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.
Equilibrium Distribution Functions: Another Look.
ERIC Educational Resources Information Center
Waite, Boyd A.
1986-01-01
Discusses equilibrium distribution functions and provides an alternative "derivation" that allows the student, with the help of a computer, to gain intuitive insight as to the nature of distributions in general and the precise nature of the dominance of the Boltzmann distribution. (JN)
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).
Oxygen and nitrogen vibration in the thermosphere. [Boltzmann distribution discrepancy
NASA Technical Reports Server (NTRS)
Walker, J. C. G.
1973-01-01
Analysis of the departure of oxygen and nitrogen molecules from the Boltzmann distribution in the thermosphere. It is concluded that the daytime production rates are too low to cause departures from the Boltzmann distribution at altitudes below about 300 km for vibrational levels containing a significant fraction of total population. It is also pointed out that diffusion cannot perturb significantly the Boltzmann distribution at altitudes below about 370 km.
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
NASA Astrophysics Data System (ADS)
Brosens, Fons; Magnus, Wim
2009-03-01
In principle, transport of charged carriers in nanometer sized solid-state devices can be fully characterized once the non- equilibrium distribution function describing the carrier ensemble is known. In this light, we have revisited the Boltzmann and the Wigner distribution functions and the framework in which they emerge from the classical respectively quantum mechanical Liouville equation. We have assessed the method of the characteristic curves as a potential workhorse to solve the time dependent Boltzmann equation for carriers propagating through spatially non-uniform systems, such as nanodevices. In order to validate the proposed solution strategy, we numerically solve the Boltzmann equation for a one- dimensional conductor mimicking the basic features of a biased low-dimensional transistor operating in the on-state. Finally, we propose a computational scheme capable of extending the benefits of the above mentioned solution strategy when it comes to solve the Wigner-Liouville equation.
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.
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
Permit allocation in emissions trading using the Boltzmann distribution
NASA Astrophysics Data System (ADS)
Park, Ji-Won; Kim, Chae Un; Isard, Walter
2012-10-01
In emissions trading, the initial allocation of permits is an intractable issue because it needs to be essentially fair to the participating countries. There are many ways to distribute a given total amount of emissions permits among countries, but the existing distribution methods, such as auctioning and grandfathering, have been debated. In this paper we describe a new method for allocating permits in emissions trading using the Boltzmann distribution. We introduce the Boltzmann distribution to permit allocation by combining it with concepts in emissions trading. We then demonstrate through empirical data analysis how emissions permits can be allocated in practice among participating countries. The new allocation method using the Boltzmann distribution describes the most probable, natural, and unbiased distribution of emissions permits among multiple countries. Simple and versatile, this new method holds potential for many economic and environmental applications.
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.
Lin, X.
1991-01-01
This paper reports the development of an object-oriented programming methodology for particle simulations. It is established on the (m reductionist) view that many physical phenomena cana be reduced to many-body problems. By doing the reduction, many seemly unrelated physical phenomena can be simulated in a systematic way and a high-level programming system can be constructed to facilitate the programming and the solution of the simulations. In the object-oriented particle simulation methodology, a hierarchy of abstract particles is defined to represent a variety of characteristics in physical system simulations. A simulation program is constructed from particles derived from the abstract particles. The object- oriented particle simulation methodology provides a unifying modeling and simulation framework for a variety of simulation applications with the use of particle methods. It allows easy composition of simulation programs from predefined software modules and facilitates software reusability. It greatly increase the productivity of simulation program constructions. Boltzmann (after Ludwig Boltzmann, 1844-1906) is a prototype programming system in the object-oriented particle simulation methodology. Boltzmann is implemented in C++ and the X Window System. It contains a library of data types and functions that support simulations in particle methods. Moreover, it provides a visualization window to support friendly user-computer interaction. Examples of the application of the Boltzmann programming system are presented. The effectiveness of the object-oriented particle simulation methodology is demonstrated. A user's manual is included in the appendix.
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
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.
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.
Li, Q; He, Y L; Wang, Y; Tao, W Q
2007-11-01
A coupled double-distribution-function lattice Boltzmann method is developed for the compressible Navier-Stokes equations. Different from existing thermal lattice Boltzmann methods, this method can recover the compressible Navier-Stokes equations with a flexible specific-heat ratio and Prandtl number. In the method, a density distribution function based on a multispeed lattice is used to recover the compressible continuity and momentum equations, while the compressible energy equation is recovered by an energy distribution function. The energy distribution function is then coupled to the density distribution function via the thermal equation of state. In order to obtain an adjustable specific-heat ratio, a constant related to the specific-heat ratio is introduced into the equilibrium energy distribution function. Two different coupled double-distribution-function lattice Boltzmann models are also proposed in the paper. Numerical simulations are performed for the Riemann problem, the double-Mach-reflection problem, and the Couette flow with a range of specific-heat ratios and Prandtl numbers. The numerical results are found to be in excellent agreement with analytical and/or other solutions.
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.
Study of nonequilibrium work distributions from a fluctuating lattice Boltzmann model.
Nasarayya Chari, S Siva; Murthy, K P N; Inguva, Ramarao
2012-04-01
A system of ideal gas is switched from an initial equilibrium state to a final state not necessarily in equilibrium, by varying a macroscopic control variable according to a well-defined protocol. The distribution of work performed during the switching process is obtained. The equilibrium 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 ones that "violate" the second law of thermodynamics. A fluctuating lattice Boltzmann model has been employed to carry out the simulation of the switching experiment. Our results show that the probability of violation of the second law increases with the increase of switching time (τ) and tends to one-half in the reversible limit of τ→∞.
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.…
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.
Calibration of Boltzmann distribution priors in Bayesian data analysis.
Mechelke, Martin; Habeck, Michael
2012-12-01
The Boltzmann distribution is commonly used as a prior probability in Bayesian data analysis. Examples include the Ising model in statistical image analysis and the canonical ensemble based on molecular dynamics force fields in protein structure calculation. These models involve a temperature or weighting factor that needs to be inferred from the data. Bayesian inference stipulates to determine the temperature based on the model evidence. This is challenging because the model evidence, a ratio of two high-dimensional normalization integrals, cannot be calculated analytically. We outline a replica-exchange Monte Carlo scheme that allows us to estimate the model evidence by use of multiple histogram reweighting. The method is illustrated for an Ising model and examples in protein structure determination.
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.
Discrete Boltzmann equation for microfluidics.
Li, Baoming; Kwok, Daniel Y
2003-03-28
We propose a discrete Boltzmann model for microfluidics based on the Boltzmann equation with external forces using a single relaxation time collision model. Considering the electrostatic interactions in microfluidics systems, we introduce an equilibrium distribution function that differs from the Maxwell-Boltzmann distribution by an exponential factor to represent the action of an external force field. A statistical mechanical approach is applied to derive the equivalent external acceleration force exerting on the lattice particles based on a mean-field approximation, resulting from the electro-static potential energy and intermolecular potential energy between fluid-fluid and fluid-substrate interactions.
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.
Caveat on the Boltzmann distribution function use in biology.
Sevcik, Carlos
2017-08-01
Sigmoid semilogarithmic functions with shape of Boltzmann equations, have become extremely popular to describe diverse biological situations. Part of the popularity is due to the easy availability of software which fits Boltzmann functions to data, without much knowledge of the fitting procedure or the statistical properties of the parameters derived from the procedure. The purpose of this paper is to explore the plasticity of the Boltzmann function to fit data, some aspects of the optimization procedure to fit the function to data and how to use this plastic function to differentiate the effect of treatment on data and to attest the statistical significance of treatment effect on the data. Copyright © 2017. Published by Elsevier Ltd.
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.
Boltzmann-Langevin approach to pre-equilibrium correlations in nuclear collisions
NASA Astrophysics Data System (ADS)
Gavin, Sean; Moschelli, George; Zin, Christopher
2017-06-01
Correlations born before the onset of hydrodynamic flow can leave observable traces on the final-state particles. Measurement of these correlations yield important information on the isotropization and thermalization processes. Starting from a Boltzmann-like kinetic theory in the presence of dynamic Langevin noise, we derive a new partial differential equation for the two-particle correlation function that respects the microscopic conservation laws. To illustrate how these equations can be used, we study the effect of thermalization on long-range correlations. We show quite generally that two-particle correlations at early times depend on S , the average probability that a parton suffers no interactions. We extract S from transverse momentum fluctuations measured in nucleus-nucleus collisions and predict the degree of partial thermalization in proton-nucleus experiments.
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.
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…
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…
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…
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…
NASA Astrophysics Data System (ADS)
Saveliev, V. L.
2011-05-01
Pair collisions is the main interaction process in the Boltzmann gas dynamics. By making use of exactly the same physical assumptions as was used by Ludwig Boltzmann we write the kinetic equation for two-particle distribution function of molecules in the gas mixtures. Instead of the collision integral, there are the linear scattering operator and the chaos projector in the right part of this equation. Because the scattering operator is more simple then Boltzmann collision integral this equation opens new opportunities for mathematical description of the Boltzmann gas dynamics.
Parametric lattice Boltzmann method
NASA Astrophysics Data System (ADS)
Shim, Jae Wan
2017-06-01
The discretized equilibrium distributions of the lattice Boltzmann method are presented by using the coefficients of the Lagrange interpolating polynomials that pass through the points related to discrete velocities and using moments of the Maxwell-Boltzmann distribution. The ranges of flow velocity and temperature providing positive valued distributions vary with regulating discrete velocities as parameters. New isothermal and thermal compressible models are proposed for flows of the level of the isothermal and thermal compressible Navier-Stokes equations. Thermal compressible shock tube flows are simulated by only five on-lattice discrete velocities. Two-dimensional isothermal and thermal vortices provoked by the Kelvin-Helmholtz instability are simulated by the parametric models.
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.…
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
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.
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.
Equilibrium distribution of heavy quarks in fokker-planck dynamics
Walton; Rafelski
2000-01-03
We obtain an explicit generalization, within Fokker-Planck dynamics, of Einstein's relation between drag, diffusion, and the equilibrium distribution for a spatially homogeneous system, considering both the transverse and longitudinal diffusion for dimension n>1. We provide a complete characterization of the equilibrium distribution in terms of the drag and diffusion transport coefficients. We apply this analysis to charm quark dynamics in a thermal quark-gluon plasma for the case of collisional equilibration.
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
Equilibrium distributions in entropy driven balanced processes
NASA Astrophysics Data System (ADS)
Biró, Tamás S.; Néda, Zoltán
2017-05-01
For entropy driven balanced processes we obtain final states with Poisson, Bernoulli, negative binomial and Pólya distributions. We apply this both for complex networks and particle production. For random networks we follow the evolution of the degree distribution, Pn, in a system where a node can activate k fixed connections from K possible partnerships among all nodes. The total number of connections, N, is also fixed. For particle physics problems Pn is the probability of having n particles (or other quanta) distributed among k states (phase space cells) while altogether a fixed number of N particles reside on K states.
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.
Yuan, Wu-Zhi; Zhang, Li-Zhi
2017-01-24
Superhydrophobic surfaces have attracted much attention in environmental control because of their excellent water-repellent properties. A successful design of superhydrophobic surfaces requires a correct understanding of the influences of surface roughness on water-repellent behaviors. Here, a new approach, a mesoscale lattice Boltzmann simulation approach, is proposed and used to model the dynamic behavior of droplets impacting on surfaces with randomly distributed rough microstructures. The fast Fourier transformation method is used to generate non-Gaussian randomly distributed rough surfaces, with the skewness and kurtosis obtained from real surfaces. Then, droplets impacting on the rough surfaces are modeled. It is found that the shape of droplet spreading is obviously affected by the distributions of surface asperity. Decreasing the skewness and keeping the kurtosis around 3 is an effective method to enhance the ability of droplet rebound. The new approach gives more detailed insights into the design of superhydrophobic surfaces.
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.
USDA-ARS?s Scientific Manuscript database
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...
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.
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.
Q Value-Based Dynamic Programming with Boltzmann Distribution in Large Scale Road Network
NASA Astrophysics Data System (ADS)
Yu, Shanqing; Xu, Yelei; Mabu, Shingo; Mainali, Manoj Kanta; Shimada, Kaoru; Hirasawa, Kotaro
In this paper, a global optimal traffic assignment strategy, i.e., Q value-based Dynamic Programming with Boltzmann Distribution is applied to the Kitakyushu City traffic system. The main idea of the proposed traffic assignment strategy is to calculate the expected traveling time for each origin-destination pair and the probability of selecting the next section, then to generate a considerable number of route candidates for the drivers based on the calculated probability. In the simulation, how to select the temperature parameter and the number of the route candidates is discussed in detail. The comparison between the proposed method and the shortest path algorithms indicates that the proposed method could reduce the risk of the traffic congestion occurrence and save the traveling cost effectively. In addition, the computation time is given to reveal the feasibility of the proposed method in large scale networks.
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.
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.
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,…
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,…
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)
Lattice Boltzmann model for compressible fluids
NASA Technical Reports Server (NTRS)
Alexander, F. J.; Chen, H.; Chen, S.; Doolen, G. D.
1992-01-01
A lattice Boltzmann model is derived which simulates compressible fluids. By choosing the parameters of the equilibrium distribution appropriately, the sound speed (which may be set arbitrarily low), bulk viscosity, and kinematic viscosity can be selected. This model simulates compressible flows and can include shocks. With a proper rescaling and zero-sound speed, this model simulates Burgers's equation. The viscosity determined by a Chapman-Enskog expansion compares well with that measured form simulations. The exact solutions of Burgers's equation on the unit circle are compared to solutions of lattice Boltzmann model finding reasonable agreement.
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.
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.
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.
Teixeira, Vitor H; Cunha, Carlos A; Machuqueiro, Miguel; Oliveira, A Sofia F; Victor, Bruno L; Soares, Cláudio M; Baptista, António M
2005-08-04
Poisson-Boltzmann (PB) models are a fast and common tool for studying electrostatic processes in proteins, particularly their ionization equilibrium (protonation and/or reduction), often yielding quite good results when compared with more detailed models. Yet, they are conceptually very simple and necessarily approximate, their empirical character being most evident when it comes to the choice of the dielectric constant assigned to the protein region. The present study analyzes several factors affecting the ability of PB-based methods to model protein ionization equilibrium. We give particular attention to a suggestion made by Warshel and co-workers (e.g., Sham et al. J. Phys. Chem. B 1997, 101, 4458) of using different protein dielectric constants for computing the individual (site) and the pairwise (site-site) terms of the ionization free energies. Our prediction of pK(a) values for several proteins indicates that no advantage is obtained by such a procedure, even for sites that are buried and/or display large pK(a) shifts relative to the solution values. In particular, the present methodology gives the best predictions using a dielectric constant around 20, for shifted/buried and nonshifted/exposed sites alike. The similarities and differences between the PB model and Warshel's PDLD/S model are discussed, as well as the reasons behind their apparently discrepant results. The present PB model is shown to predict also good reduction potentials in redox proteins.
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.
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.
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
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.
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.
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.
NASA Astrophysics Data System (ADS)
Aquilanti, Vincenzo; Coutinho, Nayara Dantas; Carvalho-Silva, Valter Henrique
2017-03-01
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 increase of temperature
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.
Detailed balance, quantum distribution functions, and equilibrium of mixtures
NASA Astrophysics Data System (ADS)
Lawrence, W. E.
1999-12-01
We consider systems of nearly free particles (or quasiparticles) interacting by scattering, emission and absorption of radiation, or by physical or chemical transformation. The condition of detailed balance yields the appropriate distribution function for each species, the equality of their temperatures, and a relation for their chemical potentials associated with particle transformations. For example, antiparticles coexisting in equilibrium have opposite chemical potentials, and excitations above the Bose-Einstein condensate have zero chemical potential. For mixtures of classical ideal gases, the law of mass action is obtained.
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 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.
Truncated Thermal Equilibrium Distribution for Intense Beam Propagation
Ronald C. Davidson; Hong Qin; Steven M. Lund
2003-02-26
An intense charged-particle beam with directed kinetic energy ({lambda}{sub b}-1)m{sub b}c{sup 2} propagates in the z-direction through an applied focusing field with transverse focusing force modeled by F{sub foc} = -{lambda}{sub b}m{sub b}{omega}{sub beta}{sup 2} {perpendicular} x {perpendicular} in the smooth focusing approximation. This paper examines properties of the axisymmetric, truncated thermal equilibrium distribution F(sub)b(r,p perpendicular) = A exp (-H Perpendicular/T perpendicular (sub)b) = (H perpendicular-E(sub)b), where A, T perpendicular (sub)b, and E (sub)b are positive constants, and H perpendicular is the Hamiltonian for transverse particle motion. The equilibrium profiles for beam number density, n(sub)b(r) = * d{sup 2}pF(sub)b(r,p perpendicular), and transverse temperature, T perpendicular (sub)b(r) = * d{sup 2}p(p{sup 2} perpendicular/2 lambda (sbu)bm (sub)b)F(sub)b(r,p perpendicular), are calculated self-consistently including space-charge effects. Several properties of the equilibrium profiles are noteworthy. For example, the beam has a sharp outer edge radius r(sub)b with n(sub)b(r greater than or equal to rb) = 0, where r(sub)b depends on the value of E(sub)b/T (sub)perpendicular(sub)b. In addition, unlike the choice of a semi-Gaussian distribution, F{sup SG}(sub)b = A exp (-p{sup 2}(sub)perpendicular/2lambda(sub)bm(sub)bTperpendicular(sub)b) = (r-r(sub)b), the truncated thermal equilibrium distribution F(sub)b(r,p) depends on (r,p) only through the single-particle constant of the motion Hperpendiuclar and is therefore a true steady-state solution (*/*t = 0) of the nonlinear Vlasov-Maxwell equations.
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.
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
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.
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
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.
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.
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.
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.
The Approach to Equilibrium: Detailed Balance and the Master Equation
Hall, G.E.; Alexander, M.H.; Dagdigian, P.J.
2011-08-18
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 equation is presented and shows that the equilibrium distribution is the Boltzmann distribution. This solution is applied to the master equation involving collisions of rotational states of a diatomic molecule with a monatomic bath gas.
Nonequilibrium thermodynamics of restricted Boltzmann machines
NASA Astrophysics Data System (ADS)
Salazar, Domingos S. P.
2017-08-01
In this work, we analyze the nonequilibrium thermodynamics of a class of neural networks known as restricted Boltzmann machines (RBMs) in the context of unsupervised learning. We show how the network is described as a discrete Markov process and how the detailed balance condition and the Maxwell-Boltzmann equilibrium distribution are sufficient conditions for a complete thermodynamics description, including nonequilibrium fluctuation theorems. Numerical simulations in a fully trained RBM are performed and the heat exchange fluctuation theorem is verified with excellent agreement to the theory. We observe how the contrastive divergence functional, mostly used in unsupervised learning of RBMs, is closely related to nonequilibrium thermodynamic quantities. We also use the framework to interpret the estimation of the partition function of RBMs with the annealed importance sampling method from a thermodynamics standpoint. Finally, we argue that unsupervised learning of RBMs is equivalent to a work protocol in a system driven by the laws of thermodynamics in the absence of labeled data.
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.
IS THE SIZE DISTRIBUTION OF URBAN AEROSOLS DETERMINED BY THERMODYNAMIC EQUILIBRIUM? (R826371C005)
A size-resolved equilibrium model, SELIQUID, is presented and used to simulate the size–composition distribution of semi-volatile inorganic aerosol in an urban environment. The model uses the efflorescence branch of aerosol behavior to predict the equilibrium partitioni...
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.
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.
Effect of pre-equilibrium spin distribution on neutron-induced reaction cross sections
Dashdorj, D.; Mitchell, G. E.; Becker, J. A.; Wu, C. Y.; Chadwick, M. B.; Devlin, M.; Fotiades, N.; Kawano, T.; Nelson, R. O.; Garrett, P. E.; Kunieda, S.
2008-04-17
Cross section measurements were made of prompt gamma-ray production as a function of neutron energy using the germanium array for neutron induced excitations (GEANIE) at LANSCE. Measuring the prompt reaction gamma rays as a function of incident neutron energy provides more precise understanding of the spins populated by the pre-equilibrium reaction. The effect of the spin distribution in pre-equilibrium reactions has been investigated using the GNASH reaction code. Widely used classical theories such as the exciton model usually assume that the spin distribution of the pre-equilibrium reaction is the same as the spin distribution of the compound nucleus reaction mechanism. In the present approach, the pre-equilibrium reaction spin distribution was calculated using the quantum mechanical theory of Feshbach, Kerman, and Koonin (FKK). This pre-equilibrium spin distribution was incorporated into the GNASH code and the gamma-ray production cross sections were calculated and compared with experimental data. Spin distributions peak at lower spin when calculated with the FKK formulation than with the Compound Nuclear theory. The measured partial gamma-ray cross sections reflect this spin difference. Realistic treatment of the spin distribution improves the accuracy of calculations of gamma-ray production cross sections.
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+.
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
Geometry of the equilibrium distribution of interacting particles
NASA Astrophysics Data System (ADS)
Rebesh, A. P.; Lev, B. I.
2017-08-01
The formations of structures in all systems of interacting particles at different temperatures and particle concentrations have the same physical nature and therefore they could be described geometrically the same way. We propose a geometric description of a thermodynamically stable distribution of interacting particles. The character and intensity of interaction between particles determine the effective geometry of the medium which is provided by the minimum of the total free energy with non-linearity. A realization of spaces with possible particle distributions of arbitrary-order symmetries (including fifth, seventh etc. orders) which cannot occur in the ordinary Euclidean space, is described.
NASA Astrophysics Data System (ADS)
Salimi, M. R.; Taeibi-Rahni, M.
2015-12-01
Due to superior accuracy and stability of multiple relaxation time (MRT) collision operator over its single relaxation time (SRT) counterpart, new lifting relations are proposed here to construct single particle distribution functions for MRT-LBM from macroscopic variables. Using these lifting relations, a new hybrid FVM-LB method is presented (called Finite type-LB hybrid method), which is consistent with MRT-LBM. In this new hybrid method, single-particle distribution functions in MRT-LBM sub-domain boundaries are computed, using equilibrium and non-equilibrium moments. These moments are computed in Navier-Stokes/FVM sub-domain boundaries, using macroscopic variables and their derivatives. The new method is validated by solving three benchmark problems, i.e., two- and three-dimensional lid driven cavity flows and two-dimensional unsteady flow around a squared section cylinder. These problems are analyzed with pure FVM, pure LBM, and Finite type-LB hybrid method (FTLBHM) and the related results are compared with each other and with benchmark data. These comparisons clearly demonstrate the accuracy of the present novel methodology for simulating steady/unsteady flow fields in two and three dimensions.
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…
Treatment of Chemical Equilibrium without Using Thermodynamics or Statistical Mechanics.
ERIC Educational Resources Information Center
Nelson, P. G.
1986-01-01
Discusses the conventional approaches to teaching about chemical equilibrium in advanced physical chemistry courses. Presents an alternative approach to the treatment of this concept by using Boltzmann's distribution law. Lists five advantages to using this method as compared with the other approaches. (TW)
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.
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
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.
Electron Energy Distribution and Transfer Phenomena in Non-Equilibrium Gases
2016-09-01
AFRL-RQ-WP-TR-2016-0130 ELECTRON ENERGY DISTRIBUTION AND TRANSFER PHENOMENA IN NON-EQUILIBRIUM GASES Steven F. Adams and Bradley S...2. REPORT TYPE 3. DATES COVERED (From - To) September 2016 Final 15 March 2010 – 16 September 2016 4. TITLE AND SUBTITLE ELECTRON ENERGY DISTRIBUTION...and ultimately control the distribution of electronic and kinetic energies within low temperature plasmas and enhance the understanding of phenomena
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.
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.
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.
Lattice Boltzmann model for nonlinear convection-diffusion equations.
Shi, Baochang; Guo, Zhaoli
2009-01-01
A lattice Boltzmann model for convection-diffusion equation with nonlinear convection and isotropic-diffusion terms is proposed through selecting equilibrium distribution function properly. The model can be applied to the common real and complex-valued nonlinear evolutionary equations, such as the nonlinear Schrödinger equation, complex Ginzburg-Landau equation, Burgers-Fisher equation, nonlinear heat conduction equation, and sine-Gordon equation, by using a real and complex-valued distribution function and relaxation time. Detailed simulations of these equations are performed, and it is found that the numerical results agree well with the analytical solutions and the numerical solutions reported in previous studies.
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.
Park, H M; Lee, J S; Kim, T W
2007-11-15
In the analysis of electroosmotic flows, the internal electric potential is usually modeled by the Poisson-Boltzmann equation. The Poisson-Boltzmann equation is derived from the assumption of thermodynamic equilibrium where the ionic distributions are not affected by fluid flows. Although this is a reasonable assumption for steady electroosmotic flows through straight microchannels, there are some important cases where convective transport of ions has nontrivial effects. In these cases, it is necessary to adopt the Nernst-Planck equation instead of the Poisson-Boltzmann equation to model the internal electric field. In the present work, the predictions of the Nernst-Planck equation are compared with those of the Poisson-Boltzmann equation for electroosmotic flows in various microchannels where the convective transport of ions is not negligible.
Approximation method for the kinetic Boltzmann equation
NASA Technical Reports Server (NTRS)
Shakhov, Y. M.
1972-01-01
The further development of a method for approximating the Boltzmann equation is considered and a case of pseudo-Maxwellian molecules is treated in detail. A method of approximating the collision frequency is discussed along with a method for approximating the moments of the Boltzmann collision integral. Since the return collisions integral and the collision frequency are expressed through the distribution function moments, use of the proposed methods make it possible to reduce the Boltzmann equation to a series of approximating equations.
Lattice Boltzmann model for incompressible flows through porous media.
Guo, Zhaoli; Zhao, T S
2002-09-01
In this paper a lattice Boltzmann model is proposed for isothermal incompressible flow in porous media. The key point is to include the porosity into the equilibrium distribution, and add a force term to the evolution equation to account for the linear and nonlinear drag forces of the medium (the Darcy's term and the Forcheimer's term). Through the Chapman-Enskog procedure, the generalized Navier-Stokes equations for incompressible flow in porous media are derived from the present lattice Boltzmann model. The generalized two-dimensional Poiseuille flow, Couette flow, and lid-driven cavity flow are simulated using the present model. It is found the numerical results agree well with the analytical and/or the finite-difference solutions.
Detecting temperature fluctuations at equilibrium.
Dixit, Purushottam D
2015-05-21
The Gibbs and the Boltzmann definition of temperature agree only in the macroscopic limit. The ambiguity in identifying the equilibrium temperature of a finite-sized 'small' system exchanging energy with a bath is usually understood as a limitation of conventional statistical mechanics. We interpret this ambiguity as resulting from a stochastically fluctuating temperature coupled with the phase space variables giving rise to a broad temperature distribution. With this ansatz, we develop the equilibrium statistics and dynamics of small systems. Numerical evidence using an analytically tractable model shows that the effects of temperature fluctuations can be detected in the equilibrium and dynamical properties of the phase space of the small system. Our theory generalizes statistical mechanics to small systems relevant in biophysics and nanotechnology.
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.
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.
Lattice Boltzmann model with nearly constant density.
Fang, Hai-ping; Wan, Rong-zheng; Lin, Zhi-fang
2002-09-01
An improved lattice Boltzmann model is developed to simulate fluid flow with nearly constant fluid density. The ingredient is to incorporate an extra relaxation for fluid density, which is realized by introducing a feedback equation in the equilibrium distribution functions. The pressure is dominated by the moving particles at a node, while the fluid density is kept nearly constant and explicit mass conservation is retained as well. Numerical simulation based on the present model for the (steady) plane Poiseuille flow and the (unsteady) two-dimensional Womersley flow shows a great improvement in simulation results over the previous models. In particular, the density fluctuation has been reduced effectively while achieving a relatively large pressure gradient.
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.
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.
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
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.
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.
Zheng, Xiliang; Wang, Jin
2015-01-01
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. PMID:25885453
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.
Boltzmann's Approach to Statistical Mechanics
NASA Astrophysics Data System (ADS)
Goldstein, Sheldon
In the last quarter of the nineteenth century, Ludwig Boltzmann explained how irreversible macroscopic laws, in particular the second law of thermodynamics, originate in the time-reversible laws of microscopic physics. Boltzmann's analysis, the essence of which I shall review here, is basically correct. The most famous criticisms of Boltzmann's later work on the subject have little merit. Most twentieth century innovations - such as the identification of the state of a physical system with a probability distribution \\varrho on its phase space, of its thermodynamic entropy with the Gibbs entropy of \\varrho, and the invocation of the notions of ergodicity and mixing for the justification of the foundations of statistical mechanics - are thoroughly misguided.
NASA Astrophysics Data System (ADS)
Dzifčáková, Elena; Dudík, Jaroslav
2015-12-01
We use the latest available atomic data to calculate the ionisation and recombination rates for the non-Maxwellian n-distributions, which were shown previously to provide a good fit to the enhanced intensities of dielectronic satellite lines during solar flares. The ionisation and recombination coefficients are subsequently used to derive the ionisation equilibrium. To do so, we consider odd values of n ranging from 1 to 19, i.e., from Maxwellian to strongly non-Maxwellian cases. These calculations involve all elements with proton number up to 30, i.e., H to Zn. The n-distributions modify both the ionisation and the recombination rates. The ionisation rates decrease more steeply at lower pseudo-temperatures, while the radiative recombination rate is reduced due to a lower number of low-energy electrons. The peaks of the dielectronic recombination rates become narrower. These changes are reflected in the ionisation equilibrium. Ion abundance peaks become narrower and can also be shifted, mostly towards higher temperatures. The He-like ions are an important exception, as they are formed in a larger temperature range than that for the Maxwellian distribution. The ions Si xiii - xiv used previously for the diagnostics of the n-distributions are affected only weakly, confirming the determination of n. The ionisation equilibria are available as the electronic supplementary material in a format compatible with the CHIANTI database.
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.
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.
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.
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)
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.
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.
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.
Equilibrium charge distribution on a finite straight one-dimensional wire
NASA Astrophysics Data System (ADS)
Batle, Josep; Ciftja, Orion; Abdalla, Soliman; Elhoseny, Mohamed; Alkhambashi, Majid; Farouk, Ahmed
2017-09-01
The electrostatic properties of uniformly charged regular bodies are prominently discussed on college-level electromagnetism courses. However, one of the most basic problems of electrostatics that deals with how a continuous charge distribution reaches equilibrium is rarely mentioned at this level. In this work we revisit the problem of equilibrium charge distribution on a straight one-dimensional (1D) wire with finite length. The majority of existing treatments in the literature deal with the 1D wire as a limiting case of a higher-dimensional structure that can be treated analytically for a Coulomb interaction potential between point charges. Surprisingly, different models (for instance, an ellipsoid or a cylinder model) may lead to different results, thus there is even some ambiguity on whether the problem is well-posed. In this work we adopt a different approach where we do not start with any higher-dimensional body that reduces to a 1D wire in the appropriate limit. Instead, our starting point is the obvious one, a finite straight 1D wire that contains charge. However, the new tweak in the model is the assumption that point charges interact with each other via a non-Coulomb power-law interaction potential. This potential is well-behaved, allows exact analytical results and approaches the standard Coulomb interaction potential as a limit. The results originating from this approach suggest that the equilibrium charge distribution for a finite straight 1D wire is a uniform charge density when the power-law interaction potential approaches the Coulomb interaction potential as a suitable limit. We contrast such a finding to results obtained using a different regularised logarithmic interaction potential which allows exact treatment in 1D. The present self-contained material may be of interest to instructors teaching electromagnetism as well as students who will discover that simple-looking problems may sometimes pose important scientific challenges.
Diffusion and near-equilibrium distribution of MRI and CT contrast agents in articular cartilage.
Silvast, Tuomo S; Kokkonen, Harri T; Jurvelin, Jukka S; Quinn, Thomas M; Nieminen, Miika T; Töyräs, Juha
2009-11-21
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((R)), 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.
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.
Equilibrium Beam Distribution and Quantum Lifetime in the Presence of a Single Nonlinear Resonance
Chao, A
2003-12-09
In the proximity of a nonlinear resonance {nu} {approx} m/n, n , the beam distribution in a storage ring is distorted depending on how close by is the resonance and how strong is the resonance strength. In the 1-dimensional case, it is well known that the particle motion near the resonance can be described in a smooth approximation by a Hamiltonian of the form ({nu} - m/n) J + D{sub {nu}}(J) + f{sub 1}({phi}, J), where ({phi}, J) are the phase space angle and action variables, D{sub {nu}} is the detuning function, and f{sub 1} is an oscillating resonance term. In a proton storage ring, the equilibrium beam distribution is readily solved to be any function exclusively of the Hamiltonian. For an electron beam, this is not true and the equilibrium distribution is more complicated. This paper solves the Fokker-Planck equation near a single resonance for an electron beam in a storage ring. The result is then applied to obtain the quantum lifetime of an electron beam in the presence of this resonance. Resonances due to multipole fields and due to the beam-beam force are considered as examples.
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.
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.
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.
High-order regularization in lattice-Boltzmann equations
NASA Astrophysics Data System (ADS)
Mattila, Keijo K.; Philippi, Paulo C.; Hegele, Luiz A.
2017-04-01
A lattice-Boltzmann equation (LBE) is the discrete counterpart of a continuous kinetic model. It can be derived using a Hermite polynomial expansion for the velocity distribution function. Since LBEs are characterized by discrete, finite representations of the microscopic velocity space, the expansion must be truncated and the appropriate order of truncation depends on the hydrodynamic problem under investigation. Here we consider a particular truncation where the non-equilibrium distribution is expanded on a par with the equilibrium distribution, except that the diffusive parts of high-order non-equilibrium moments are filtered, i.e., only the corresponding advective parts are retained after a given rank. The decomposition of moments into diffusive and advective parts is based directly on analytical relations between Hermite polynomial tensors. The resulting, refined regularization procedure leads to recurrence relations where high-order non-equilibrium moments are expressed in terms of low-order ones. The procedure is appealing in the sense that stability can be enhanced without local variation of transport parameters, like viscosity, or without tuning the simulation parameters based on embedded optimization steps. The improved stability properties are here demonstrated using the perturbed double periodic shear layer flow and the Sod shock tube problem as benchmark cases.
Ambruş, Victor Eugen; Sofonea, Victor
2014-04-01
The Gauss-Laguerre quadrature method is used on the Cartesian semiaxes in the momentum space to construct a family of lattice Boltzmann models. When all quadrature orders Qx, Qy, Qz equal N+1, the Laguerre lattice Boltzmann model LLB(Qx,Qy,Qz) exactly recovers all moments up to order N of the Maxwell-Boltzmann equilibrium distribution function f(eq), calculated over any Cartesian octant of the three-dimensional momentum space. Results of Couette flow simulations at Kn=0.1, 0.5, 1.0 and in the ballistic regime are reported. Specific microfluidic effects (velocity slip, temperature jump, longitudinal heat flux) are well captured up to Kn=0.5, as demonstrated by comparison to direct simulation Monte Carlo results. Excellent agreement with analytic results is obtained in the ballistic regime.
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.
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. PMID:23558425
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.
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.
NASA Astrophysics Data System (ADS)
Batle, Josep; Ciftja, Orion; Naseri, Mosayeb; Ghoranneviss, Mahmood; Farouk, Ahmed; Elhoseny, Mohamed
2017-05-01
We study the minimum energy equilibrium configurations of a classical two-dimensional system of point charges confined by a triangular, square and disk region with a hard-wall boundary. It is assumed that the point charges interact via a repulsive Coulomb interaction potential. Monte Carlo simulations with the annealing algorithm suggest that the equilibrium configurations of a given system are strongly influenced by the external (isotropic/anisotropic) geometry of the hard-wall boundary. The numerically obtained energies extrapolated in the bulk limit converge to the expected continuum equilibrium values (when known). It is found that the equilibrium charge distribution is non-uniform in the continuum limit for all the hard-wall confining regions considered in this work. Since the continuum equilibrium charge distribution is not known for the case of an equilateral triangle or a square domain we choose to compare the numerically obtained bulk energy results to corresponding values for a uniformly charged system. We calculated exactly the electrostatic energy of various uniformly charged planar objects and used the results to assess the discrepancy between such results and the numerically obtained equilibrium bulk energy values for the cases of equilateral triangle and square hard-wall boundaries. These estimates help us understand how an anisotropic boundary with the shape of an equilateral triangle or square influences the energy of an equilibrium charge distribution. The results indicate that the energy discrepancy between equilibrium and uniform charge distributions in the continuum limit is not very large. It is found that the order of magnitude of the relative deviation of the energy for all three different planar domains considered here is approximately the same.
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
Non equilibrium spectra of degenerate relic neutrinos
NASA Astrophysics Data System (ADS)
Esposito, S.; Miele, G.; Pastor, S.; Peloso, M.; Pisanti, O.
2000-12-01
We calculate the exact kinetic evolution of cosmic neutrinos until complete decoupling, in the case when a large neutrino asymmetry exists. While not excluded by present observations, this large asymmetry can have relevant cosmological consequences and in particular may be helpful in reconciling Primordial Nucleosynthesis with a high baryon density as suggested by the most recent observations of the Cosmic Microwave Background Radiation. By solving numerically the Boltzmann kinetic equations for the neutrino distribution functions, we find the momentum-dependent corrections to the equilibrium spectra and briefly discuss their phenomenological implications.
Ye, Xiang; Cai, Qin; Yang, Wei; Luo, Ray
2009-07-22
The wide use of lattice-sum strategies in biomolecular simulations has raised many questions on potential artifacts in these strategies. One interesting question is the artifacts in the counterion distributions of highly charged systems. As one would anticipate, Coulombic interactions under the periodic boundary condition may deviate noticeably from those under the free boundary condition in the highly charged systems, significantly influencing their counterion distributions. On the other hand, the electrostatic screening due to water molecules and mobile ions may effectively damp the possible periodic distortions in Coulombic interactions. Therefore, the magnitude of periodicity-induced artifacts in counterion distributions is not straightforward to dissect without detailed analyses. In this study, we have developed a hybrid explicit counterion/implicit salt representation of mobile ions to address this question. We have chosen a well-studied DNA for easy validation of the minimal hybrid ion representation. Our detailed analysis of continuum ion distributions, explicit ion distributions, radial counterion distribution functions, and sequence-dependent counterion distributions, however, indicates that periodicity artifacts are not apparent at the surface of the tested DNA. Nevertheless, influence of boundary conditions does show up starting at the second solvation shell and becomes apparent at the cell boundary.
The statistical distribution of money and the rate of money transference
NASA Astrophysics Data System (ADS)
Ferrero, Juan C.
2004-10-01
The distribution of money is analysed in connection with the Boltzmann distribution of energy in the degenerate states of molecules. Plots of the population density of income distribution for various countries are well reproduced by a Gamma function, confirming the validity of the statistical distribution at equilibrium. The equilibrium state is reached through pair wise money transference processes, independently of the shape of the initial distribution and also of the detailed nature of the money transactions between the economic agents.
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.
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.
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.
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.
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.
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.
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.
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).
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.
Pierson, Nicholas A.; Valentine, Stephen J.; Clemmer, David E.
2010-01-01
Multidimensional ion mobility spectrometry coupled with mass spectrometry (IMS–IMS-MS) techniques are used to select and activate six different gas-phase conformations of bradykinin [M+3H]3+ ions. Drift time distributions as a function of activation voltage show that at low voltages selected structures undergo conformational transitions in what appears to be a pathway dependent fashion. Over a relatively wide range of intermediate activation voltages a distribution of states that is independent of the initial conformation selected for activation (as well as the activation voltage in this intermediate region) is established. This distribution appears to represent an equilibrium distribution of gas-phase structures that is reached prior to the energy required for dissociation. Establishment of a quasi-equilibrium prior to dissociation results in identical dissociation patterns for different selected conformations. A discussion of the transition from solution-like to gas-phase structures is provided. PMID:20469905
Local non-equilibrium thermodynamics.
Jinwoo, Lee; Tanaka, Hajime
2015-01-16
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.
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
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.
Boltzmann's H-function and diffusion processes.
Hubbard, Joseph B; Lund, Steven P; Halter, Michael
2013-10-24
There exists a generalization of Boltzmann's H-function that allows for nonuniformly populated stationary states, which may exist far from thermodynamic equilibrium. Here we describe a method for obtaining a generalized or collective diffusion coefficient D directly from this H-function, the only constraints being that the relaxation process is Markov (short memory), continuous in the reaction coordinate, and local in the sense of a flux/force relationship. As an application of this H-function method, we simulate the self-consistent extraction of D via Langevin/Fokker-Planck (L/FP) dynamics on various potential energy landscapes. We observe that the initial epoch of relaxation, which is far removed from the stationary state, provides the most reliable estimates of D. The construction of an H-function that guarantees conformity with the second law of thermodynamics has been generalized to allow for diffusion coefficients that may depend on both the reaction coordinate and time, and the extension to an arbitrary number of reaction coordinates is straightforward. For this multidimensional case, the diffusion tensor must be positive definite in the sense that its eigenvalues must be real and positive. To illustrate the behavior of the proposed collective diffusion coefficient, we simulate the H-function for a variety of Langevin systems. In particular, the impacts on H and D of landscape shape, sample size, selection of an initial distribution, finite dynamic observation range, stochastic correlations, and short/long-term memory effects are examined.
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.
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.
NASA Astrophysics Data System (ADS)
Chen, Z.; Shu, C.; Tan, D.
2017-05-01
In this paper, a three-dimensional simplified and unconditionally stable lattice Boltzmann method (3D-USLBM) is proposed for simulating incompressible isothermal/thermal flows. This method is developed by reconstructing solutions to the macroscopic governing equations recovered from the lattice Boltzmann equation and resolved in a predictor-corrector scheme. The final formulations of 3D-USLBM only involve the equilibrium and the non-equilibrium distribution functions. Among them, the former is calculated from the macroscopic variables and the latter is evaluated from the difference between two equilibrium distribution functions at different locations and time levels. Thus, 3D-USLBM directly tracks the evolution of macroscopic variables, which yields lower cost in virtual memory and facilitates the implementation of physical boundary conditions. A von Neumann stability analysis was performed on the present method to theoretically prove its unconditional stability. By imposing a regular Lagrange interpolation algorithm, this method can be flexibly extended to a non-uniform Cartesian mesh or body-fitted mesh with curved boundaries. Four numerical tests, that is, plane Poiseuille flow, 3D lid-driven cavity flow and 3D natural convection in a cubic cavity, and concentric annulus, were conducted to verify the stability, accuracy, and flexibility of the presented method.
Lattice Boltzmann modeling and simulation of liquid jet breakup
NASA Astrophysics Data System (ADS)
Saito, Shimpei; Abe, Yutaka; Koyama, Kazuya
2017-07-01
A three-dimensional color-fluid lattice Boltzmann model for immiscible two-phase flows is developed in the framework of a three-dimensional 27-velocity (D3Q27) lattice. The collision operator comprises the D3Q27 versions of three suboperators: a multiple-relaxation-time (MRT) collision operator, a generalized Liu-Valocchi-Kang perturbation operator, and a Latva-Kokko-Rothman recoloring operator. A D3Q27 version of an enhanced equilibrium distribution function is also incorporated into this model to improve the Galilean invariance. Three types of numerical tests, namely, a static droplet, an oscillating droplet, and the Rayleigh-Taylor instability, show a good agreement with analytical solutions and numerical simulations. Following these numerical tests, this model is applied to liquid-jet-breakup simulations. The simulation conditions are matched to the conditions of the previous experiments. In this case, numerical stability is maintained throughout the simulation, although the kinematic viscosity for the continuous phase is set as low as 1.8 ×10-4 , in which case the corresponding Reynolds number is 3.4 ×103 ; the developed lattice Boltzmann model based on the D3Q27 lattice enables us to perform the simulation with parameters directly matched to the experiments. The jet's liquid column transitions from an asymmetrical to an axisymmetrical shape, and entrainment occurs from the side of the jet. The measured time history of the jet's leading-edge position shows a good agreement with the experiments. Finally, the reproducibility of the regime map for liquid-liquid systems is assessed. The present lattice Boltzmann simulations well reproduce the characteristics of predicted regimes, including varicose breakup, sinuous breakup, and atomization.
Lattice Boltzmann model for a steady radiative transfer equation.
Yi, Hong-Liang; Yao, Feng-Ju; Tan, He-Ping
2016-08-01
A complete lattice Boltzmann model (LBM) is proposed for the steady radiative transfer equation (RTE). The RTE can be regarded as a pure convection equation with a source term. To derive the expressions for the equilibrium distribution function and the relaxation time, an artificial isotropic diffusion term is introduced to form a convection-diffusion equation. When the dimensionless relaxation time has a value of 0.5, the lattice Boltzmann equation (LBE) is exactly applicable to the original steady RTE. We also perform a multiscale analysis based on the Chapman-Enskog expansion to recover the macroscopic RTE from the mesoscopic LBE. The D2Q9 model is used to solve the LBE, and the numerical results obtained by the LBM are comparable to the results obtained by other methods or analytical solutions, which demonstrates that the proposed model is highly accurate and stable in simulating multidimensional radiative transfer. In addition, we find that the convergence rate of the LBM depends on the transport properties of RTE: for diffusion-dominated RTE with a large optical thickness, the LBM shows a second-order convergence rate in space, while for convection-dominated RTE with a small optical thickness, a lower convergence rate is observed.
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.
Hydrodynamic interactions in colloidal ferrofluids: a lattice Boltzmann study.
Kim, Eunhye; Stratford, Kevin; Camp, Philip J; Cates, Michael E
2009-03-26
We use lattice Boltzmann simulations, in conjunction with Ewald summation methods, to investigate the role of hydrodynamic interactions in colloidal suspensions of dipolar particles, such as ferrofluids. Our work addresses volume fractions phi of up to 0.20 and dimensionless dipolar interaction parameters lambda of up to 8. We compare quantitatively with Brownian dynamics simulations, in which many-body hydrodynamic interactions are absent. Monte Carlo data are also used to check the accuracy of static properties measured with the lattice Boltzmann technique. At equilibrium, hydrodynamic interactions slow down both the long-time and the short-time decays of the intermediate scattering function S(q, t), for wavevectors close to the peak of the static structure factor S(q), by a factor of roughly two. The long-time slowing is diminished at high interaction strengths, whereas the short-time slowing (quantified via the hydrodynamic factor H(q)) is less affected by the dipolar interactions, despite their strong effect on the pair distribution function arising from cluster formation. Cluster formation is also studied in transient data following a quench from lambda = 0; hydrodynamic interactions slow the formation rate, again by a factor of roughly two.
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)
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.
Boltzmann equations for a binary one-dimensional ideal gas.
Boozer, A D
2011-09-01
We consider a time-reversal invariant dynamical model of a binary ideal gas of N molecules in one spatial dimension. By making time-asymmetric assumptions about the behavior of the gas, we derive Boltzmann and anti-Boltzmann equations that describe the evolution of the single-molecule velocity distribution functions for an ensemble of such systems. We show that for a special class of initial states of the ensemble one can obtain an exact expression for the N-molecule velocity distribution function, and we use this expression to rigorously prove that the time-asymmetric assumptions needed to derive the Boltzmann and anti-Boltzmann equations hold in the limit of large N. Our results clarify some subtle issues regarding the origin of the time asymmetry of Boltzmann's H theorem.
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.
Molecular kinetic analysis of a local equilibrium Carnot cycle
NASA Astrophysics Data System (ADS)
Izumida, Yuki; Okuda, Koji
2017-07-01
We identify a velocity distribution function of ideal gas particles that is compatible with the local equilibrium assumption and the fundamental thermodynamic relation satisfying the endoreversibility. We find that this distribution is a Maxwell-Boltzmann distribution with a spatially uniform temperature and a spatially varying local center-of-mass velocity. We construct the local equilibrium Carnot cycle of an ideal gas, based on this distribution, and show that the efficiency of the present cycle is given by the endoreversible Carnot efficiency using the molecular kinetic temperatures of the gas. We also obtain an analytic expression of the efficiency at maximum power of our cycle under a small temperature difference. Our theory is also confirmed by a molecular dynamics simulation.
Hu, Kainan; Zhang, Hongwu; Geng, Shaojuan
2016-10-01
A decoupled scheme based on the Hermite expansion to construct lattice Boltzmann models for the compressible Navier-Stokes equations with arbitrary specific heat ratio is proposed. The local equilibrium distribution function including the rotational velocity of particle is decoupled into two parts, i.e., the local equilibrium distribution function of the translational velocity of particle and that of the rotational velocity of particle. From these two local equilibrium functions, two lattice Boltzmann models are derived via the Hermite expansion, namely one is in relation to the translational velocity and the other is connected with the rotational velocity. Accordingly, the distribution function is also decoupled. After this, the evolution equation is decoupled into the evolution equation of the translational velocity and that of the rotational velocity. The two evolution equations evolve separately. The lattice Boltzmann models used in the scheme proposed by this work are constructed via the Hermite expansion, so it is easy to construct new schemes of higher-order accuracy. To validate the proposed scheme, a one-dimensional shock tube simulation is performed. The numerical results agree with the analytical solutions very well.
Lattice Boltzmann Stokesian dynamics.
Ding, E J
2015-11-01
Lattice Boltzmann Stokesian dynamics (LBSD) is presented for simulation of particle suspension in Stokes flows. This method is developed from Stokesian dynamics (SD) with resistance and mobility matrices calculated using the time-independent lattice Boltzmann algorithm (TILBA). TILBA is distinguished from the traditional lattice Boltzmann method (LBM) in that a background matrix is generated prior to the calculation. The background matrix, once generated, can be reused for calculations for different scenarios, thus the computational cost for each such subsequent calculation is significantly reduced. The LBSD inherits the merits of the SD where both near- and far-field interactions are considered. It also inherits the merits of the LBM that the computational cost is almost independent of the particle shape.
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.
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.
Sensitivity of an exothermic chemical wave front to a departure from local equilibrium.
Nowakowski, B; Lemarchand, A
2007-11-07
We study the propagation of an exothermic chemical wave front in a reactive dilute gas and show that the particle velocity distribution departs from the Maxwellian form in the front zone. The analytical corrections to the balance equations for concentrations, temperature, and stream velocity induced by the departure from local equilibrium are derived from a perturbative solution of the Boltzmann equation. Our analytical predictions of the front properties, including its propagation speed, compare well with microscopic simulations of the particle dynamics.
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)
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.
NASA Astrophysics Data System (ADS)
Tang, M.; Zhang, S.; Liu, Y.
2015-12-01
Several important equilibrium Si isotope fractionation factors among minerals, organic molecules and the H4SiO4 solution are complemented to facilitate explanation of distributions of Si isotope in the Earth's surface environments. The results reveal that heavy Si isotopes will be significantly enriched in the secondary silicate minerals in comparison to aqueous H4SiO4. On the contrary, quadra-coordinated organosilicon complexes are enriched in light silicon isotope relative to the solution. The extent of 28Si-enrichment in hyper-coordinated organosilicon complexes is found the largest. In addition, the large kinetic isotope effect associated with the polymerization of monosilicic acid and dimer is calculated and the result supports previous statement that highly 28Si-enrichment in the formation of amorphous quartz precursor contributes to the discrepancy between theoretical calculations and field observations. With equilibrium Si isotope fractionation factors provided here, Si isotope distributions in many surface systems of the Earth can be explained. For example, the change of bulk soil δ30Si can be predicted as a concave pattern with respect to weathering degree, with the minimum value where allophane completely dissolves and the total amount of sesqui-oxides and poorly crystalline minerals reaches its maximum. When well-crystallized clays start to precipitate from pore solutions under equilibrium conditions, the bulk soil δ30Si will increase again and reach a constant value. Similarly, the precipitation of crystalline smectite and the dissolution of poorly crystalline kaolinite may explain δ30Si variations in the ground water profile. Equilibrium Si isotope fractionations among quadra-coordinated organosilicon complexes and the H4SiO4 solution may also shed the light on the Si isotope distributions in Si-accumulating plants.
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.
Preserving the Boltzmann ensemble in replica-exchange molecular dynamics.
Cooke, Ben; Schmidler, Scott C
2008-10-28
We consider the convergence behavior of replica-exchange molecular dynamics (REMD) [Sugita and Okamoto, Chem. Phys. Lett. 314, 141 (1999)] based on properties of the numerical integrators in the underlying isothermal molecular dynamics (MD) simulations. We show that a variety of deterministic algorithms favored by molecular dynamics practitioners for constant-temperature simulation of biomolecules fail either to be measure invariant or irreducible, and are therefore not ergodic. We then show that REMD using these algorithms also fails to be ergodic. As a result, the entire configuration space may not be explored even in an infinitely long simulation, and the simulation may not converge to the desired equilibrium Boltzmann ensemble. Moreover, our analysis shows that for initial configurations with unfavorable energy, it may be impossible for the system to reach a region surrounding the minimum energy configuration. We demonstrate these failures of REMD algorithms for three small systems: a Gaussian distribution (simple harmonic oscillator dynamics), a bimodal mixture of Gaussians distribution, and the alanine dipeptide. Examination of the resulting phase plots and equilibrium configuration densities indicates significant errors in the ensemble generated by REMD simulation. We describe a simple modification to address these failures based on a stochastic hybrid Monte Carlo correction, and prove that this is ergodic.
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.
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
NASA Astrophysics Data System (ADS)
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.
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.
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
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.
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.
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.
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.
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.
Zhao, Yong; Abreu, Eladio; Kim, Jinyong; Stadler, Guido; Eskiocak, Ugur; Terns, Michael P.; Terns, Rebecca M.; Shay, Jerry W.; Wright, Woodring E.
2011-01-01
SUMMARY Specific information about how telomerase acts in vivo is necessary for understanding telomere dynamics in human tumor cells. Our results imply that under homeostatic telomere length-maintenance conditions only one molecule of telomerase acts at each telomere during every cell division and processively adds ~60 nt to each end. In contrast, multiple molecules of telomerase act at each telomere when telomeres are elongating (non-equilibrium conditions). Telomerase extension is less processive during the first few weeks following the reversal of long-term treatment with the telomerase inhibitor GRN163L, a time when Cajal bodies fail to deliver telomerase RNA to telomeres. This result implies that processing of telomerase by Cajal bodies may affect its processivity. Overexpressed telomerase is also less processive than the endogenously expressed telomerase. These findings reveal two major distinct extension modes adopted by telomerase in vivo. PMID:21549308
Equilibrium distributions and relaxation times in gaslike economic models: An analytical derivation
NASA Astrophysics Data System (ADS)
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.
The fundamental and universal nature of Boltzmann`s constant
Biedenharn, L.C.; Solem, J.C.
1996-07-01
The nature of Boltzmann`s constant is very unclear in the physics literature. In the first part of this paper, on general considerations, the authors examine this situation in detail and demonstrate the conclusion that Boltzmann`s constant is indeed both fundamental and universal. As a consequence of their development they find there is an important implication of this work for the problem of the entropy of information. In the second part they discuss, Szilard`s famous construction showing in detail how his result is incompatible with the demonstrations in both parts 1 and 2.
On the dispute between Boltzmann and Gibbs entropy
Buonsante, Pierfrancesco; Franzosi, Roberto Smerzi, Augusto
2016-12-15
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.
Accurate deterministic solutions for the classic Boltzmann shock profile
NASA Astrophysics Data System (ADS)
Yue, Yubei
The Boltzmann equation or Boltzmann transport equation is a classical kinetic equation devised by Ludwig Boltzmann in 1872. It is regarded as a fundamental law in rarefied gas dynamics. Rather than using macroscopic quantities such as density, temperature, and pressure to describe the underlying physics, the Boltzmann equation uses a distribution function in phase space to describe the physical system, and all the macroscopic quantities are weighted averages of the distribution function. The information contained in the Boltzmann equation is surprisingly rich, and the Euler and Navier-Stokes equations of fluid dynamics can be derived from it using series expansions. Moreover, the Boltzmann equation can reach regimes far from the capabilities of fluid dynamical equations, such as the realm of rarefied gases---the topic of this thesis. Although the Boltzmann equation is very powerful, it is extremely difficult to solve in most situations. Thus the only hope is to solve it numerically. But soon one finds that even a numerical simulation of the equation is extremely difficult, due to both the complex and high-dimensional integral in the collision operator, and the hyperbolic phase-space advection terms. For this reason, until few years ago most numerical simulations had to rely on Monte Carlo techniques. In this thesis I will present a new and robust numerical scheme to compute direct deterministic solutions of the Boltzmann equation, and I will use it to explore some classical gas-dynamical problems. In particular, I will study in detail one of the most famous and intrinsically nonlinear problems in rarefied gas dynamics, namely the accurate determination of the Boltzmann shock profile for a gas of hard spheres.
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.
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
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.
Bean, William T; Stafford, Robert; Butterfield, H Scott; Brashares, Justin S
2014-01-01
Species distributions are known to be limited by biotic and abiotic factors at multiple temporal and spatial scales. Species distribution models, however, frequently assume a population at equilibrium in both time and space. Studies of habitat selection have repeatedly shown the difficulty of estimating resource selection if the scale or extent of analysis is incorrect. Here, we present a multi-step approach to estimate the realized and potential distribution of the endangered giant kangaroo rat. First, we estimate the potential distribution by modeling suitability at a range-wide scale using static bioclimatic variables. We then examine annual changes in extent at a population-level. We define "available" habitat based on the total suitable potential distribution at the range-wide scale. Then, within the available habitat, model changes in population extent driven by multiple measures of resource availability. By modeling distributions for a population with robust estimates of population extent through time, and ecologically relevant predictor variables, we improved the predictive ability of SDMs, as well as revealed an unanticipated relationship between population extent and precipitation at multiple scales. At a range-wide scale, the best model indicated the giant kangaroo rat was limited to areas that received little to no precipitation in the summer months. In contrast, the best model for shorter time scales showed a positive relation with resource abundance, driven by precipitation, in the current and previous year. These results suggest that the distribution of the giant kangaroo rat was limited to the wettest parts of the drier areas within the study region. This multi-step approach reinforces the differing relationship species may have with environmental variables at different scales, provides a novel method for defining "available" habitat in habitat selection studies, and suggests a way to create distribution models at spatial and temporal scales
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.
[Welding arc temperature field measurements based on Boltzmann spectrometry].
Si, Hong; Hua, Xue-Ming; Zhang, Wang; Li, Fang; Xiao, Xiao
2012-09-01
Arc plasma, as non-uniform plasma, has complicated energy and mass transport processes in its internal, so plasma temperature measurement is of great significance. Compared with absolute spectral line intensity method and standard temperature method, Boltzmann plot measuring is more accurate and convenient. Based on the Boltzmann theory, the present paper calculates the temperature distribution of the plasma and analyzes the principle of lines selection by real time scanning the space of the TIG are measurements.
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
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.
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.
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.
ON QUIET-TIME SOLAR WIND ELECTRON DISTRIBUTIONS IN DYNAMICAL EQUILIBRIUM WITH LANGMUIR TURBULENCE
Zaheer, S.; Yoon, P. H.
2013-10-01
A recent series of papers put forth a self-consistent theory of an asymptotically steady-state electron distribution function and Langmuir turbulence intensity. The theory was developed in terms of the κ distribution which features Maxwellian low-energy electrons and a non-Maxwellian energetic power-law tail component. The present paper discusses a generalized κ distribution that features a Davydov-Druyvesteyn type of core component and an energetic power-law tail component. The physical motivation for such a generalization is so that the model may reflect the influence of low-energy electrons interacting with low-frequency kinetic Alfvénic turbulence as well as with high-frequency Langmuir turbulence. It is shown that such a solution and the accompanying Langmuir wave spectrum rigorously satisfy the balance requirement between the spontaneous and induced emission processes in both the particle and wave kinetic equations, and approximately satisfy the similar balance requirement between the spontaneous and induced scattering processes, which are nonlinear. In spite of the low velocity modification of the electron distribution function, it is shown that the resulting asymptotic velocity power-law index α, where f{sub e} ∼ v {sup –α} is close to the average index observed during the quiet-time solar wind condition, i.e., α ∼ O(6.5) whereas α{sub average} ∼ 6.69, according to observation.
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…
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.
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.
Skrdla, Peter J
2009-08-20
The potential applications of dispersive kinetic models range from solid-state conversions to gas-phase chemical physics and to microbiology. Here, the derivation and application of two such models, for use in solid-state applications, is presented. The models are based on the concept of a Maxwell-Boltzmann distribution of activation energies. The ability of the models to fit/explain an assortment of asymmetric, sigmoidal conversion-versus-time transients presented in the recent literature, as well as to provide physicochemical interpretations of the kinetics via the two fit parameters, alpha and beta, makes them a powerful tool for understanding nucleation/denucleation rate-limited processes that are involved in many phase transformations, dissolutions and crystallizations.
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.
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.
Koga, S; Shibata, T; Terasaki, R; Kameyama, N; Hatayama, A; Bacal, M; Tsumori, K
2012-02-01
In negative ion sources for the neutral beam injection, it is important to calculate H atom flux onto the plasma grid (PG) surface for the evaluation of H(-) production on the PG surface. We have developed a neutral (H(2) molecules and H atoms) transport code. In the present study, the neutral transport code is applied to the analysis of the H(2) and H transport in a NIFS-R&D ion source in order to calculate the flux onto the PG surface. Taking into account non-equilibrium feature of the electron energy distribution function (EEDF), i.e., the fast electron component, we have done the neutral transport simulation. The results suggest that the precise evaluation of the EEDF, especially in the energy range 15 eV < E < 30 eV is important for the dissociation rate of H(2) molecules by the electron impact collision and the resultant H atom flux on the PG.
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
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.
Fluctuating multicomponent lattice Boltzmann model.
Belardinelli, D; Sbragaglia, M; Biferale, L; Gross, M; Varnik, F
2015-02-01
Current implementations of fluctuating lattice Boltzmann equations (FLBEs) describe single component fluids. In this paper, a model based on the continuum kinetic Boltzmann equation for describing multicomponent fluids is extended to incorporate the effects of thermal fluctuations. The thus obtained fluctuating Boltzmann equation is first linearized to apply the theory of linear fluctuations, and expressions for the noise covariances are determined by invoking the fluctuation-dissipation theorem directly at the kinetic level. Crucial for our analysis is the projection of the Boltzmann equation onto the orthonormal Hermite basis. By integrating in space and time the fluctuating Boltzmann equation with a discrete number of velocities, the FLBE is obtained for both ideal and nonideal multicomponent fluids. Numerical simulations are specialized to the case where mean-field interactions are introduced on the lattice, indicating a proper thermalization of the system.
Schuck, Peter; Gillis, Richard B.; Besong, Tabot M.D.; Almutairi, Fahad; Adams, Gary G.; Rowe, Arthur J.; Harding, Stephen E.
2014-01-01
Sedimentation equilibrium (analytical ultracentrifugation) is one of the most inherently suitable methods for the determination of average molecular weights and molecular weight distributions of polymers, because of its absolute basis (no conformation assumptions) and inherent fractionation ability (without the need for columns or membranes and associated assumptions over inertness). With modern instrumentation it is also possible to run up to 21 samples simultaneously in a single run. Its application has been severely hampered because of difficulties in terms of baseline determination (incorporating estimation of the concentration at the air/solution meniscus) and complexity of the analysis procedures. We describe a new method for baseline determination based on a smart-smoothing principle and built into the highly popular platform SEDFIT for the analysis of the sedimentation behavior of natural and synthetic polymer materials. The SEDFIT-MSTAR procedure – which takes only a few minutes to perform - is tested with four synthetic data sets (including a significantly non-ideal system) a naturally occurring protein (human IgG1) and two naturally occurring carbohydrate polymers (pullulan and λ–carrageenan) in terms of (i) weight average molecular weight for the whole distribution of species in the sample (ii) the variation in “point” average molecular weight with local concentration in the ultracentrifuge cell and (iii) molecular weight distribution. PMID:24244936
Schuck, Peter; Gillis, Richard B; Besong, Tabot M D; Almutairi, Fahad; Adams, Gary G; Rowe, Arthur J; Harding, Stephen E
2014-01-07
Sedimentation equilibrium (analytical ultracentrifugation) is one of the most inherently suitable methods for the determination of average molecular weights and molecular weight distributions of polymers, because of its absolute basis (no conformation assumptions) and inherent fractionation ability (without the need for columns or membranes and associated assumptions over inertness). With modern instrumentation it is also possible to run up to 21 samples simultaneously in a single run. Its application has been severely hampered because of difficulties in terms of baseline determination (incorporating estimation of the concentration at the air/solution meniscus) and complexity of the analysis procedures. We describe a new method for baseline determination based on a smart-smoothing principle and built into the highly popular platform SEDFIT for the analysis of the sedimentation behavior of natural and synthetic polymer materials. The SEDFIT-MSTAR procedure - which takes only a few minutes to perform - is tested with four synthetic data sets (including a significantly non-ideal system), a naturally occurring protein (human IgG1) and two naturally occurring carbohydrate polymers (pullulan and λ-carrageenan) in terms of (i) weight average molecular weight for the whole distribution of species in the sample (ii) the variation in "point" average molecular weight with local concentration in the ultracentrifuge cell and (iii) molecular weight distribution.
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.
Lattice Boltzmann simulation of droplet formation in T-junction geometries
NASA Astrophysics Data System (ADS)
Busuioc, Sergiu; Ambruş, Victor E.; Sofonea, Victor
2017-01-01
The formation of droplets in T-junction configurations is investigated using a two-dimensional Lattice Boltzmann model for liquid-vapor systems. We use an expansion of the equilibrium distribution function with respect to Hermite polynomials and an off-lattice velocity set. To evolve the distribution functions we use the second order corner transport upwind numerical scheme and a third order scheme is used to compute the gradient operators in the force term. The droplet formation successfully recovers the squeezing, dripping and jetting regimes. We find that the droplet length decreases proportionally with the flow rate of the continuous phase and increases with the flow rate of the dispersed phase in all simulation configurations and has a linear dependency on the surface tension parameter κ.
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.
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.
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.
Lattice Boltzmann method for electromagnetic wave propagation
NASA Astrophysics Data System (ADS)
Hanasoge, S. M.; Succi, S.; Orszag, S. A.
2011-10-01
We present a new Lattice Boltzmann (LB) formulation to solve the Maxwell equations for electromagnetic (EM) waves propagating in a heterogeneous medium. By using a pseudo-vector discrete Boltzmann distribution, the scheme is shown to reproduce the continuum Maxwell equations. The technique compares well with a pseudo-spectral method at solving for two-dimensional wave propagation in a heterogeneous medium, which by design contains substantial contrasts in the refractive index. The extension to three dimensions follows naturally and, owing to the recognized efficiency of LB schemes for parallel computation in irregular geometries, it gives a powerful method to numerically simulate a wide range of problems involving EM wave propagation in complex media.
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.
On the Non-Equilibrium Population Distribution of E-Methanol in Dark Clouds
NASA Astrophysics Data System (ADS)
Wollman, Emma
2007-12-01
The goal of this project was to determine the typical distribution of rotational level populations in the k=0 ladder of E-methanol in dark clouds in order to provide another observational test for theoretical models of pumping. We used our own observations of several sources with the 12-m ARO telescope on Kitt Peak as well as the published observational results by Slysh et al. (1999). The relative level populations (excitation temperatures) were determined from the measured intensity ratios of a series of the J(0)-J(-1) transitions of E-methanol under the assumption of spontaneous, optically thin emission. We observed the J(0)-J(-1) lines in six sources: W75N, DR21N, DR21, and three positions at DR21OH. The J=1 to J=5 lines were observed for all sources and the J=7 line was observed for W75N, DR21N, and one position in DR21OH. We also used Slysh et al.'s results for the J=1 through 4 lines in 52 sources, for the J=5 line in 50 sources, for the J=6 line in 15 sources, and for the J=7 and 8 lines in 2 sources. We determined the excitation temperatures of the involved levels in the k=0 ladder relative to the 1(0) level for each source and averaged the results over the sources. The average excitation temperatures demonstrate strong evidence of overcooling in the k=0 ladder - the excitation temperature increases linearly with increasing energy, from 8 K to 35 K. Our observations confirm this tendency of overcooling. We will discuss the agreement of these results with the predictions of the current models of methanol pumping. The author thanks the technical staff of the 12-m ARO telescope for help with the observations. This project was supported by the NSF/REU grant AST-0354056 and the Nantucket Maria Mitchell Association.
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.
Jacxsens, L; Devlieghere, F; Debevere, J
2002-03-01
The impact of temperature fluctuations in a simulated cold distribution chain, typical of commercial practice, was investigated on both the microbial and sensorial quality of equilibrium modified atmosphere (EMA) packaged minimally processed vegetables. The internal O2 concentration of the designed packages could be predicted for the different steps of the simulated distribution chain by applying an integrated mathematical system. The internal atmosphere in the packages remained in its aerobic range during storage in the chain due to the application of high permeable packaging films for O2 and CO2. Spoilage microorganisms were proliferating fast on minimally processed bell peppers and lettuce. Yeasts showed to be the shelf-life limiting group. Visual properties limited the sensorial shelf-life. Listeria monocytogenes was able to multiply on cucumber slices, survived on minimally processed lettuce and decreased in number on bell peppers due to the combination of low pH and refrigeration. Aeromonas caviae was multiplying on both cucumber slices and mixed lettuce, but was as well inhibited by the low pH of bell peppers. Storage temperature control was found to be of paramount importance for the microbial (spoilage and safety) and sensorial quality evaluation of EMA-packaged minimally processed vegetables.
Thermodynamic consistency of liquid-gas lattice Boltzmann simulations.
Wagner, A J
2006-11-01
Lattice Boltzmann simulations have been very successful in simulating liquid-gas and other multiphase fluid systems. However, the underlying second-order analysis of the equation of motion has long been known to be insufficient to consistently derive the fourth-order terms that are necessary to represent an extended interface. These same terms are also responsible for thermodynamic consistency--i.e., to obtain a true equilibrium solution with both a constant chemical potential and a constant pressure. In this article we present an equilibrium analysis of nonideal lattice Boltzmann methods of sufficient order to identify those higher-order terms that lead to a lack of thermodynamic consistency. We then introduce a thermodynamically consistent forcing method.
Observation of a Persistent Non-Equilibrium State in an Extremely Isotropic Harmonic Potential
NASA Astrophysics Data System (ADS)
Lobser, D. S.
Ludwig Boltzmann made tremendously important contributions to the problem of con- necting macroscopic, empirical phenomena with microscopic, atomistic dynamics. At the end of the nineteenth century, Boltzmann was confronted with various strong objections to his work. For example, Boltzmann's atomistic explanations presuppose the reality of atoms, a notion that was vigorously rejected in some circles [14, 38]. Then too, there was the critique by Loschmidt that Boltzmann's H-theorem, put forth as a microscopic explanation for the Second Law of Thermodynamics, could hardly account for irreversible physics when the individual two-atom collisions were each reversible [18, 42]. Still intriguing today is the existence of special cases of the Boltzmann equation in which time-varying distributions of atoms re- sist the imperative of equilibration, even in the presence of collisions. Boltzmann discussed such situations in a paper dedicated to responding to Loschmidt's critique [7, 4]. Perhaps Boltzmann's motivation was to enumerate special cases where his famous H value does not relax as it should, and by enumerating them, point out their nonnaturalness, their artificiality. Damping, or relaxation to equilibrium, of a time-invariant phase-space distribution, is an all-but universal result predicted by the Boltzmann equation. Such improbable systems of atoms have only very recently been realized experimentally. Kinoshita et al. [36] experimentally confirmed that atoms constrained to move in a quasi one-dimensional potential, an atomistic Newtons cradle, exhibit vastly suppressed relaxation. Chevy et al. [15] observed long-lived breathe-mode oscillations in highly elongated but still 3D geometries. Perhaps one of the more interesting cases is the vanishing damping of the monopole breathe-mode oscillation in a spherically symmetric harmonic oscillator [29], where a cloud of atoms experiences undamped temperature oscillations, causing the cloud to expand and contract as if it
Coupling lattice Boltzmann model for simulation of thermal flows on standard lattices.
Li, Q; Luo, K H; He, Y L; Gao, Y J; Tao, W Q
2012-01-01
In this paper, a coupling lattice Boltzmann (LB) model for simulating thermal flows on the standard two-dimensional nine-velocity (D2Q9) lattice is developed in the framework of the double-distribution-function (DDF) approach in which the viscous heat dissipation and compression work are considered. In the model, a density distribution function is used to simulate the flow field, while a total energy distribution function is employed to simulate the temperature field. The discrete equilibrium density and total energy distribution functions are obtained from the Hermite expansions of the corresponding continuous equilibrium distribution functions. The pressure given by the equation of state of perfect gases is recovered in the macroscopic momentum and energy equations. The coupling between the momentum and energy transports makes the model applicable for general thermal flows such as non-Boussinesq flows, while the existing DDF LB models on standard lattices are usually limited to Boussinesq flows in which the temperature variation is small. Meanwhile, the simple structure and general features of the DDF LB approach are retained. The model is tested by numerical simulations of thermal Couette flow, attenuation-driven acoustic streaming, and natural convection in a square cavity with small and large temperature differences. The numerical results are found to be in good agreement with the analytical solutions and/or other numerical results reported in the literature.
Maxwell-Boltzmann type Hawking radiation
NASA Astrophysics Data System (ADS)
Yoon, Youngsub
2017-04-01
Twenty years ago, Rovelli proposed that the degeneracy of black hole (i.e. the exponential of the Bekenstein-Hawking entropy) is given by the number of ways the black hole horizon area can be expressed as a sum of unit areas. However, when counting the sum, one should treat the area quanta on the black hole horizon as distinguishable. This distinguishability of area quanta is noted in Rovelli’s paper. Building on this idea, we derive that the Hawking radiation spectrum is not given by Planck radiation spectrum (i.e. Bose-Einstein distribution) but given by Maxwell-Boltzmann distribution.
Electron Boltzmann equation in nonthermal plasmas
NASA Technical Reports Server (NTRS)
Kunc, J. A.; Soon, W. H.
1991-01-01
Numerical and analytical solutions of the electron Boltzmann equation for a two-temperature steady-state He plasma are examined in a broad range of conditions, i.e., atom temperature ranging from 5000 K to 20,000 K; electron temperature ranging from 10,000 K to 20,000 K; and atom density ranging from 10 to the 10th to 10 to the 18th per cu cm. The WKB analytical solution is shown to be satisfactory in most situations. Attention is also given to the deviation of the electron distribution from Maxwellian, and to the possibility of raising the tail of the distribution.
Koshka, Yaroslav; Perera, Dilina; Hall, Spencer; Novotny, M A
2017-07-01
The possibility of using a quantum computer D-Wave 2X with more than 1000 qubits to determine the global minimum of the energy landscape of trained restricted Boltzmann machines is investigated. In order to overcome the problem of limited interconnectivity in the D-Wave architecture, the proposed RBM embedding combines multiple qubits to represent a particular RBM unit. The results for the lowest-energy (the ground state) and some of the higher-energy states found by the D-Wave 2X were compared with those of the classical simulated annealing (SA) algorithm. In many cases, the D-Wave machine successfully found the same RBM lowest-energy state as that found by SA. In some examples, the D-Wave machine returned a state corresponding to one of the higher-energy local minima found by SA. The inherently nonperfect embedding of the RBM into the Chimera lattice explored in this work (i.e., multiple qubits combined into a single RBM unit were found not to be guaranteed to be all aligned) and the existence of small, persistent biases in the D-Wave hardware may cause a discrepancy between the D-Wave and the SA results. In some of the investigated cases, introduction of a small bias field into the energy function or optimization of the chain-strength parameter in the D-Wave embedding successfully addressed difficulties of the particular RBM embedding. With further development of the D-Wave hardware, the approach will be suitable for much larger numbers of RBM units.
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
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.
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.
Brownian motion from Boltzmann's equation.
NASA Technical Reports Server (NTRS)
Montgomery, D.
1971-01-01
Two apparently disparate lines of inquiry in kinetic theory are shown to be equivalent: (1) Brownian motion as treated by the (stochastic) Langevin equation and Fokker-Planck equation; and (2) Boltzmann's equation. The method is to derive the kinetic equation for Brownian motion from the Boltzmann equation for a two-component neutral gas by a simultaneous expansion in the density and mass ratios.
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)
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.
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.
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
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.
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
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.
Privacy-Preserving Restricted Boltzmann Machine
Li, Yu
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. PMID:25101139
NASA Astrophysics Data System (ADS)
Izzo, Dario; Petazzi, Lorenzo
2006-08-01
We present a satellite path planning technique able to make identical spacecraft aquire a given configuration. The technique exploits a behaviour-based approach to achieve an autonomous and distributed control over the relative geometry making use of limited sensorial information. A desired velocity is defined for each satellite as a sum of different contributions coming from generic high level behaviours: forcing the final desired configuration the behaviours are further defined by an inverse dynamic calculation dubbed Equilibrium Shaping. We show how considering only three different kind of behaviours it is possible to acquire a number of interesting formations and we set down the theoretical framework to find the entire set. We find that allowing a limited amount of communication the technique may be used also to form complex lattice structures. Several control feedbacks able to track the desired velocities are introduced and discussed. Our results suggest that sliding mode control is particularly appropriate in connection with the developed technique.
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
Lattice Boltzmann model for agrochemical transport in soils.
Zhang, Xiaoxian; Ren, Li
2003-12-01
Agrochemical transport in soils is complicated and involves physical, chemical and biochemical reactions; its mathematical modelling remains a challenging task. This paper presents a lattice Boltzmann model to simulate the agrochemical movement. The lattice Boltzmann model is a microscopic and process-based model, simulating the transport process by tracking chemical particles. The model presented in this paper is for one-dimensional vertical leaching and assumes that the chemical particles at the microscopic level move in five directions: one stagnant, two in vertical direction and two in an internal horizontal direction bounded by two reactive walls. Reactions at the walls are assumed to take place at two different rates, one in fast rate where the chemicals in the solution and on the wall are in an instant equilibrium, and the other in slow rate where the mass exchange rate between the chemicals in the solution and on the wall is a first-order kinetic. The reactions on both walls are assumed to occur instantly when the chemical particles moving in the internal direction hit the walls. To test the model, we measured the leaching of atrazine through soil columns in the laboratory. The results simulated with the lattice Boltzmann model are compared with the measured breakthrough curves and the non-equilibrium two-site convection-dispersion model; they all show close agreement. The transport parameters needed in the models are obtained from the measurement of adsorption isotherm of atrazine, bromide leaching in the same soil columns and calibration.
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.
NASA Technical Reports Server (NTRS)
Yoshikawa, K. K.
1978-01-01
Theoretical results pertaining to internally excited translational-rotational energy relaxation in a spatially uniform diatomic gas far removed from solid boundaries are obtained by solving the Boltzmann equation by means of the Monte Carlo direct simulation method. The analysis is based on calculations involving three different types of initial conditions: equilibrium, nonequilibrium-equipartition (i.e., equipartition is satisfied, but the distributions are perturbed), and nonequilibrium-nonequipartition (i.e., both equipartition and the distributions are perturbed). Results of monatomic-gas simulations are also included to facilitate comparisons with the coupled translational-rotational relaxation simulations, and some simulations for a normal shock-wave structure are briefly examined. The results show that: (1) single-step transitions are the significant mechanisms of intermodal energy transfer; (2) translational-rotational transitions are coupled most efficiently for low-lying states of rotationally excited molecules and least efficiently for highly rotationally excited molecules; and (3) relaxation occurs via a successive set of distributions that are not Maxwell-Boltzmann (nonlocal Maxwellian).
Zermelo, Boltzmann, and the recurrence paradox
NASA Astrophysics Data System (ADS)
Steckline, Vincent S.
1983-10-01
The papers exchanged by Ludwig Boltzmann and Ernst Zermelo concerning the recurrence paradox are summarized. The historical context of the paradox, Zermelo's proof of the paradox, his opinions of its consequences, Boltzmann's reply, and the ensuing discussion are described.
Estimation of Boltzmann damping coefficients in beam models
NASA Technical Reports Server (NTRS)
Banks, H. T.; Fabiano, R. H.; Wang, Y.
1988-01-01
A distributed parameter model of a flexible structure with Boltzmann type viscoelastic damping is discussed. A computational method for the estimation of the damping parameters is developed, and theoretical convergence results are given. An example is presented in which actual experimental data is used, demonstrating the efficacy of the computational method and the plausibility of the model for predicting response in damped structures.
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.
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.
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.
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.
NASA Astrophysics Data System (ADS)
Beretta, Gian P.
2008-09-01
A rate equation for a discrete probability distribution is discussed as a route to describe smooth relaxation towards the maximum entropy distribution compatible at all times with one or more linear constraints. The resulting dynamics follows the path of steepest entropy ascent compatible with the constraints. The rate equation is consistent with the Onsager theorem of reciprocity and the fluctuation-dissipation theorem. The mathematical formalism was originally developed to obtain a quantum theoretical unification of mechanics and thermodinamics. It is presented here in a general, non-quantal formulation as a part of an effort to develop tools for the phenomenological treatment of non-equilibrium problems with applications in engineering, biology, sociology, and economics. The rate equation is also extended to include the case of assigned time-dependences of the constraints and the entropy, such as for modeling non-equilibrium energy and entropy exchanges.
NASA Astrophysics Data System (ADS)
Suleimenov, I.; Aushev, V.; Adamov, T.; Vasiliev, I.
Modern investigations show that the effect of acoustic and acoustic-gravity waves amplification strongly influence on the temperature balance in atmosphere. These waves may be amplified due to the transformation of energy of chemically active (or ionized) components into the energy of wave motion, i.e. the nature of the effect is the same as the amplification of sound in other non-equilibrium gas media (for example, in gas discharge plasma). Recently Jiyao Xu (1999) reported that the theory of such waves might be developed in the same way as the theory of acoustic-gravity waves. It is shown that the influence of inhomogeneous altitude distribution of temperature should be taken into account for the correct interpretation of temperature balance in the atmosphere. In other words, the self-agreed problem have to be solved: transformation of chemical energy into energy of wave motion change the vertical profile of the atmosphere temperature, but the profile of the temperature itself determine the amplification coefficient of the wave. The results of analytical solution of the problem are reported. We show that the sign of temperature gradient strongly influence on the behavior of amplified acoustic and acoustic-gravity waves. The regime of amplification is stable when the second derivative of the temperature is negative (temperature has minimum at some point). In other words the stable channel of amplification of the waves may exist, for example, in the tube when the temperature of the walls is higher than the temperature of the gas at the axe. The different instabilities appear in the opposite case when the temperature in the reference point has a maximum. In particular, it means that the amplification of acoustic waves in gas discharge tubes cannot be stable. Moreover, our results show that self-generation of acoustic-gravity in middle atmosphere due to photochemical reactions cannot be stable process too. This conclusion is in accordance with known experimental
Anisotropic Flow from Non-equilibrium Initial Condition with a Saturation Scale
NASA Astrophysics Data System (ADS)
Greco, V.; Ruggieri, M.; Scardina, F.; Plumari, S.; Puglisi, A.
2014-03-01
A current goal of relativistic heavy ion collisions experiments is to understand the impact of initial non-equilibrium on final observables. A Color Glass Condensate (CGC) as the limiting state of QCD matter at very high density implies initial non-thermal distribution at least for momenta below the saturation scale. In viscous hydrodynamics simulations, a standard Glauber initial condition leads to estimate 4πη/s ~ 1, while employing the Kharzeev-Levin-Nardi (KLN) modeling of the CGC leads to at least a factor of 2 larger η/s. Within a kinetic theory approach based on a relativistic Boltzmann-like transport simulation, our main result is that the out-of-equilibrium initial distribution in p-space reduces the effciency in building-up the elliptic flow. At RHIC energy we find the available data on υ2 are in agreement with a 4πη/s ~ 1 also for KLN initial conditions.
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
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.
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.
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.
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.
A method for direct numerical integration of the Boltzmann equation
NASA Technical Reports Server (NTRS)
Cheremisin, F. G.
1972-01-01
The principal difficulties in numerical solution of the Boltzmann equation are considered. The study is aimed at formulating a numerical solution in such a manner that it contains a minimum amount of excess information at the distribution function level. It is pointed out that the accurate calculation of the distribution function at each point in phase space requires a tremendous number of operations, due to the necessity of solving five-fold quadratures in the collision integral. This results in the operational memory of the digital computer being insufficient to store all the data on the distribution functions at the necessary points in phase space. An algorithm is constructed involving successive iterations of the Boltzmann equation which does not require storage of each step of the new distribution function.
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.
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.
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.
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.
Ambruş, Victor Eugen; Sofonea, Victor
2012-07-01
We use the spherical coordinate system in the momentum space and an appropriate discretization procedure to derive a hierarchy of lattice Boltzmann (LB) models with variable temperature. The separation of the integrals in the momentum space into angular and radial parts allows us to compute the moments of the equilibrium distribution function by means of Gauss-Legendre and Gauss-Laguerre quadratures, as well as to find the elements of the discrete momentum set for each LB model in the hierarchy. The capability of the high-order models in this hierarchy to capture specific effects in microfluidics is investigated through a computer simulation of Couette flow by using the Shakhov collision term to get the right value of the Prandtl number.
A fast iterative scheme for the linearized Boltzmann equation
NASA Astrophysics Data System (ADS)
Wu, Lei; Zhang, Jun; Liu, Haihu; Zhang, Yonghao; Reese, Jason M.
2017-06-01
Iterative schemes to find steady-state solutions to the Boltzmann equation are efficient for highly rarefied gas flows, but can be very slow to converge in the near-continuum flow regime. In this paper, a synthetic iterative scheme is developed to speed up the solution of the linearized Boltzmann equation by penalizing the collision operator L into the form L = (L + Nδh) - Nδh, where δ is the gas rarefaction parameter, h is the velocity distribution function, and N is a tuning parameter controlling the convergence rate. The velocity distribution function is first solved by the conventional iterative scheme, then it is corrected such that the macroscopic flow velocity is governed by a diffusion-type equation that is asymptotic-preserving into the Navier-Stokes limit. The efficiency of this new scheme is assessed by calculating the eigenvalue of the iteration, as well as solving for Poiseuille and thermal transpiration flows. We find that the fastest convergence of our synthetic scheme for the linearized Boltzmann equation is achieved when Nδ is close to the average collision frequency. The synthetic iterative scheme is significantly faster than the conventional iterative scheme in both the transition and the near-continuum gas flow regimes. Moreover, due to its asymptotic-preserving properties, the synthetic iterative scheme does not need high spatial resolution in the near-continuum flow regime, which makes it even faster than the conventional iterative scheme. Using this synthetic scheme, with the fast spectral approximation of the linearized Boltzmann collision operator, Poiseuille and thermal transpiration flows between two parallel plates, through channels of circular/rectangular cross sections and various porous media are calculated over the whole range of gas rarefaction. Finally, the flow of a Ne-Ar gas mixture is solved based on the linearized Boltzmann equation with the Lennard-Jones intermolecular potential for the first time, and the difference
Maeda, K.; Shibata, T.; Ejiri, H.; Sakai, H.
1983-08-01
Pre-equilibrium and equilibrium deexcitation processes for (..cap alpha..,xn ..gamma..) reactions induced by 50--120 MeV ..cap alpha.. particles were studied. Reaction channels were identified by measuring rotational ..gamma.. rays characteristic of the reaction residues. The branching of the reaction channels gave neutron multiplicity distributions. Features characteristic of the pre-equilibrium process were seen in the reaction channels with small neutron multiplicity x. An exciton model calculation code was developed so as to incorporate both multiparticle emission at the fast pre-equilibrium stage and multiparticle p evaporation at the slow equilibrium stage. The calculation reproduced the neutron multiplicity distributions in the whole range of E/sub ..cap alpha../ = 50--120 MeV. The pre-equilibrium fractions and the entry lines from the pre-equilibrium stage to the equilibrium one were deduced. The pre-equilibrium fractions were found to be approximately 40--60 %, being rather independent of the individual reaction channel. The entry lines slowly increase from 25 to 35 MeV with increasing projectile energy.
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
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.
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.
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.
Diffusion, sedimentation equilibrium, and harmonic trapping of run-and-tumble nanoswimmers.
Wang, Zhengjia; Chen, Hsuan-Yi; Sheng, Yu-Jane; Tsao, Heng-Kwong
2014-05-14
The diffusion of self-propelling nanoswimmers is explored by dissipative particle dynamics in which a nanoswimmer swims by forming an instantaneous force dipole with one of its nearest neighboring solvent beads. Our simulations mimic run-and-tumble behavior by letting the swimmer run for a time τ, then it randomly changes its direction for the next run period. Our simulations show that the swimming speed (ν(a)) of a nanoswimmer is proportional to the propulsion force and the mobility of a pusher is the same as that of a puller. The effective diffusivity is determined by three methods: mean squared displacement, velocity autocorrelation function, and sedimentation equilibrium. The active colloid undergoes directed propulsion at short time scales but changes to random motion at long time scales. The velocity autocorrelation function decreases with time and becomes zero beyond the run time. Under gravity, the concentration profile of active colloids follows Boltzmann distribution with a sedimentation length consistent with that acquired from the drift-diffusion equation. In our simulation, all three methods yield the same result, the effective diffusivity of an active colloid is the sum of the diffusivity of a passive colloid and ν(a)²τ/6. When the active colloids are confined by a harmonic well, they are trapped within a confinement length defined by the balance between the swimmer active force and restoring force of the well. When the confinement length is large compared to the run length, the stationary density profile follows the Boltzmann distribution. However, when the run length exceeds the confinement length, the density distribution is no longer described by Boltzmann distribution, instead we found a bimodal distribution.
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.
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.
Lattice Boltzmann algorithm to simulate isotropic-nematic emulsions.
Sulaiman, N; Marenduzzo, D; Yeomans, J M
2006-10-01
We present lattice Boltzmann simulations of the dynamical equations of motion of a drop of isotropic fluid in a nematic liquid crystal solvent, both in the absence and in the presence of an electric field. The coupled equations we solve are the Beris-Edward equations for the dynamics of the tensor order parameter describing the nematic solvent, the Cahn-Hilliard equation for the concentration evolution, and the Navier-Stokes equations for the determination of the instantaneous velocity field. We implement the lattice Boltzmann algorithm to ensure that spurious velocities are close to zero in equilibrium. We first study the effects of the liquid crystal elastic constant, K, anchoring strength, W, and surface tension, sigma, on the shape of the droplet and on the director field texture in equilibrium. We then consider how the drop behaves as the director field is switched by an applied electric field. We also show that the algorithm allows us to follow the motion of a drop of isotropic fluid placed in a liquid crystal cell with a tilted director field at the boundaries.
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.
The relativistic Boltzmann equation on a spherically symmetric gravitational field
NASA Astrophysics Data System (ADS)
Takou, Etienne; Ciake Ciake, Fidèle L.
2017-10-01
In this paper, we consider the Cauchy problem for the relativistic Boltzmann equation with near vacuum initial data where the distribution function depends on the time, the position and the impulsion. We consider this equation on a spherically symmetric gravitational field spacetime. The collision kernel considered here is for the hard potentials case. We prove the existence of a unique global (in time) mild solution in a suitable weighted space.
NASA Technical Reports Server (NTRS)
Roussel-Dupre, R.
1979-01-01
Non-Maxwellian electron velocity distribution functions, previously computed for Dupree's model of the solar transition region are used to calculate ionization rates for ions of carbon, nitrogen, and oxygen. Ionization equilibrium populations for these ions are then computed and compared with similar calculations assuming Maxwellian distribution functions for the electrons. The results show that the ion populations change (compared to the values computed with a Maxwellian) in some cases by several orders of magnitude depending on the ion and its temperature of formation.
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.
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.
NASA Astrophysics Data System (ADS)
Bhakta, Subrata; Sarkar, Susmita
2017-07-01
In this paper, we have investigated the effect of secondary electron emission on nonlinear propagation of dust acoustic waves in a complex plasma considering equilibrium dust charge positive and compared the results with those obtained in our recently published paper [Bhakta et al., Phys. Plasmas 24, 023704 (2017)] where the equilibrium dust charge was negative. In both papers, primary and secondary electrons are assumed to follow Boltzmann distribution with separate electron temperatures, ions are also Boltzmann distributed, and charged dust grains are inertial. Change in the nature of dust charge (negative to positive) gives rise to some opposite behaviour of wave propagation characteristics in dusty plasma when dust grains are charged by secondary electron emission mechanism. Both adiabatic and nonadiabatic dust charge variations have been separately considered in both the papers. The investigation in this paper shows that compressive dust acoustic soliton propagates in case of adiabatic dust charge variation whose amplitude increases and width decreases with an increase in the strength of the secondary electron emission. This is in contrast to the case of negative equilibrium dust charge which confirms the existence of rarefied dust acoustic soliton with decreasing amplitude and increasing width for an increase in the strength of the secondary electron emission. Nonadiabaticity of dust charge variation in both cases generates dust acoustic shock wave which is oscillatory for weak nonadiabaticity and monotonic for strong nonadiabaticity. For positive equilibrium dust charge, the amplitude of both oscillatory and monotonic shocks increases and oscillation of the oscillatory shock persists longer for stronger secondary electron emission. On the other hand for negative equilibrium dust charge, the amplitude of both the oscillatory and monotonic shocks diminishes with increasing secondary electron emission.
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.
A Lattice Boltzmann Method for Turbomachinery Simulations
NASA Technical Reports Server (NTRS)
Hsu, A. T.; Lopez, I.
2003-01-01
Lattice Boltzmann (LB) Method is a relatively new method for flow simulations. The start point of LB method is statistic mechanics and Boltzmann equation. The LB method tries to set up its model at molecular scale and simulate the flow at macroscopic scale. LBM has been applied to mostly incompressible flows and simple geometry.
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…
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.
NASA Astrophysics Data System (ADS)
Tripathy, Sushanta; Khuntia, Arvind; Tiwari, Swatantra Kumar; Sahoo, Raghunath
2017-05-01
In the continuation of our previous work, the transverse-momentum (pT) spectra and nuclear modification factor (R_{AA}) are derived using the relaxation time approximation of Boltzmann Transport Equation (BTE). The initial pT-distribution used to describe p + p collisions has been studied with the perturbative-Quantum Chromodynamics (pQCD) inspired power-law distribution, Hagedorn's empirical formula and with the Tsallis non-extensive statistical distribution. The non-extensive Tsallis distribution is observed to describe the complete range of the transverse-momentum spectra. The Boltzmann-Gibbs Blast Wave (BGBW) distribution is used as the equilibrium distribution in the present formalism, to describe the pT-distribution and nuclear modification factor in nucleus-nucleus collisions. The experimental data for Pb+Pb collisions at √{s_{NN}} = 2.76 TeV at the Large Hadron Collider at CERN have been analyzed for pions, kaons, protons, K^{\\ast0} and φ. It is observed that the present formalism while explaining the transverse-momentum spectra up to 5 GeV/ c, explains the nuclear modification factor very well up to 8 GeV/ c in pT for all these particles except for protons. R_{AA} is found to be independent of the degree of non-extensivity, q_{pp} after pT ˜ 8 GeV/ c.
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
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
Montaigne, F.; Lacour, D.; Chioar, I. A.; Rougemaille, N.; Louis, D.; Murtry, S. Mc; Riahi, H.; Burgos, B. Santos; Menteş, T. O.; Locatelli, A.; Canals, B.; Hehn, M.
2014-01-01
A crystal of emerging magnetic charges is expected in the phase diagram of the dipolar kagomé spin ice. An observation of charge crystallites in thermally demagnetized artificial spin ice arrays has been recently reported by S. Zhang and coworkers1 and explained through the thermodynamics of the system as it approaches a charge-ordered state. Following a similar approach, we have generated a partial order of magnetic charges in an artificial kagomé spin ice lattice made out of ferrimagnetic material having a Curie temperature of 475 K. A statistical study of the size of the charge domains reveals an unconventional sawtooth distribution. This distribution is in disagreement with the predictions of the thermodynamic model and is shown to be a signature of the kinetic process governing the remagnetization. PMID:25029620
Montaigne, F; Lacour, D; Chioar, I A; Rougemaille, N; Louis, D; Mc Murtry, S; Riahi, H; Burgos, B Santos; Menteş, T O; Locatelli, A; Canals, B; Hehn, M
2014-07-16
A crystal of emerging magnetic charges is expected in the phase diagram of the dipolar kagomé spin ice. An observation of charge crystallites in thermally demagnetized artificial spin ice arrays has been recently reported by S. Zhang and coworkers and explained through the thermodynamics of the system as it approaches a charge-ordered state. Following a similar approach, we have generated a partial order of magnetic charges in an artificial kagomé spin ice lattice made out of ferrimagnetic material having a Curie temperature of 475 K. A statistical study of the size of the charge domains reveals an unconventional sawtooth distribution. This distribution is in disagreement with the predictions of the thermodynamic model and is shown to be a signature of the kinetic process governing the remagnetization.
Deviations from Boltzmann-Gibbs Statistics in Confined Optical Lattices.
Dechant, Andreas; Kessler, David A; Barkai, Eli
2015-10-23
We investigate the semiclassical phase-space probability distribution P(x,p) of cold atoms in a Sisyphus cooling lattice with an additional harmonic confinement. We pose the question of whether this nonequilibrium steady state satisfies the equivalence of energy and probability. This equivalence is the foundation of Boltzmann-Gibbs and generalized thermostatic statistics, and a prerequisite for the description in terms of a temperature. At large energies, P(x,p) depends only on the Hamiltonian H(x,p) and the answer to the question is yes. In distinction to the Boltzmann-Gibbs state, the large-energy tails are power laws P(x,p)∝H(x,p)(-1/D), where D is related to the depth of the optical lattice. At intermediate energies, however, P(x,p) cannot be expressed as a function of the Hamiltonian and the equivalence between energy and probability breaks down. As a consequence the average potential and kinetic energy differ and no well-defined temperature can be assigned. The Boltzmann-Gibbs state is regained only in the limit of deep optical lattices. For strong confinement relative to the damping, we derive an explicit expression for the stationary phase-space distribution.
Quadrature-based lattice Boltzmann model for relativistic flows
NASA Astrophysics Data System (ADS)
Blaga, Robert; Ambruş, Victor E.
2017-01-01
A quadrature-based finite-difference lattice Boltzmann model is developed that is suitable for simulating relativistic flows of massless particles. We briefly review the relativistc Boltzmann equation and present our model. The quadrature is constructed such that the stress-energy tensor is obtained as a second order moment of the distribution function. The results obtained with our model are presented for a particular instance of the Riemann problem (the Sod shock tube). We show that the model is able to accurately capture the behavior across the whole domain of relaxation times, from the hydrodynamic to the ballistic regime. The property of the model of being extendable to arbitrarily high orders is shown to be paramount for the recovery of the analytical result in the ballistic regime.
Solving the homogeneous Boltzmann equation with arbitrary scattering kernel
NASA Astrophysics Data System (ADS)
Hohenegger, A.
2009-03-01
With applications in astroparticle physics in mind, we generalize a method for the solution of the nonlinear, space-homogeneous Boltzmann equation with an isotropic distribution function to arbitrary matrix elements. The method is based on the expansion of the scattering kernel in terms of two cosines of the “scattering angles.” The scattering functions used by previous authors in particle physics for matrix elements in the Fermi approximation are retrieved as lowest order results in this expansion. The method is designed for the unified treatment of reactive mixtures of particles obeying different scattering laws, including the quantum statistical terms for blocking or stimulated emission, in possibly large networks of Boltzmann equations. Although our notation is the relativistic one, as it is used in astroparticle physics, the results can also be applied in the classical case.
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.
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
Solution of the Boltzmann kinetic equation for the relaxation of a gas mixture
NASA Technical Reports Server (NTRS)
Rykov, V. A.; Chukanova, T. I.
1972-01-01
The temporal behavior is considered of a quiescent mixture of gases of different temperatures with spatially uniform distribution. The process of heating a cold gas by a hot gas is treated on the basis of the Boltzmann kinetic equation. The mixture is assumed to be composed of absolutely hard smooth spheres, and the initial distribution functions for each gas is taken to the Maxwellian. With such a choice of initial distribution functions, it is shown that the solution of the Boltzmann kinetic equation depends only on the velocity modulus and the time.
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.
Boltzmann: The Genius of Disorder
NASA Astrophysics Data System (ADS)
Mussardo, G.; Merlone, A.
2010-07-01
The tragedy and greatness of the contribution of Ludwig Boltzmann cannot be understood without taking into account for the relevant scientific developments that took place in the nineteenth century, one of the most eventful periods in the history of science. The kinetic theory opened a new theoretical perspective in understanding natural phenomena. The introduction of new categories of order and disorder changed radically the point of view of those physicists that accepted Boltzmann’s thesis and led, at the same time, to strong opposition to the Viennese Scientist. In this article, we present the academic situation, scientific theories, and disputes involving the Boltzmann’s theories. A short introduction on the birth of the atomistic theories opens the article, while a view on the evolution of the concept of temperature and the definition of its unit quantity closes it.
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.
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.
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.
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.
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.
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.
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.
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.
Maximum-entropy reconstruction method for moment-based solution of the Boltzmann equation
NASA Astrophysics Data System (ADS)
Summy, Dustin; Pullin, Dale
2013-11-01
We describe a method for a moment-based solution of the Boltzmann equation. This starts with moment equations for a 10 + 9 N , N = 0 , 1 , 2 . . . -moment representation. The partial-differential equations (PDEs) for these moments are unclosed, containing both higher-order moments and molecular-collision terms. These are evaluated using a maximum-entropy construction of the velocity distribution function f (c , x , t) , using the known moments, within a finite-box domain of single-particle-velocity (c) space. Use of a finite-domain alleviates known problems (Junk and Unterreiter, Continuum Mech. Thermodyn., 2002) concerning existence and uniqueness of the reconstruction. Unclosed moments are evaluated with quadrature while collision terms are calculated using a Monte-Carlo method. This allows integration of the moment PDEs in time. Illustrative examples will include zero-space- dimensional relaxation of f (c , t) from a Mott-Smith-like initial condition toward equilibrium and one-space dimensional, finite Knudsen number, planar Couette flow. Comparison with results using the direct-simulation Monte-Carlo method will be presented.
Revised Chapman-Enskog analysis for a class of forcing schemes in the lattice Boltzmann method
NASA Astrophysics Data System (ADS)
Li, Q.; Zhou, P.; Yan, H. J.
2016-10-01
In the lattice Boltzmann (LB) method, the forcing scheme, which is used to incorporate an external or internal force into the LB equation, plays an important role. It determines whether the force of the system is correctly implemented in an LB model and affects the numerical accuracy. In this paper we aim to clarify a critical issue about the Chapman-Enskog analysis for a class of forcing schemes in the LB method in which the velocity in the equilibrium density distribution function is given by u =∑αeαfα / ρ , while the actual fluid velocity is defined as u ̂=u +δtF / (2 ρ ) . It is shown that the usual Chapman-Enskog analysis for this class of forcing schemes should be revised so as to derive the actual macroscopic equations recovered from these forcing schemes. Three forcing schemes belonging to the above class are analyzed, among which Wagner's forcing scheme [A. J. Wagner, Phys. Rev. E 74, 056703 (2006), 10.1103/PhysRevE.74.056703] is shown to be capable of reproducing the correct macroscopic equations. The theoretical analyses are examined and demonstrated with two numerical tests, including the simulation of Womersley flow and the modeling of flat and circular interfaces by the pseudopotential multiphase LB model.
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
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.
Quantifying Non-Equilibrium in Hypersonic Flows Using Entropy Generation
2007-03-01
more generalized Liouville equation is actually an equation for the N-particle distribution function, a much more broad and exact representation of...statistical mechanics. The Boltzmann equation is limited compared to the Liouville , namely that it is only appropriate for electrically neutral, low...appropriate to discuss the Boltzmann equation for position and velocity in some detail. It can be derived from the Liouville equation as (Vincenti and
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
Poisson-Boltzmann-Nernst-Planck model.
Zheng, Qiong; Wei, Guo-Wei
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
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
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 sampling by re-weighting non-equilibrium simulation trajectories
NASA Astrophysics Data System (ADS)
Yang, Cheng; Wan, Biao; Xu, Shun; Wang, Yanting
2015-12-01
With the traditional equilibrium molecular simulations, it is usually difficult to efficiently visit the whole conformational space in complex systems, which are separated into some metastable conformational regions by high free energy barriers. The applied non-equilibrium process in simulations could enhance the transitions among these conformational regions, and the associated non-equilibrium effects can be removed by employing the Jarzynski equality (JE), then the global equilibrium distribution can be reproduced. However, the original JE requires the initial distribution of the non-equilibrium process is equilibrium, which largely limits the application of the non-equilibrium method in equilibrium sampling. By extending the previous method, the reweighted ensemble dynamics (RED), which re-weights many equilibrium simulation trajectories from arbitrary initial distribution to reproduce the global equilibrium, to non-equilibrium simulations, we present a method, named as re-weighted non-equilibrium ensemble dynamics (RNED), to generalize the JE in the non-equilibrium trajectories started from an arbitrary initial distribution, thus provide an efficient method to reproduce the equilibrium distribution based on multiple independent (short) non-equilibrium trajectories. We have illustrated the validity of the RNED in a one-dimensional toy model and in a Lennard-Jones system to detect the liquid-solid phase coexistence.
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.
On a derivation of the Boltzmann equation in Quantum Field Theory
NASA Astrophysics Data System (ADS)
Leiler, Gregor
The Boltzmann equation (BE) is a commonly used tool for the study of non-equilibrium many particle systems. It has been introduced in 1872 by Ludwig Boltzmann and has been widely generalized throughout the years. Today it is commonly used in physical applications, from the study of ordinary fluids to problems in particle Cosmology where Quantum Field Theoretical techniques are essential. Despite its numerous experimental successes, the conceptual basis of the BE is not entirely clear. For instance, it is well known that it is not a fundamental equation of physics like, say, the Heisenberg equation (HE). A natural question then arises whether it is possible to derive the BE from physical first principles, i.e. the Heisenberg equation in Quantum Field Theory. In this work we attempted to answer this question and succeeded in deriving the BE from the HE, thus further clarifying its conceptual status. In particular, the results we have obtained are as follows. Firstly, we establish the non-perturbative validity of what we call the "pre-Boltzmann equation". The crucial point here is that this latter equation is equivalent to the Heisenberg equation. Secondly, we proceed to consider various limits of the pre-Boltzmann equation, namly the "low density" and the "weak coupling" limits, to obtain two equations that can be considered as generalizations of the BE. These limits are always taken together with the "long time" limit, which allows us to interpret the BE as an appropriate long time limit of the HE. The generalization we obtain consists in additional "correction" terms to the usual Boltzmann collision factor, and can be associated to multiple particle scattering. Unlike the pre-Boltzmann equation, these latter results are only valid pertubatively. Finally, we briefly consider the possibility to extend these results beyond said limits and outline some important aspects in this case.
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.
Critical properties and phase separation in lattice Boltzmann fluid mixtures.
Martys, N S; Douglas, J F
2001-03-01
Basic equilibrium properties of lattice Boltzmann (LB) fluid mixtures (coexistence curve, surface tension, interfacial profile, correlation length) are calculated to characterize the critical phenomena occurring in these model liquids and to establish a reduced variable description allowing a comparison with real fluid mixtures. We observe mean-field critical exponents and amplitudes so that the LB model may be useful for modeling high molecular weight polymer blends and other fluid mixtures approximated over a wide temperature range by mean-field theory. We also briefly consider phase separation under quiescent and shearing conditions and point out the strong influence of interacting boundaries on the qualitative form of the late-stage phase-separation morphology.
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.
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.
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
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
NASA Astrophysics Data System (ADS)
Hong, Liu; Yang, Zaibao; Zhu, Yi; Yong, Wen-An
2015-12-01
In this article, we propose a novel approach to construct macroscopic balance equations and constitutive equations describing various irreversible phenomena. It is based on the general principles of non-equilibrium thermodynamics and consists of four basic steps: picking suitable state variables, choosing a strictly concave entropy function, properly separating entropy fluxes and production rates, and determining a dissipation matrix. Our approach takes advantage of both extended irreversible thermodynamics and GENERIC formalisms and shows a direct correspondence with Levermore's moment-closure hierarchies for the Boltzmann equation. As a direct application, a new ten-moment model beyond the classical hierarchies is constructed and is shown to recover the Euler equations in the equilibrium state. These interesting results may put various macroscopic modeling approaches, starting from the general principles of non-equilibrium thermodynamics, on a solid microscopic foundation based on the Boltzmann equation.
NASA Astrophysics Data System (ADS)
Taitano, W. T.; Chacón, L.; Simakov, A. N.
2017-06-01
The Fokker-Planck collision operator is an advection-diffusion operator which describe dynamical systems such as weakly coupled plasmas [1,2], photonics in high temperature environment [3,4], biological [5], and even social systems [6]. For plasmas in the continuum, the Fokker-Planck collision operator supports such important physical properties as conservation of number, momentum, and energy, as well as positivity. It also obeys the Boltzmann's H-theorem [7-11], i.e., the operator increases the system entropy while simultaneously driving the distribution function towards a Maxwellian. In the discrete, when these properties are not ensured, numerical simulations can either fail catastrophically or suffer from significant numerical pollution [12,13]. There is strong emphasis in the literature on developing numerical techniques to solve the Fokker-Planck equation while preserving these properties [12-24]. In this short note, we focus on the analytical equilibrium preserving property, meaning that the Fokker-Planck collision operator vanishes when acting on an analytical Maxwellian distribution function. The equilibrium preservation property is especially important, for example, when one is attempting to capture subtle transport physics. Since transport arises from small O (ɛ) corrections to the equilibrium [25] (where ɛ is a small expansion parameter), numerical truncation error present in the equilibrium solution may dominate, overwhelming transport dynamics.
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.
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.
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.
Lattice-Boltzmann Simulation of Coalescence-Driven Island Coarsening
Hakan Basagaoglu; Christopher T. Green; Paul Meakin; Benjamin J. McCoy
2004-10-01
A two-dimensional lattice-Boltzmann model (LBM) with fluid-fluid interactions was used to simulate first-order phase separation in a thin fluid film. The intermediate asymptotic time dependence of the mean island size, island number concentration, and polydispersity were determined and compared with the predictions of the distribution-kinetics model. The comparison revealed that the combined effects of growth, coalescence, and Ostwald ripening control the phase transition process in the LBM simulations. However, the overall process is dominated by coalescence, which is independent of island mass. As the phase transition advances, the mean island size increases, the number of islands decrease, and the polydispersity approaches unity, which conforms to the predictions of the distribution-kinetics model. The effects of the domain size on the intermediate asymptotic island size distribution, scaling form of the island size distribution, and the crossover to the long-term asymptotic behavior were elucidated. (C) 2004 American Institute of Physics.
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.
Hierarchical Boltzmann simulations and model error estimation
NASA Astrophysics Data System (ADS)
Torrilhon, Manuel; Sarna, Neeraj
2017-08-01
A hierarchical simulation approach for Boltzmann's equation should provide a single numerical framework in which a coarse representation can be used to compute gas flows as accurately and efficiently as in computational fluid dynamics, but a subsequent refinement allows to successively improve the result to the complete Boltzmann result. We use Hermite discretization, or moment equations, for the steady linearized Boltzmann equation for a proof-of-concept of such a framework. All representations of the hierarchy are rotationally invariant and the numerical method is formulated on fully unstructured triangular and quadrilateral meshes using a implicit discontinuous Galerkin formulation. We demonstrate the performance of the numerical method on model problems which in particular highlights the relevance of stability of boundary conditions on curved domains. The hierarchical nature of the method allows also to provide model error estimates by comparing subsequent representations. We present various model errors for a flow through a curved channel with obstacles.
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.
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.
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.
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.
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.
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.
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 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.
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 Poisson Boltzmann equation. The PB-AM software package includes the generation of outputs files appropriate for visualization using VMD, 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.
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.
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 (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.
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
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.
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.
A time-reversal lattice Boltzmann method
NASA Astrophysics Data System (ADS)
Vergnault, E.; Malaspinas, O.; Sagaut, P.
2011-09-01
In this paper we address the time-reversed simulation of viscous flows by the lattice Boltzmann method (LB). The theoretical derivation of the reversed LB from the Boltzmann equation is detailed, and the method implemented for weakly compressible flows using the D2Q9 scheme. The implementation of boundary conditions is also discussed. The accuracy and stability are illustrated by four test cases, namely the propagation of an acoustic wave in a medium at rest and in an uniform mean flow, the Taylor-Green vortex decay and the vortex pair-wall collision.
Grid refinement for entropic lattice Boltzmann models.
Dorschner, B; Frapolli, N; Chikatamarla, S S; Karlin, I V
2016-11-01
We propose a multidomain grid refinement technique with extensions to entropic incompressible, thermal, and compressible lattice Boltzmann models. Its validity and accuracy are assessed by comparison to available direct numerical simulation and experiment for the simulation of isothermal, thermal, and viscous supersonic flow. In particular, we investigate the advantages of grid refinement for the setups of turbulent channel flow, flow past a sphere, Rayleigh-Bénard convection, as well as the supersonic flow around an airfoil. Special attention is paid to analyzing the adaptive features of entropic lattice Boltzmann models for multigrid simulations.
Far-from-equilibrium processes without net thermal exchange via energy sorting.
Vilar, Jose M G; Rubi, J Miguel
2012-02-14
Many important processes at the microscale require far-from-equilibrium conditions to occur, as in the functioning of mesoscopic bioreactors, nanoscopic rotors, and nanoscale mass conveyors. Achieving such conditions, however, is typically based on energy inputs that strongly affect the thermal properties of the environment and the controllability of the system itself. Here, we present a general class of far-from-equilibrium processes that suppress the net thermal exchange with the environment by maintaining the Maxwell-Boltzmann velocity distribution intact. This new phenomenon, referred to as ghost equilibrium, results from the statistical cancellation of superheated and subcooled nonequilibrated degrees of freedom that are autonomously generated through a microscale energy sorting process. We provide general conditions to observe this phenomenon and study its implications for manipulating energy at the microscale. The results are applied explicitly to two mechanistically different cases, an ensemble of rotational dipoles and a gas of trapped particles, which encompass a great variety of common situations involving both rotational and translational degrees of freedom. © 2012 American Institute of Physics
NASA Astrophysics Data System (ADS)
Shao, Chun; Zhao, Wenwen; Chen, Weifang
2016-11-01
The inner shock wave structure with non-equilibrium effect is difficult to be accurately simulated due to the great gradient of density and temperature. In this paper, simplified conventional Burnett (SCB) equations were formulated for the study of hypersonic shock wave structure in continuum-transition regime. The conventional Burnett equations were derived by using the second-order Chapman-Enskog expansion of the velocity distribution function in Boltzmann equation. By neglecting conventional Burnett terms which are inversely proportional to Mach number, the constitutive relations in SCB equations were simplified specifically for hypersonic flow. The rotational and vibrational energy balance equations were also introduced into the governing equations to study the non-equilibrium relaxation processes inside shock waves. Meanwhile, generalized Rankine-Hugoniot relations were established to obtain the post-shock flow parameters in non-equilibrium flow. The numerical methods included three-order Runge-Kutta time-splitting method and AUSMPW+ flux-difference splitting method with MUSCL scheme. One-dimensional Nitrogen shock wave structure at different Mach numbers was simulated using SCB and NS equations respectively. The results indicate that the SCB equations can capture the shock waves structures more precisely and the flow variables are in better agreement with the DSMC results than NS equations in high Mach number cases.
Reaction and internal energy relaxation rates in viscous thermochemically non-equilibrium gas flows
Kustova, E. V.; Oblapenko, G. P.
2015-01-15
In the present paper, reaction and energy relaxation rates as well as the normal stress are studied for viscous gas flows with vibrational and chemical non-equilibrium. Using the modified Chapman-Enskog method, multi-temperature models based on the Treanor and Boltzmann vibrational distributions are developed for the general case taking into account all kinds of vibrational energy transitions, exchange reactions, dissociation, and recombination. Integral equations specifying the first-order corrections to the normal mean stress and reaction rates are derived, as well as approximate systems of linear equations for their numerical computation. Generalized thermodynamic driving forces associated with all non-equilibrium processes are introduced. It is shown that normal stresses and rates of non-equilibrium processes can be expressed in terms of the same driving forces; the symmetry of kinetic coefficients in these expressions is proven. The developed general model is applied to a particular case of a pure N{sub 2} viscous flow with slow VT relaxation. Normal stress and rates of vibrational relaxation are studied for various ratios of vibrational and translational temperatures. The cross effects between different vibrational transitions in viscous flows are evaluated, along with the influence of anharmonicity and flow compressibility on the first-order corrections to the relaxation rate. Limits of validity for the widely used Landau–Teller model of vibrational relaxation are indicated.
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.
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.
Increase of Boltzmann entropy in a quantum forced harmonic oscillator
NASA Astrophysics Data System (ADS)
Campisi, Michele
2008-11-01
Recently, a quantum-mechanical proof of the increase of Boltzmann entropy in quantum systems that are coupled to an external classical source of work has been given. Here we illustrate this result by applying it to a forced quantum harmonic oscillator. We show plots of the actual temporal evolution of work and entropy for various forcing protocols. We note that entropy and work can be partially or even fully returned to the source, although both work and entropy balances are non-negative at all times in accordance with the minimal work principle and the Clausius principle, respectively. A necessary condition for the increase of entropy is that the initial distribution is decreasing (e.g., canonical). We show evidence that for a nondecreasing distribution (e.g., microcanonical), the quantum expectation of entropy may decrease slightly. Interestingly, the classical expectation of entropy cannot decrease, irrespective of the initial distribution, in the forced harmonic oscillator.
Increase of Boltzmann entropy in a quantum forced harmonic oscillator.
Campisi, Michele
2008-11-01
Recently, a quantum-mechanical proof of the increase of Boltzmann entropy in quantum systems that are coupled to an external classical source of work has been given. Here we illustrate this result by applying it to a forced quantum harmonic oscillator. We show plots of the actual temporal evolution of work and entropy for various forcing protocols. We note that entropy and work can be partially or even fully returned to the source, although both work and entropy balances are non-negative at all times in accordance with the minimal work principle and the Clausius principle, respectively. A necessary condition for the increase of entropy is that the initial distribution is decreasing (e.g., canonical). We show evidence that for a nondecreasing distribution (e.g., microcanonical), the quantum expectation of entropy may decrease slightly. Interestingly, the classical expectation of entropy cannot decrease, irrespective of the initial distribution, in the forced harmonic oscillator.
Fluctuating lattice Boltzmann method for the diffusion equation.
Wagner, Alexander J; Strand, Kyle
2016-09-01
We derive a fluctuating lattice Boltzmann method for the diffusion equation. The derivation removes several shortcomings of previous derivations for fluctuating lattice Boltzmann methods for hydrodynamic systems. The comparative simplicity of this diffusive system highlights the basic features of this first exact derivation of a fluctuating lattice Boltzmann method.
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.
A Fokker-Planck Model of the Boltzmann Equation with Correct Prandtl Number for Polyatomic Gases
NASA Astrophysics Data System (ADS)
Mathiaud, J.; Mieussens, L.
2017-09-01
We propose an extension of the Fokker-Planck model of the Boltzmann equation to get a correct Prandtl number in the Compressible Navier-Stokes asymptotics for polyatomic gases. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model is obtained from the Bathnagar-Gross-Krook model of the Boltzmann equation, and by adding a diffusion term for the internal energy. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman-Enskog analysis shows how to compute the transport coefficients of our model. Some numerical tests are performed to illustrate that a correct Prandtl number can be obtained.
A Fokker-Planck Model of the Boltzmann Equation with Correct Prandtl Number for Polyatomic Gases
NASA Astrophysics Data System (ADS)
Mathiaud, J.; Mieussens, L.
2017-07-01
We propose an extension of the Fokker-Planck model of the Boltzmann equation to get a correct Prandtl number in the Compressible Navier-Stokes asymptotics for polyatomic gases. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model is obtained from the Bathnagar-Gross-Krook model of the Boltzmann equation, and by adding a diffusion term for the internal energy. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman-Enskog analysis shows how to compute the transport coefficients of our model. Some numerical tests are performed to illustrate that a correct Prandtl number can be obtained.
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.
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
Quantum Heat Engine and Negative Boltzmann Temperature
NASA Astrophysics Data System (ADS)
Xi, Jing-Yi; Quan, Hai-Tao
2017-09-01
To clarify the ambiguity on negative Boltzmann temperature in literature, we study the Carnot and the Otto cycle with one of the heat reservoirs at the negative Boltzmann temperature based on a canonical ensemble description. The work extraction, entropy production and the efficiency of these cycles are explored. Conditions for constructing and properties of these thermodynamic cycles are elucidated. We find that the apparent “violation” of the second law of thermodynamics in these cycles are due to the fact that the traditional definition of thermodynamic efficiency is inappropriate in this situation. When properly understanding the efficiency and the adiabatic processes, in which the system crosses over “absolute ZERO” in a limit sense, the Carnot cycle with one of the heat reservoirs at a negative Boltzmann temperature can be understood straightforwardly, and it contradicts neither the second nor the third law of thermodynamics. Hence, negative Boltzmann temperature is a consistent concept in thermodynamics. We use a two-level system and an Ising spin system to illustrate our central results. Support from the National Science Foundation of China under Grants Nos. 11375012, 11534002, and The Recruitment Program of Global Youth Experts of China
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.
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.
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.
Equilibrium theory of island biogeography: A review
Angela D. Yu; Simon A. Lei
2001-01-01
The topography, climatic pattern, location, and origin of islands generate unique patterns of species distribution. The equilibrium theory of island biogeography creates a general framework in which the study of taxon distribution and broad island trends may be conducted. Critical components of the equilibrium theory include the species-area relationship, island-...
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
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.
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.
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.
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
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.
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.
Casa, G; Castrillo, A; Galzerano, G; Wehr, R; Merlone, A; Di Serafino, D; Laporta, P; Gianfrani, L
2008-05-23
We report on a new optical implementation of primary gas thermometry based on laser-absorption spectrometry in the near infrared. The method consists in retrieving the Doppler broadening from highly accurate observations of the line shape of the R(12) nu1+2nu2(0)+nu3 transition in CO2 gas at thermodynamic equilibrium. Doppler width measurements as a function of gas temperature, ranging between the triple point of water and the gallium melting point, allowed for a spectroscopic determination of the Boltzmann constant with a relative accuracy of approximately 1.6 x 10(-4).
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.
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.
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.
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.
Coupling relativistic viscous hydrodynamics to Boltzmann descriptions
Pratt, Scott; Torrieri, Giorgio
2010-10-15
Models of relativistic heavy-ion collisions typically involve both a hydrodynamic module to describe the high-density liquidlike phase and a Boltzmann module to simulate the low-density breakup phase, which is gaslike. Coupling the prescriptions is more complicated for viscous prescriptions if one wants to maintain continuity of the entire stress-energy tensor and currents. Derivations for the viscosity for a gas are reviewed, which then lead to expressions for changes in the phase-space occupation based on simple relaxation-time pictures of viscosity. These expressions are shown to consistently reproduce the nonequilibrium components of the stress-energy tensor. An algorithm for generating a Monte Carlo sampling of particles with which to initiate the Boltzmann calculations is also presented.
[Boltzmann's principle and Einstein's first quantum theories].
Navarro Veguillas, Luis; Pérez Canals, Enric
2002-01-01
The crucial role played by statistical mechanics in Einstein's work on quantum theory has been repeatedly stressed. Nevertheless, in this paper we argue that Einstein's attitude to Boltzmann's principle was more complex than is usually understood. In fact, there are significant differences and nuances that in our opinion have yet to be sufficiently considered, in the various interpretations and uses Einstein made of this principle in his work on quantum theory, more specifically between 1905 and the First Solvay Conference, in 1911.
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.
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.
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.
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.}
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.
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
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.
``Thermal'' and ``superthermal'' two-class structure of the personal income distribution
NASA Astrophysics Data System (ADS)
Yakovenko, Victor
2005-03-01
In Ref. [1] we proposed an analogy between the thermal Boltzmann-Gibbs probability distribution of energy in physics and the probability distribution of money in economics in statistical equilibrium. In Ref. [2] we find that the probability distribution of personal income 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 the income data for 1983--2001 from IRS, we show that the ``thermal'' part is stationary in time, save for a gradual increase of the effective temperature, whereas the nonequilibrium ``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 the US population. [] [1] A. Dragulescu and V. M. Yakovenko, ``Statistical mechanics of money'', Eur. Phys. J. B 17, 723--729 (2000). [cond-mat/0001432] [] [2] A. C. Silva and V. M. Yakovenko, ``Temporal evolution of the `thermal' and `superthermal' income classes in the USA during 1983--2001'', accepted to Europhysics Letters. [cond- mat/0406385
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.
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.
Consistent lattice Boltzmann equations for phase transitions.
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.
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.
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.
Christopher Litvay; Alan Rudie; Peter Hart
2003-01-01
An Excel spreadsheet developed to solve the ion-exchange equilibrium in wood pulps has been linked by dynamic data exchange to WinGEMS and used to model the non-process elements in the hardwood bleach plant of the Mead/Westvaco Evandale mill. Pulp and filtrate samples were collected from the diffusion washers and final wash press of the bleach plant. A WinGEMS model of...
NASA Astrophysics Data System (ADS)
Wang, Yahui; Yan, Liming; Ma, Yu
2017-06-01
Applications of the transient Boltzmann transport equation (BTE) have undergone much investigation, such as radiative heat transfer and neutron transport. This paper provides a lattice Boltzmann model to efficiently resolve the multidimensional transient BTE. For a higher angular resolution, enough transport directions are considered while the transient BTE in each direction is treated as a conservation law equation and solved independently. Both macroscopic equations recovered from a Chapman-Enskog expansion and simulated results of typical benchmark problems show not only the second-order accuracy but also the flexibility and applicability of the proposed lattice Boltzmann model. This approach may contribute a powerful technique for the parallel simulation of large-scale engineering and some alternative perspectives for solving the nonlinear transport problem further.
Wang, Yahui; Yan, Liming; Ma, Yu
2017-06-01
Applications of the transient Boltzmann transport equation (BTE) have undergone much investigation, such as radiative heat transfer and neutron transport. This paper provides a lattice Boltzmann model to efficiently resolve the multidimensional transient BTE. For a higher angular resolution, enough transport directions are considered while the transient BTE in each direction is treated as a conservation law equation and solved independently. Both macroscopic equations recovered from a Chapman-Enskog expansion and simulated results of typical benchmark problems show not only the second-order accuracy but also the flexibility and applicability of the proposed lattice Boltzmann model. This approach may contribute a powerful technique for the parallel simulation of large-scale engineering and some alternative perspectives for solving the nonlinear transport problem further.
Linearized lattice Boltzmann method for micro- and nanoscale flow and heat transfer.
Shi, Yong; Yap, Ying Wan; Sader, John E
2015-07-01
Ability to characterize the heat transfer in flowing gases is important for a wide range of applications involving micro- and nanoscale devices. Gas flows away from the continuum limit can be captured using the Boltzmann equation, whose analytical solution poses a formidable challenge. An efficient and accurate numerical simulation of the Boltzmann equation is thus highly desirable. In this article, the linearized Boltzmann Bhatnagar-Gross-Krook equation is used to develop a hierarchy of thermal lattice Boltzmann (LB) models based on half-space Gaussian-Hermite (GH) quadrature ranging from low to high algebraic precision, using double distribution functions. Simplified versions of the LB models in the continuum limit are also derived, and are shown to be consistent with existing thermal LB models for noncontinuum heat transfer reported in the literature. Accuracy of the proposed LB hierarchy is assessed by simulating thermal Couette flows for a wide range of Knudsen numbers. Effects of the underlying quadrature schemes (half-space GH vs full-space GH) and continuum-limit simplifications on computational accuracy are also elaborated. The numerical findings in this article provide direct evidence of improved computational capability of the proposed LB models for modeling noncontinuum flows and heat transfer at small length scales.
Simulation of plume dynamics by the Lattice Boltzmann Method
NASA Astrophysics Data System (ADS)
Mora, Peter; Yuen, David A.
2017-09-01
The Lattice Boltzmann Method (LBM) is a semi-microscopic method to simulate fluid mechanics by modelling distributions of particles moving and colliding on a lattice. We present 2-D simulations using the LBM of a fluid in a rectangular box being heated from below, and cooled from above, with a Rayleigh of Ra = 108, similar to current estimates of the Earth's mantle, and a Prandtl number of 5000. At this Prandtl number, the flow is found to be in the non-inertial regime where the inertial terms denoted I ≪ 1. Hence, the simulations presented lie within the regime of relevance for geodynamical problems. We obtain narrow upwelling plumes with mushroom heads and chutes of downwelling fluid as expected of a flow in the non-inertial regime. The method developed demonstrates that the LBM has great potential for simulating thermal convection and plume dynamics relevant to geodynamics, albeit with some limitations.
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.
Transmission-Reflection Coefficient in the Lattice Boltzmann Method
NASA Astrophysics Data System (ADS)
Yoshida, Hiroaki; Hayashi, Hidemitsu
2014-04-01
We consider the permeable bounce-back scheme in the lattice Boltzmann (LB) method for incompressible flows, in which a fraction of the distribution function is bounced back and the remainder travels to the neighboring lattice points. An asymptotic analysis of the scheme is carried out in order to show that the fractional coefficient, referred to as the transmission-reflection coefficient, relates the pressure drop to the flow velocity. The derived relation, which clarifies the role played by the transmission-reflection coefficient in the macroscopic description, is helpful in using the scheme to simulate flows involving a pressure drop or gradient. The scheme is compared with the existing methods in which the transmission-reflection coefficient is employed, and the difference is clarified. As an application of the permeable bounce-back scheme, we perform an LB simulation for flows through porous media described by the Brinkman model.
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.
Emergence of Compositional Representations in Restricted Boltzmann Machines
NASA Astrophysics Data System (ADS)
Tubiana, J.; Monasson, R.
2017-03-01
Extracting automatically the complex set of features composing real high-dimensional data is crucial for achieving high performance in machine-learning tasks. Restricted Boltzmann machines (RBM) are empirically known to be efficient for this purpose, and to be able to generate distributed and graded representations of the data. We characterize the structural conditions (sparsity of the weights, low effective temperature, nonlinearities in the activation functions of hidden units, and adaptation of fields maintaining the activity in the visible layer) allowing RBM to operate in such a compositional phase. Evidence is provided by the replica analysis of an adequate statistical ensemble of random RBMs and by RBM trained on the handwritten digits data set MNIST.
Ionic size effects on the Poisson-Boltzmann theory
NASA Astrophysics Data System (ADS)
Colla, Thiago; Nunes Lopes, Lucas; dos Santos, Alexandre P.
2017-07-01
In this paper, we develop a simple theory to study the effects of ionic size on ionic distributions around a charged spherical particle. We include a correction to the regular Poisson-Boltzmann equation in order to take into account the size of ions in a mean-field regime. The results are compared with Monte Carlo simulations and a density functional theory based on the fundamental measure approach and a second-order bulk expansion which accounts for electrostatic correlations. The agreement is very good even for multivalent ions. Our results show that the theory can be applied with very good accuracy in the description of ions with highly effective ionic radii and low concentration, interacting with a colloid or a nanoparticle in an electrolyte solution.
LUDWIG: A parallel Lattice-Boltzmann code for complex fluids
NASA Astrophysics Data System (ADS)
Desplat, Jean-Christophe; Pagonabarraga, Ignacio; Bladon, Peter
2001-03-01
This paper describes Ludwig, a versatile code for the simulation of Lattice-Boltzmann (LB) models in 3D on cubic lattices. In fact, Ludwig is not a single code, but a set of codes that share certain common routines, such as I/O and communications. If Ludwig is used as intended, a variety of complex fluid models with different equilibrium free energies are simple to code, so that the user may concentrate on the physics of the problem, rather than on parallel computing issues. Thus far, Ludwig's main application has been to symmetric binary fluid mixtures. We first explain the philosophy and structure of Ludwig which is argued to be a very effective way of developing large codes for academic consortia. Next we elaborate on some parallel implementation issues such as parallel I/O, and the use of MPI to achieve full portability and good efficiency on both MPP and SMP systems. Finally, we describe how to implement generic solid boundaries, and look in detail at the particular case of a symmetric binary fluid mixture near a solid wall. We present a novel scheme for the thermodynamically consistent simulation of wetting phenomena, in the presence of static and moving solid boundaries, and check its performance.
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.
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
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.
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.
Rodero, A.; Garcia, M.C.; Gamero, A.
1995-12-31
The spectroscopy method based on the Boltzmann-plot of emission lines has been usually employed for measuring the excitation temperature (T{sub exc}) in high pressure plasmas. In the present work, it is shown that this method can produce great errors in the temperature determination when equilibrium separation exists. In this way, the suitability of this determination is tested comparing with other alternative methods in a high pressure helium plasma and also studying its separation from the equilibrium situation, via the absolute population measurements of atomic levels and the estimation of its atomic state distribution function (ASDF). We have made this study using a new excitation structure, the axial injection torch (Torche A Injection Axiale or T.I.A.), which produces a high power microwave plasma at atmospheric pressure. The measurements were carried out at the beginning of the flame (the highest line intensity zone) for a 300-900 W power range at 2.45 GHz and 71/min. of helium gas flow.
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.
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.
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.
Thermal Lattice Boltzmann Simulations for Vapor-Liquid Two-Phase Flows in Two Dimensions
NASA Astrophysics Data System (ADS)
Wei, Yikun; Qian, Yuehong
2011-11-01
A lattice Boltzmann model with double distribution functions is developed to simulate thermal vapor-liquid two-phase flows. In this model, the so-called mesoscopic inter-particle pseudo-potential for the single component multi-phase lattice Boltzmann model is used to simulate the fluid dynamics and the internal energy field is simulated by using a energy distribution function. Theoretical results for large-scale dynamics including the internal energy equation can be derived and numerical results for the coexistence curve of vapor-liquid systems are in good agreement with the theoretical predictions. It is shown from numerical simulations that the model has the ability to mimic phase transitions, bubbly flows and slugging flows. This research is support in part by the grant of Education Ministry of China IRT0844 and the grant of Shanghai CST 11XD1402300.
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
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.
2006-07-10
the two-dimensional finite difference LB model with multiple speeds of Watari and Tsutahara [14], which allows the correct recovery of mass, momentum...conditions in the thermal LB model of Watari and Tsutahara relies on the redistribution (thermalization) of the particle distribution functions in...14] M. Watari and M. Tsutahara, Two-dimensional thermal model of the finite- difference lattice Boltzmann method with high spatial isotropy
An analysis of the asymmetric part of electron-electron Boltzmann integral
NASA Technical Reports Server (NTRS)
Kunc, J. A.
1983-01-01
A numerical analysis of the asymmetric part of electon-electron collision integral is presented. The results are given in the form of graphs for two commonly considered plasma situations: the collision-dominated case (symmetric part of electron disribution is Maxwellian) and the field-dominated case (symmetric part of electron distribution is Druyvesteynian). The importance of the asymmetric part of e-e collision integral in the Boltzmann equation is also discussed.
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.
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.
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.
Ion-conserving Poisson-Boltzmann theory.
Sugioka, Hideyuki
2012-07-01
It is well known that the Poisson-Nernst-Planck (PNP) theory and the classical Gouy-Chapman theory are inconsistent at a high applied voltage. For solving this problem, we propose an ion-conserving Poisson-Boltzmann theory, which shows remarkable agreement with the numerical PNP solutions, even at a high applied voltage. In other words, we have found the exact analytical solutions for steady PNP equations; we believe that this finding greatly contributes to understanding surface science between solids and liquids.
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 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.
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
Lattice Boltzmann formulation for conjugate heat transfer in heterogeneous media.
Karani, Hamid; Huber, Christian
2015-02-01
In this paper, we propose an approach for studying conjugate heat transfer using the lattice Boltzmann method (LBM). The approach is based on reformulating the lattice Boltzmann equation for solving the conservative form of the energy equation. This leads to the appearance of a source term, which introduces the jump conditions at the interface between two phases or components with different thermal properties. The proposed source term formulation conserves conductive and advective heat flux simultaneously, which makes it suitable for modeling conjugate heat transfer in general multiphase or multicomponent systems. The simple implementation of the source term approach avoids any correction of distribution functions neighboring the interface and provides an algorithm that is independent from the topology of the interface. Moreover, our approach is independent of the choice of lattice discretization and can be easily applied to different advection-diffusion LBM solvers. The model is tested against several benchmark problems including steady-state convection-diffusion within two fluid layers with parallel and normal interfaces with respect to the flow direction, unsteady conduction in a three-layer stratified domain, and steady conduction in a two-layer annulus. The LBM results are in excellent agreement with analytical solution. Error analysis shows that our model is first-order accurate in space, but an extension to a second-order scheme is straightforward. We apply our LBM model to heat transfer in a two-component heterogeneous medium with a random microstructure. This example highlights that the method we propose is independent of the topology of interfaces between the different phases and, as such, is ideally suited for complex natural heterogeneous media. We further validate the present LBM formulation with a study of natural convection in a porous enclosure. The results confirm the reliability of the model in simulating complex coupled fluid and thermal dynamics
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.
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'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-Langevin transport model for heavy-ion collisions
Ayik, S. |
1994-06-01
Heavy-ion collisions at intermediate energies exhibit catastrophic phenomena which requires descriptions based on stochastic transport models. First, the Boltzmann-Langevin model, which provides an example of such stochastic approaches, is briefly described. Then, a projection method for obtaining numerical solutions of the Boltzmann-Langevin equation is discussed. Finally, some applications of the model to heavy-ion collisions are presented.
Exploiting Restricted Boltzmann Machines and Deep Belief Networks in Compressed Sensing
NASA Astrophysics Data System (ADS)
Polania, Luisa F.; Barner, Kenneth E.
2017-09-01
This paper proposes a CS scheme that exploits the representational power of restricted Boltzmann machines and deep learning architectures to model the prior distribution of the sparsity pattern of signals belonging to the same class. The determined probability distribution is then used in a maximum a posteriori (MAP) approach for the reconstruction. The parameters of the prior distribution are learned from training data. The motivation behind this approach is to model the higher-order statistical dependencies between the coefficients of the sparse representation, with the final goal of improving the reconstruction. The performance of the proposed method is validated on the Berkeley Segmentation Dataset and the MNIST Database of handwritten digits.
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.
The Einstein-Boltzmann equations revisited
NASA Astrophysics Data System (ADS)
Nadkarni-Ghosh, Sharvari; Refregier, Alexandre
2017-10-01
The linear Einstein-Boltzmann (E-B) equations describe the evolution of perturbations in the universe and its numerical solutions play a central role in cosmology. We revisit this system of differential equations and present a detailed investigation of its mathematical properties. For this purpose, we focus on a simplified set of equations aimed at describing the broad features of the matter power spectrum. We first perform an eigenvalue analysis and study the onset of oscillations in the system signalled by the transition from real to complex eigenvalues. We then provide a stability criterion of different numerical schemes for this linear system and estimate the associated step size. We elucidate the stiffness property of the E-B system and show how it can be characterized in terms of the eigenvalues. While the parameters of the system are time dependent making it non-autonomous, we define an adiabatic regime where the parameters vary slowly enough for the system to be quasi-autonomous. We summarize the different regimes of the system for these different criteria as function of wavenumber k and scalefactor a. We also provide a compendium of analytic solutions for all perturbation variables in six limits on the k-a plane and express them explicitly in terms of initial conditions. These results are aimed to help the further development and testing of numerical cosmological Boltzmann solvers.
Rashba torque beyond the Boltzmann regime
NASA Astrophysics Data System (ADS)
Xiao, Cong; Niu, Qian
2017-07-01
We study spin torques induced by Rashba spin-orbit coupling in two-dimensional ferromagnets under the good-metal condition ɛFτ /ℏ ≫1 (ɛF the Fermi energy, τ the electron lifetime) by employing the Kubo formula. We find that in the presence of spin-dependent disorder the Rashba torque changes greatly as the system evolves out of the weak-disorder limit where ℏ /τ is much smaller than any intrinsic energy scale characterizing the multiband structure. The antidamping-like component of Rashba torque can be comparable to and larger than the field-like one out of the weak-disorder limit. The semiclassical Boltzmann theory produces the same results as microscopic linear response calculations only in the weak-disorder limit. Our analysis indicates that rich behaviors of various nonequilibrium phenomena beyond the Boltzmann theory may also be present even when ɛFτ /ℏ ≫1 in multiband systems where ɛF is not the unique intrinsic energy scale.
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.
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.