundary orientation and motion with respect to the mesh. Validation studies are carried out in four cases. The first is to simulate fluid dynamics in syringes focusing on how MCVLBM captures the underlying physics of flow driven by a willfully moving piston. The second and third cases are two-dimensional (2D) peristaltic flow and three-dimensional (3D) pipe flow, respectively. In each case, we compare the MCVLBM simulation result with the analytical solution and achieve quantitatively good agreements. The fourth case is to simulate blood flow in human aortic arteries with a very complicated irregular boundary. We study steady flow in two dimensions and unsteady flow via the pulsation of the cardiac cycle in three dimensions. In the 2D case, both vector (velocity) and
Calculation of momentum distribution function of a non-thermal fermionic dark matter
NASA Astrophysics Data System (ADS)
Biswas, Anirban; Gupta, Aritra
2017-03-01
The most widely studied scenario in dark matter phenomenology is the thermal WIMP scenario. Inspite of numerous efforts to detect WIMP, till now we have no direct evidence for it. A possible explanation for this non-observation of dark matter could be because of its very feeble interaction strength and hence, failing to thermalise with the rest of the cosmic soup. In other words, the dark matter might be of non-thermal origin where the relic density is obtained by the so-called freeze-in mechanism. Furthermore, if this non-thermal dark matter is itself produced substantially from the decay of another non-thermal mother particle, then their distribution functions may differ in both size and shape from the usual equilibrium distribution function. In this work, we have studied such a non-thermal (fermionic) dark matter scenario in the light of a new type of U(1)B‑L model. The U(1)B‑L model is interesting, since, besides being anomaly free, it can give rise to neutrino mass by Type II see-saw mechanism. Moreover, as we will show, it can accommodate a non-thermal fermionic dark matter as well. Starting from the collision terms, we have calculated the momentum distribution function for the dark matter by solving a coupled system of Boltzmann equations. We then used it to calculate the final relic abundance, as well as other relevant physical quantities. We have also compared our result with that obtained from solving the usual Boltzmann (or rate) equations directly in terms of comoving number density, Y. Our findings suggest that the latter approximation is valid only in cases where the system under study is close to equilibrium, and hence should be used with caution.
Macroscopic Surface Tension in a Lattice Boltzmann BGK Model of Two Immiscible Fluids.
NASA Astrophysics Data System (ADS)
Thompson, S. P.; Halliday, I.; Care, C. M.
1997-08-01
We present a method by which an interface generating algorithm, similar to that of earlier lattice Boltzmann models of immisible fluids, may be extended to a two component, two-speed D2Q9 lattice Bhatnagar Gross Krook fluid. For two-dimensional, microcurrent-free planar interfaces between the two immiscible fluids we derive expressions for static interfacial tensions and interfacial distributions of the two fluids. Extending our analysis to curved interfaces we propose a scheme for incorporating the influence of interfacial microcurrents which is based upon general symmetry arguments and is correct to second order in lattice velocity. The analysis demonstrates that the interfacial microcurrents have only second order influence upon the macroscopic behaviour of the model. We find good agreement between our calculations and simulation results based on the microcurrent stream function and surface tension results from the pressure tensor or Laplace law.
A conservative multi-group approach to the Boltzmann equations for reactive gas mixtures
NASA Astrophysics Data System (ADS)
Bisi, M.; Rossani, A.; Spiga, G.
2015-11-01
Starting from a simple kinetic model for a quaternary mixture of gases undergoing a bimolecular chemical reaction, multi-group integro-differential equations are derived for the particle distribution functions of all species. The procedure takes advantage of a suitable probabilistic formulation, based on the underlying collision frequencies and transition probabilities, of the relevant reactive kinetic equations of Boltzmann type. Owing to an appropriate choice of a sufficiently large number of weight functions, it is shown that the proposed multi-group equations are able to fulfil exactly, at any order of approximation, the correct conservation laws that must be inherited from the original kinetic equations, where speed was a continuous variable. Future developments are also discussed.
Zhang, Jie; Liu, Xiaolin; Wen, Yanwei; Shi, Lu; Chen, Rong; Liu, Huijun; Shan, Bin
2017-01-25
Good electronic transport capacity and low lattice thermal conductivity are beneficial for thermoelectric applications. In this study, the potential use as a thermoelectric material for the recently synthesized two-dimensional TiS3 monolayer is explored by applying first-principles method combined with Boltzmann transport theory. Our work demonstrates that carrier transport in the TiS3 sheet is orientation-dependent, caused by the difference in charge density distribution at band edges. Due to a variety of Ti-S bonds with longer lengths, we find that the TiS3 monolayer shows thermal conductivity much lower compared with that of transition-metal dichalcogenides such as MoS2. Combined with a high power factor along the y-direction, a considerable n-type ZT value (3.1) can be achieved at moderate carrier concentration, suggesting that the TiS3 monolayer is a good candidate for thermoelectric applications.
Lattice Boltzmann Method for Diffusion-Reaction-Transport Processes in Heterogeneous Porous Media
NASA Astrophysics Data System (ADS)
Xu, You-Sheng; Zhong, Yi-Jun; Huang, Guo-Xiang
2004-07-01
Based on the lattice Boltzmann method and general theory of fluids flowing in porous media, a numerical model is presented for the diffusion-reaction-transport (DRT) processes in porous media. As a test, we simulate a DRT process in a two-dimensional horizontal heterogeneous porous medium. The influence of gravitation in this case can be neglected, and the DRT process can be described by a strongly heterogeneous diagnostic test strip or a thin confined piece of soil with stochastically distributing property in horizontal directions. The results obtained for the relations between reduced fluid saturation S, concentration c1, and concentration c2 are shown by using the visualization computing technique. The computational efficiency and stability of the model are satisfactory.
NASA Astrophysics Data System (ADS)
Xie, Dexuan
2014-10-01
The Poisson-Boltzmann equation (PBE) is one widely-used implicit solvent continuum model in the calculation of electrostatic potential energy for biomolecules in ionic solvent, but its numerical solution remains a challenge due to its strong singularity and nonlinearity caused by its singular distribution source terms and exponential nonlinear terms. To effectively deal with such a challenge, in this paper, new solution decomposition and minimization schemes are proposed, together with a new PBE analysis on solution existence and uniqueness. Moreover, a PBE finite element program package is developed in Python based on the FEniCS program library and GAMer, a molecular surface and volumetric mesh generation program package. Numerical tests on proteins and a nonlinear Born ball model with an analytical solution validate the new solution decomposition and minimization schemes, and demonstrate the effectiveness and efficiency of the new PBE finite element program package.
NASA Astrophysics Data System (ADS)
Zhang, Yong; Yi, Hong-Liang; Tan, He-Ping
2014-04-01
The lattice Boltzmann method (LBM) is extended to solve transient radiative transfer in one-dimensional slab containing absorbing and scattering media with graded index subjected to a short square laser irradiation. By using a fully implicit backward differencing scheme to discretize the transient term in the radiative transfer equation, a new type of lattice structure is devised. Firstly, for the case of the refractive index matched boundary, LBM solutions to transient radiative transfer in graded index medium are validated by comparison with results reported in the literature. Afterward, LBM is employed to investigate transient radiative transfer in graded index medium with a refractive index discontinuity at the boundaries. Effects of the graded index distributions, the optical thickness, and scattering phase function on transmittance and reflectance signals are investigated, and several interesting trends on the time-resolved signals are observed and analyzed.
Lattice Boltzmann simulation of alumina-water nanofluid in a square cavity
2011-01-01
A lattice Boltzmann model is developed by coupling the density (D2Q9) and the temperature distribution functions with 9-speed to simulate the convection heat transfer utilizing Al2O3-water nanofluids in a square cavity. This model is validated by comparing numerical simulation and experimental results over a wide range of Rayleigh numbers. Numerical results show a satisfactory agreement between them. The effects of Rayleigh number and nanoparticle volume fraction on natural convection heat transfer of nanofluid are investigated in this study. Numerical results indicate that the flow and heat transfer characteristics of Al2O3-water nanofluid in the square cavity are more sensitive to viscosity than to thermal conductivity. PMID:21711683
Extended Lattice Boltzmann Method with Application to Predict Aerodynamic Loads of Long Span Bridge
NASA Astrophysics Data System (ADS)
Liu, Tiancheng; Liu, Gao; Li, Yi; Ge, Yaojun
2010-05-01
The lattice Boltzmann (LB) method, a new conceptual approach to solve the fluid dynamics problem, is presented at first. The turbulence model is incorporated into the normal LB equation to simulate turbulence flow in the form of turbulence relaxation time determined by the nonequilibrium particle distribution function and Smagorinsky model. The total relaxation time is defined as the contribution of molecule viscosity and turbulence eddy viscosity. The aerodynamic forces on bridge girders are predicted by present LB method and the analysis of flow state is performed. The validity of LB method is verified through comparing the present results with the available experimental data and those obtained from the solutions of Navier-Stockes equation like Reynolds averaged Navier-Stokes (RANS) and discrete vortex method (DVM).
NASA Astrophysics Data System (ADS)
Lu, Benzhuo; Cheng, Xiaolin; Huang, Jingfang; McCammon, J. Andrew
2010-06-01
A Fortran program package is introduced for rapid evaluation of the electrostatic potentials and forces in biomolecular systems modeled by the linearized Poisson-Boltzmann equation. The numerical solver utilizes a well-conditioned boundary integral equation (BIE) formulation, a node-patch discretization scheme, a Krylov subspace iterative solver package with reverse communication protocols, and an adaptive new version of fast multipole method in which the exponential expansions are used to diagonalize the multipole-to-local translations. The program and its full description, as well as several closely related libraries and utility tools are available at http://lsec.cc.ac.cn/~lubz/afmpb.html and a mirror site at http://mccammon.ucsd.edu/. This paper is a brief summary of the program: the algorithms, the implementation and the usage. Program summaryProgram title: AFMPB: Adaptive fast multipole Poisson-Boltzmann solver Catalogue identifier: AEGB_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGB_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GPL 2.0 No. of lines in distributed program, including test data, etc.: 453 649 No. of bytes in distributed program, including test data, etc.: 8 764 754 Distribution format: tar.gz Programming language: Fortran Computer: Any Operating system: Any RAM: Depends on the size of the discretized biomolecular system Classification: 3 External routines: Pre- and post-processing tools are required for generating the boundary elements and for visualization. Users can use MSMS ( http://www.scripps.edu/~sanner/html/msms_home.html) for pre-processing, and VMD ( http://www.ks.uiuc.edu/Research/vmd/) for visualization. Sub-programs included: An iterative Krylov subspace solvers package from SPARSKIT by Yousef Saad ( http://www-users.cs.umn.edu/~saad/software/SPARSKIT/sparskit.html), and the fast multipole methods subroutines from FMMSuite ( http
Equilibrium avalanches in spin glasses
NASA Astrophysics Data System (ADS)
Le Doussal, Pierre; Müller, Markus; Wiese, Kay Jörg
2012-06-01
We study the distribution of equilibrium avalanches (shocks) in Ising spin glasses which occur at zero temperature upon small changes in the magnetic field. For the infinite-range Sherrington-Kirkpatrick (SK) model, we present a detailed derivation of the density ρ(ΔM) of the magnetization jumps ΔM. It is obtained by introducing a multicomponent generalization of the Parisi-Duplantier equation, which allows us to compute all cumulants of the magnetization. We find that ρ(ΔM)˜ΔM-τ with an avalanche exponent τ=1 for the SK model, originating from the marginal stability (criticality) of the model. It holds for jumps of size 1≪ΔM
Ca/Na selectivity coefficients from the Poisson-Boltzmann theory
NASA Astrophysics Data System (ADS)
Hedström, Magnus; Karnland, Ola
As a model for ion equilibrium in montmorillonite, the Poisson-Boltzmann (PB) equation was solved for two parallel charged surfaces in contact with an external NaCl/CaCl 2 mixed solution. The ion concentration profiles in the montmorillonite interlayer were obtained from the PB equation and integration of those gave the occupancy of Na + and Ca 2+ in the clay. That information together with the composition of the external electrolyte were then used for the calculation of the Gaines-Thomas selectivity coefficient K GT. The predictions from the model were compared to experimental data from batch as well as compacted conditions, and the agreement was generally good. With a surface layer-charge density of one unit charge per 145 Å 2, which is close to the value for Wyoming-type montmorillonite, the calculated selectivity coefficients were found to vary from about 4 in batch to 8 in compacted montmorillonite with dry density ∼1700 kg/m 3. From the point of view of assessing the evolution, with regard to sodium-calcium ion exchange, of the bentonite buffer in a repository for spent nuclear fuel, these results justify the use of data obtained in batch experiments.
Hard-thermal-loop corrections in leptogenesis II: solving the Boltzmann equations
Kießig, Clemens P.; Plümacher, Michael E-mail: pluemi@mpp.mpg.de
2012-09-01
We investigate hard-thermal-loop (HTL) corrections to the final lepton asymmetry in leptogenesis. To this end we solve the Boltzmann equations with HTL-corrected rates and CP asymmetries, which we calculated in paper I of this series. We pay special attention to the influence of the two leptonic quasiparticles that arise at non-zero temperature. We include only decays and inverse decays and allow for the lepton modes to be either decoupled from each other, or to be in chemical equilibrium by some strong interaction, simulating the interaction with gauge bosons. In two additional cases, we approximate the full HTL lepton propagators with zero-temperature propagators, where we replace the zero-temperature mass by the thermal mass of the leptons m{sub l}(T) or the asymptotic mass (2){sup 1/2} m{sub l}(T). We compare the final lepton asymmetries of the four thermal cases and the zero-temperature case for zero, thermal and dominant initial neutrino abundance. The final lepton asymmetries of the thermal cases differ considerably from the vacuum case and from each other in the weak washout regime for zero initial neutrino abundance and in the intermediate regime for dominant initial neutrino abundance. In the strong washout regime, the final lepton asymmetry can be enhanced by a factor of two in the case of strongly interacting lepton modes.
Lattice Boltzmann Method for Liquid-Gas-Particle Systems with Compact Discretization
NASA Astrophysics Data System (ADS)
Lee, Taehun; Farokhirad, Samaneh
2015-11-01
We have developed a liquid-gas-particle (LGP) lattice Boltzmann method (LBM) that utilizes only the nearest neighbor lattice sites for the computation of intermolecular forcing terms. Previous LGP-LBM requires larger number of lattice sites to model the interaction of fluid interfaces with immersed solid particles. This makes the treatment of contact line on a particle cumbersome when the partially wetting particle interacts with liquid-gas interface. The new model is capable of suppressing spurious currents at equilibrium. Many existing multi-component solvers suffer from spurious currents and the inability to employ components with sufficiently large density differences due to stability issues. Due to their finite size and wetting properties, particles deform an interface locally, which can lead to capillary interactions that dramatically alter the behavior of the system, relative to the particle-free case. We will present the liquid-gas-particle algorithm and its validations, which include two-particles on a flat liquid-gas interface approaching each other due to capillary effects, and a particle-laden drop impact with various impaction velocities.
Khajepor, Sorush; Wen, John; Chen, Baixin
2015-02-01
Pseudopotential lattice Boltzmann (LB) models have been recognized as efficient numerical tools to simulate complex fluid systems, including those at thermodynamic equilibrium states and with phase transitions. However, when the equation of state (EOS) of real fluids is implemented, the existing pseudopotential LB models suffer from thermodynamic inconsistency. This study presents a multipseudopotential interparticle interaction (MPI) scheme, which is fully consistent with thermodynamics and applicable to engineering applications. In this framework, multiple pseudopotentials are employed to represent dominant interaction potentials at different extents of the mean free path of particles. By simulating van der Waals and Carnahan-Starling fluids, it is demonstrated that the MPI scheme can correctly simulate the physical nature of two-phase systems on the lattice including the continuum predictions of liquid-vapor coexistence states and the sound speeds in liquid and vapor phases. It is also shown that the lattice interactions of the MPI scheme represent underlying molecular interactions as they vary in a broad range from strong short-distance repulsions to weak long-distance attractions during phase transitions. Consequently, the MPI is proved to be a reliable LB scheme as it avoids generating unphysical potentials in implementing the EOSs of real fluids and limiting the spurious velocities at the interface of two-phase systems. Additionally, a straightforward procedure is suggested and discussed to preset the MPI system with the two-phase properties of a selected fluid.
NASA Astrophysics Data System (ADS)
Khajepor, Sorush; Chen, Baixin
2016-01-01
A method is developed to analytically and consistently implement cubic equations of state into the recently proposed multipseudopotential interaction (MPI) scheme in the class of two-phase lattice Boltzmann (LB) models [S. Khajepor, J. Wen, and B. Chen, Phys. Rev. E 91, 023301 (2015)], 10.1103/PhysRevE.91.023301. An MPI forcing term is applied to reduce the constraints on the mathematical shape of the thermodynamically consistent pseudopotentials; this allows the parameters of the MPI forces to be determined analytically without the need of curve fitting or trial and error methods. Attraction and repulsion parts of equations of state (EOSs), representing underlying molecular interactions, are modeled by individual pseudopotentials. Four EOSs, van der Waals, Carnahan-Starling, Peng-Robinson, and Soave-Redlich-Kwong, are investigated and the results show that the developed MPI-LB system can satisfactorily recover the thermodynamic states of interest. The phase interface is predicted analytically and controlled via EOS parameters independently and its effect on the vapor-liquid equilibrium system is studied. The scheme is highly stable to very high density ratios and the accuracy of the results can be enhanced by increasing the interface resolution. The MPI drop is evaluated with regard to surface tension, spurious velocities, isotropy, dynamic behavior, and the stability dependence on the relaxation time.
Red Blood Cell Aggregation and Dissociation in Shear Flows Simulated by Lattice Boltzmann Method
Zhang, Junfeng; Johnson, Paul C.; Popel, Aleksander S.
2008-01-01
In this paper we develop a lattice Boltzmann algorithm to simulate red blood cell (RBC) behavior in shear flows. The immersed boundary method is employed to incorporate the fluid-membrane interaction between the flow field and deformable cells. The cell membrane is treated as a neo-Hookean viscoelastic material and a Morse potential is adopted to model the intercellular interaction. Utilizing the available mechanical properties of RBCs, multiple cells have been studied in shear flows using a two-dimensional approximation. These cells aggregate and form a rouleau under the action of intercellular interaction. The equilibrium configuration is related to the interaction strength. The end cells exhibit concave shapes under weak interaction and convex shapes under strong interaction. In shear flows, such a rouleau-like aggregate will rotate or be separated, depending on the relative strengths of the intercellular interaction and hydrodynamic viscous forces. These behaviors are qualitatively similar to experimental observations and show the potential of this numerical scheme for future studies of blood flow in microvessels. PMID:17888442
Zhou, L; Qu, Z G; Ding, T; Miao, J Y
2016-04-01
The gas-solid adsorption process in reconstructed random porous media is numerically studied with the lattice Boltzmann (LB) method at the pore scale with consideration of interparticle, interfacial, and intraparticle mass transfer performances. Adsorbent structures are reconstructed in two dimensions by employing the quartet structure generation set approach. To implement boundary conditions accurately, all the porous interfacial nodes are recognized and classified into 14 types using a proposed universal program called the boundary recognition and classification program. The multiple-relaxation-time LB model and single-relaxation-time LB model are adopted to simulate flow and mass transport, respectively. The interparticle, interfacial, and intraparticle mass transfer capacities are evaluated with the permeability factor and interparticle transfer coefficient, Langmuir adsorption kinetics, and the solid diffusion model, respectively. Adsorption processes are performed in two groups of adsorbent media with different porosities and particle sizes. External and internal mass transfer resistances govern the adsorption system. A large porosity leads to an early time for adsorption equilibrium because of the controlling factor of external resistance. External and internal resistances are dominant at small and large particle sizes, respectively. Particle size, under which the total resistance is minimum, ranges from 3 to 7 μm with the preset parameters. Pore-scale simulation clearly explains the effect of both external and internal mass transfer resistances. The present paper provides both theoretical and practical guidance for the design and optimization of adsorption systems.
Lattice Boltzmann simulation of gas-solid adsorption processes at pore scale level
NASA Astrophysics Data System (ADS)
Zhou, L.; Qu, Z. G.; Chen, L.; Tao, W. Q.
2015-11-01
A two-dimensional lattice Boltzmann (LB) approach was established to implement kinetic concentration boundary conditions in interfacial mass-transfer processes and to simulate the adsorption process in porous media at pore scale and mesoscopic levels. A general treatment was applied to conduct three types of concentration boundary conditions effectively and accurately. Applicability for adsorption was verified by two benchmark examples, which were representative of the interparticle mass transport and intraparticle mass transport in the adsorption system, respectively. The gas-solid adsorption process in reconstructed porous media at the pore scale level was numerically investigated. Mass-transfer processes of the adsorption reaction were simulated by executing Langmuir adsorption kinetics on surfaces of adsorbent particles. Meanwhile, the homogeneous solid diffusion model (HSDM) was used for mass transport in interior particles. The transient adsorbed amount was obtained in detail, and the impact of flow condition, porosity, and adsorbent particle size on the entire dynamic adsorption performance was investigated. The time needed to approach steady state decreased with increased fluid velocity. Transient adsorption capability and time consumption to equilibrium were nearly independent of porosity, whereas increasing pore size led to a moderating adsorption rate and more time was consumed to approach the saturation adsorption. Benefiting from the advantages of the LB method, both bulk and intraparticle mass transfer performances during adsorption can be obtained using the present pore scale approach. Thus, interparticle mass transfer and intraparticle mass transfer are the two primary segments, and intraparticle diffusion has the dominant role.
NASA Astrophysics Data System (ADS)
Wissocq, Gauthier; Gourdain, Nicolas; Malaspinas, Orestis; Eyssartier, Alexandre
2017-02-01
This paper reports the investigations done to adapt the Characteristic Boundary Conditions (CBC) to the Lattice-Boltzmann formalism for high Reynolds number applications. Three CBC formalisms are implemented and tested in an open source LBM code: the baseline local one-dimension inviscid (BL-LODI) approach, its extension including the effects of the transverse terms (CBC-2D) and a local streamline approach in which the problem is reformulated in the incident wave framework (LS-LODI). Then all implementations of the CBC methods are tested for a variety of test cases, ranging from canonical problems (such as 2D plane and spherical waves and 2D vortices) to a 2D NACA profile at high Reynolds number (Re =105), representative of aeronautic applications. The LS-LODI approach provides the best results for pure acoustics waves (plane and spherical waves). However, it is not well suited to the outflow of a convected vortex for which the CBC-2D associated with a relaxation on density and transverse waves provides the best results. As regards numerical stability, a regularized adaptation is necessary to simulate high Reynolds number flows. The so-called regularized FD (Finite Difference) adaptation, a modified regularized approach where the off-equilibrium part of the stress tensor is computed thanks to a finite difference scheme, is the only tested adaptation that can handle the high Reynolds computation.
Khajepor, Sorush; Chen, Baixin
2016-01-01
A method is developed to analytically and consistently implement cubic equations of state into the recently proposed multipseudopotential interaction (MPI) scheme in the class of two-phase lattice Boltzmann (LB) models [S. Khajepor, J. Wen, and B. Chen, Phys. Rev. E 91, 023301 (2015)]10.1103/PhysRevE.91.023301. An MPI forcing term is applied to reduce the constraints on the mathematical shape of the thermodynamically consistent pseudopotentials; this allows the parameters of the MPI forces to be determined analytically without the need of curve fitting or trial and error methods. Attraction and repulsion parts of equations of state (EOSs), representing underlying molecular interactions, are modeled by individual pseudopotentials. Four EOSs, van der Waals, Carnahan-Starling, Peng-Robinson, and Soave-Redlich-Kwong, are investigated and the results show that the developed MPI-LB system can satisfactorily recover the thermodynamic states of interest. The phase interface is predicted analytically and controlled via EOS parameters independently and its effect on the vapor-liquid equilibrium system is studied. The scheme is highly stable to very high density ratios and the accuracy of the results can be enhanced by increasing the interface resolution. The MPI drop is evaluated with regard to surface tension, spurious velocities, isotropy, dynamic behavior, and the stability dependence on the relaxation time.
Second Order Boltzmann-Gibbs Principle for Polynomial Functions and Applications
NASA Astrophysics Data System (ADS)
Gonçalves, Patrícia; Jara, Milton; Simon, Marielle
2017-01-01
In this paper we give a new proof of the second order Boltzmann-Gibbs principle introduced in Gonçalves and Jara (Arch Ration Mech Anal 212(2):597-644, 2014). The proof does not impose the knowledge on the spectral gap inequality for the underlying model and it relies on a proper decomposition of the antisymmetric part of the current of the system in terms of polynomial functions. In addition, we fully derive the convergence of the equilibrium fluctuations towards (1) a trivial process in case of super-diffusive systems, (2) an Ornstein-Uhlenbeck process or the unique energy solution of the stochastic Burgers equation, as defined in Gubinelli and Jara (SPDEs Anal Comput (1):325-350, 2013) and Gubinelli and Perkowski (Arxiv:1508.07764, 2015), in case of weakly asymmetric diffusive systems. Examples and applications are presented for weakly and partial asymmetric exclusion processes, weakly asymmetric speed change exclusion processes and hamiltonian systems with exponential interactions.
Dual FIB-SEM 3D imaging and lattice boltzmann modeling of porosimetry and multiphase flow in chalk.
Rinehart, Alex; Petrusak, Robin; Heath, Jason E.; Dewers, Thomas A.; Yoon, Hongkyu
2010-12-01
Mercury intrusion porosimetry (MIP) is an often-applied technique for determining pore throat distributions and seal analysis of fine-grained rocks. Due to closure effects, potential pore collapse, and complex pore network topologies, MIP data interpretation can be ambiguous, and often biased toward smaller pores in the distribution. We apply 3D imaging techniques and lattice-Boltzmann modeling in interpreting MIP data for samples of the Cretaceous Selma Group Chalk. In the Mississippi Interior Salt Basin, the Selma Chalk is the apparent seal for oil and gas fields in the underlying Eutaw Fm., and, where unfractured, the Selma Chalk is one of the regional-scale seals identified by the Southeast Regional Carbon Sequestration Partnership for CO2 injection sites. Dual focused ion - scanning electron beam and laser scanning confocal microscopy methods are used for 3D imaging of nanometer-to-micron scale microcrack and pore distributions in the Selma Chalk. A combination of image analysis software is used to obtain geometric pore body and throat distributions and other topological properties, which are compared to MIP results. 3D data sets of pore-microfracture networks are used in Lattice Boltzmann simulations of drainage (wetting fluid displaced by non-wetting fluid via the Shan-Chen algorithm), which in turn are used to model MIP procedures. Results are used in interpreting MIP results, understanding microfracture-matrix interaction during multiphase flow, and seal analysis for underground CO2 storage.
Equilibrium of nematic vesicles
NASA Astrophysics Data System (ADS)
Napoli, Gaetano; Vergori, Luigi
2010-11-01
A variational scheme is proposed which allows the derivation of a concise and elegant formulation of the equilibrium equations for closed fluid membranes, endowed with a nematic microstructure. The nematic order is described by an in-plane nematic director and a degree of orientation, as customary in the theory of uniaxial nematics. The only constitutive ingredient in this scheme is a free-energy density which depends on the vesicle geometry and order parameters. The stress and the couple stress tensors related to this free-energy density are provided. As an application of the proposed scheme, a certain number of special theories are deduced: soap bubbles, lipid vesicles, chiral and achiral nematic membranes, and nematics on curved substrates.
Statistical physics ""Beyond equilibrium
Ecke, Robert E
2009-01-01
The scientific challenges of the 21st century will increasingly involve competing interactions, geometric frustration, spatial and temporal intrinsic inhomogeneity, nanoscale structures, and interactions spanning many scales. We will focus on a broad class of emerging problems that will require new tools in non-equilibrium statistical physics and that will find application in new material functionality, in predicting complex spatial dynamics, and in understanding novel states of matter. Our work will encompass materials under extreme conditions involving elastic/plastic deformation, competing interactions, intrinsic inhomogeneity, frustration in condensed matter systems, scaling phenomena in disordered materials from glasses to granular matter, quantum chemistry applied to nano-scale materials, soft-matter materials, and spatio-temporal properties of both ordinary and complex fluids.
A Unified Theory of Non-Ideal Gas Lattice Boltzmann Models
NASA Technical Reports Server (NTRS)
Luo, Li-Shi
1998-01-01
A non-ideal gas lattice Boltzmann model is directly derived, in an a priori fashion, from the Enskog equation for dense gases. The model is rigorously obtained by a systematic procedure to discretize the Enskog equation (in the presence of an external force) in both phase space and time. The lattice Boltzmann model derived here is thermodynamically consistent and is free of the defects which exist in previous lattice Boltzmann models for non-ideal gases. The existing lattice Boltzmann models for non-ideal gases are analyzed and compared with the model derived here.