Kinetic energy equations for the average-passage equation system
NASA Technical Reports Server (NTRS)
Johnson, Richard W.; Adamczyk, John J.
1989-01-01
Important kinetic energy equations derived from the average-passage equation sets are documented, with a view to their interrelationships. These kinetic equations may be used for closing the average-passage equations. The turbulent kinetic energy transport equation used is formed by subtracting the mean kinetic energy equation from the averaged total instantaneous kinetic energy equation. The aperiodic kinetic energy equation, averaged steady kinetic energy equation, averaged unsteady kinetic energy equation, and periodic kinetic energy equation, are also treated.
Investigation of the kinetic model equations.
Liu, Sha; Zhong, Chengwen
2014-03-01
Currently the Boltzmann equation and its model equations are widely used in numerical predictions for dilute gas flows. The nonlinear integro-differential Boltzmann equation is the fundamental equation in the kinetic theory of dilute monatomic gases. By replacing the nonlinear fivefold collision integral term by a nonlinear relaxation term, its model equations such as the famous Bhatnagar-Gross-Krook (BGK) equation are mathematically simple. Since the computational cost of solving model equations is much less than that of solving the full Boltzmann equation, the model equations are widely used in predicting rarefied flows, multiphase flows, chemical flows, and turbulent flows although their predictions are only qualitatively right for highly nonequilibrium flows in transitional regime. In this paper the differences between the Boltzmann equation and its model equations are investigated aiming at giving guidelines for the further development of kinetic models. By comparing the Boltzmann equation and its model equations using test cases with different nonequilibrium types, two factors (the information held by nonequilibrium moments and the different relaxation rates of high- and low-speed molecules) are found useful for adjusting the behaviors of modeled collision terms in kinetic regime. The usefulness of these two factors are confirmed by a generalized model collision term derived from a mathematical relation between the Boltzmann equation and BGK equation that is also derived in this paper. After the analysis of the difference between the Boltzmann equation and the BGK equation, an attempt at approximating the collision term is proposed.
Kinetic Equations for Economic Sciences
NASA Astrophysics Data System (ADS)
Bisi, M.; Brugna, C.
2010-04-01
We discuss, both from the analytical and the numerical point of view, a kinetic model for wealth distribution in a simple market economy which models, besides binary trade interactions, also taxation and redistribution of collected wealth.
Turbulence kinetic energy equation for dilute suspensions
NASA Technical Reports Server (NTRS)
Abou-Arab, T. W.; Roco, M. C.
1989-01-01
A multiphase turbulence closure model is presented which employs one transport equation, namely the turbulence kinetic energy equation. The proposed form of this equation is different from the earlier formulations in some aspects. The power spectrum of the carrier fluid is divided into two regions, which interact in different ways and at different rates with the suspended particles as a function of the particle-eddy size ratio and density ratio. The length scale is described algebraically. A mass/time averaging procedure for the momentum and kinetic energy equations is adopted. The resulting turbulence correlations are modeled under less retrictive assumptions comparative to previous work. The closures for the momentum and kinetic energy equations are given. Comparisons of the predictions with experimental results on liquid-solid jet and gas-solid pipe flow show satisfactory agreement.
Spectrum Analysis of Some Kinetic Equations
NASA Astrophysics Data System (ADS)
Yang, Tong; Yu, Hongjun
2016-11-01
We analyze the spectrum structure of some kinetic equations qualitatively by using semigroup theory and linear operator perturbation theory. The models include the classical Boltzmann equation for hard potentials with or without angular cutoff and the Landau equation with {γ≥q-2}. As an application, we show that the solutions to these two fundamental equations are asymptotically equivalent (mod time decay rate {t^{-5/4}}) as {tto∞} to that of the compressible Navier-Stokes equations for initial data around an equilibrium state.
Kinetic equations for a nonideal quantum system
NASA Astrophysics Data System (ADS)
Bornath, Th.; Kremp, D.; Kraeft, W. D.; Schlanges, M.
1996-10-01
In the framework of real-time Green's functions, the general kinetic equations are investigated in a first-order gradient expansion. Within this approximation, the problem of the reconstruction of the two-time correlation functions from the one-time Wigner function was solved. For the Wigner function, a cluster expansion is found in terms of a quasiparticle distribution function. In equilibrium, this expansion leads to the well-known generalized Beth-Uhlenbeck expression of the second virial coefficient. As a special case, the T-matrix approximation for the self-energy is investigated. The quantum kinetic equation derived thus has, besides the (Markovian) Boltzmann collision integral, additional terms due to the retardation expansion which reflect memory effects. Special interest is paid to the case that bound states exist in the system. It is shown that the bound state contribution, which can be introduced via a bilinear expansion of the two-particle T matrix, follows from the first-order retardation term in the general kinetic equation. The full Wigner function is now a sum of one function describing the unbound particles and another one for the bound state contribution. The latter two functions have to be determined from a coupled set of kinetic equations. In contrast to the quantum Boltzmann equation, energy and density of a nonideal system are conserved.
Integral kinetic equation in dechanneling problem
NASA Astrophysics Data System (ADS)
Ryabov, V.
1989-11-01
A version of dechanneling theory, based on using an integral kinetic equation in both the phase and transverse energy space, is described. It is derived from the binary collision model and it takes into account consistently the thermal multiple and single scattering of axial and planar channeled particles. The connection between the method developed and that of Oshiyama and of Gartner is discussed.
Stochastic thermodynamics for linear kinetic equations
NASA Astrophysics Data System (ADS)
Van den Broeck, C.; Toral, R.
2015-07-01
Stochastic thermodynamics is formulated for variables that are odd under time reversal. The invariance under spatial rotation of the collision rates due to the isotropy of the heat bath is shown to be a crucial ingredient. An alternative detailed fluctuation theorem is derived, expressed solely in terms of forward statistics. It is illustrated for a linear kinetic equation with kangaroo rates.
Quantum kinetic equation for nonequilibrium dense systems
NASA Astrophysics Data System (ADS)
Morozov, V. G.; Röpke, G.
1995-02-01
Using the density matrix method in the form developed by Zubarev, equations of motion for nonequilibrium quantum systems with continuous short range interactions are derived which describe kinetic and hydrodynamic processes in a consistent way. The T-matrix as well as the two-particle density matrix determining the nonequilibrium collision integral are obtained in the ladder approximation including the Hartree-Fock corrections and the Pauli blocking for intermediate states. It is shown that in this approximation the total energy is conserved. The developed approach to the kinetic theory of dense quantum systems is able to reproduce the virial corrections consistent with the generalized Beth-Uhlenbeck approximation in equilibrium. The contribution of many-particle correlations to the drift term in the quantum kinetic equation for dense systems is discussed.
On a Kinetic Equation for Coalescing Particles
NASA Astrophysics Data System (ADS)
Escobedo, Miguel; Laurençot, Philippe; Mischler, Stéphane
Existence of global weak solutions to a spatially inhomogeneous kinetic model for coalescing particles is proved, each particle being identified by its mass, momentum and position. The large time convergence to zero is also shown. The cornestone of our analysis is that, for any nonnegative and convex function, the associated Orlicz norm is a Liapunov functional. Existence and asymptotic behaviour then rely on weak and strong compactness methods in L1 in the spirit of the DiPerna-Lions theory for the Boltzmann equation.
Neutrino quantum kinetic equations: The collision term
Blaschke, Daniel N.; Cirigliano, Vincenzo
2016-08-01
We derive the collision term relevant for neutrino quantum kinetic equations in the early universe and compact astrophysical objects, displaying its full matrix structure in both flavor and spin degrees of freedom. We include in our analysis neutrino-neutrino processes, scattering and annihilation with electrons and positrons, and neutrino scattering off nucleons (the latter in the low-density limit). After presenting the general structure of the collision terms, we take two instructive limiting cases. The one-flavor limit highlights the structure in helicity space and allows for a straightforward interpretation of the off-diagonal entries in terms of the product of scattering amplitudes ofmore » the two helicity states. As a result, the isotropic limit is relevant for studies of the early universe: in this case the terms involving spin coherence vanish and the collision term can be expressed in terms of two-dimensional integrals, suitable for computational implementation.« less
Neutrino quantum kinetic equations: The collision term
Blaschke, Daniel N.; Cirigliano, Vincenzo
2016-08-01
We derive the collision term relevant for neutrino quantum kinetic equations in the early universe and compact astrophysical objects, displaying its full matrix structure in both flavor and spin degrees of freedom. We include in our analysis neutrino-neutrino processes, scattering and annihilation with electrons and positrons, and neutrino scattering off nucleons (the latter in the low-density limit). After presenting the general structure of the collision terms, we take two instructive limiting cases. The one-flavor limit highlights the structure in helicity space and allows for a straightforward interpretation of the off-diagonal entries in terms of the product of scattering amplitudes of the two helicity states. As a result, the isotropic limit is relevant for studies of the early universe: in this case the terms involving spin coherence vanish and the collision term can be expressed in terms of two-dimensional integrals, suitable for computational implementation.
Neutrino quantum kinetic equations: The collision term
Blaschke, Daniel N.; Cirigliano, Vincenzo
2016-08-01
We derive the collision term relevant for neutrino quantum kinetic equations in the early universe and compact astrophysical objects, displaying its full matrix structure in both flavor and spin degrees of freedom. We include in our analysis neutrino-neutrino processes, scattering and annihilation with electrons and positrons, and neutrino scattering off nucleons (the latter in the low-density limit). After presenting the general structure of the collision terms, we take two instructive limiting cases. The one-flavor limit highlights the structure in helicity space and allows for a straightforward interpretation of the off-diagonal entries in terms of the product of scattering amplitudes of the two helicity states. As a result, the isotropic limit is relevant for studies of the early universe: in this case the terms involving spin coherence vanish and the collision term can be expressed in terms of two-dimensional integrals, suitable for computational implementation.
Kinetic equation for spin-polarized plasmas
Cowley, S.C.; Kulsrud, R.M.; Valeo, E.
1984-07-01
The usual kinetic description of a plasma is extended to include variables to describe the spin. The distribution function, over phase-space and the new spin variables, provides a sufficient description of a spin-polarized plasma. The evolution equation for the distribution function is given. The equations derived are used to calculate depolarization due to four processes, inhomogeneous fields, collisions, collisions in inhomogeneous fields, and waves. It is found that depolarization by field inhomogeneity on scales large compared with the gyroradius is totally negligible. The same is true for collisional depolarization. Collisions in inhomogeneous fields yield a depolarization rate of order 10/sup -4/S/sup -1/ for deuterons and a negligible rate for tritons in a typical fusion reactor design. This is still sufficiently small on reactor time scales. However, small amplitude magnetic fluctuations (of order one gauss) resonant with the spin precession frequency can lead to significant depolarization (depolarises triton in ten seconds and deuteron in a hundred seconds.)
Kinetic Approach for Quantum Hydrodynamic Equations
NASA Astrophysics Data System (ADS)
Tessarotto, M.; Ellero, M.; Nicolini, P.
2008-12-01
A striking feature of standard quantum mechanics (SQM) is its analogy with classical fluid dynamics. In particular it is well known the Schrödinger equation can be viewed as describing a classical compressible and non-viscous fluid, described by two (quantum) fluid fields {ρ,V}, to be identified with the quantum probability density and velocity field. This feature has suggested the construction of a phase-space hidden-variable description based on a suitable inverse kinetic theory (IKT; Tessarotto et al., 2007). The discovery of this approach has potentially important consequences since it permits to identify the classical dynamical system which advances in time the quantum fluid fields. This type of approach, however requires the identification of additional fluid fields. These can be generally identified with suitable directional fluid temperatures TQM,i (for i = 1,2,3), to be related to the expectation values of momentum fluctuations appearing in the Heisenberg inequalities. Nevertheless the definition given previously for them (Tessarotto et al., 2007) is non-unique. In this paper we intend to propose a criterion, based on the validity of a constant H-theorem, which provides an unique definition for the quantum temperatures.
Moment equations for chromatography based on Langmuir type reaction kinetics.
Miyabe, Kanji
2014-08-22
Moment equations were derived for chromatography, in which the reaction kinetics between solute molecules and functional ligands on the stationary phase was represented by the Langmuir type rate equation. A set of basic equations of the general rate model of chromatography representing the mass balance, mass transfer rate, and reaction kinetics in the column were analytically solved in the Laplace domain. The moment equations for the first absolute moment and the second central moment in the real time domain were derived from the analytical solution in the Laplace domain. The moment equations were used for predicting the chromatographic behavior under hypothetical HPLC conditions. The influence of the parameters relating to the adsorption equilibrium and to the reaction kinetics on the chromatographic behavior was quantitatively evaluated. It is expected that the moment equations are effective for a detailed analysis of the influence of the mass transfer rates and of the Langmuir type reaction kinetics on the column efficiency.
Drift-free kinetic equations for turbulent dispersion.
Bragg, A; Swailes, D C; Skartlien, R
2012-11-01
The dispersion of passive scalars and inertial particles in a turbulent flow can be described in terms of probability density functions (PDFs) defining the statistical distribution of relevant scalar or particle variables. The construction of transport equations governing the evolution of such PDFs has been the subject of numerous studies, and various authors have presented formulations for this type of equation, usually referred to as a kinetic equation. In the literature it is often stated, and widely assumed, that these PDF kinetic equation formulations are equivalent. In this paper it is shown that this is not the case, and the significance of differences among the various forms is considered. In particular, consideration is given to which form of equation is most appropriate for modeling dispersion in inhomogeneous turbulence and most consistent with the underlying particle equation of motion. In this regard the PDF equations for inertial particles are considered in the limit of zero particle Stokes number and assessed against the fully mixed (zero-drift) condition for fluid points. A long-standing question regarding the validity of kinetic equations in the fluid-point limit is answered; it is demonstrated formally that one version of the kinetic equation (derived using the Furutsu-Novikov method) provides a model that satisfies this zero-drift condition exactly in both homogeneous and inhomogeneous systems. In contrast, other forms of the kinetic equation do not satisfy this limit or apply only in a limited regime.
A classical but new kinetic equation for hydride transfer reactions.
Zhu, Xiao-Qing; Deng, Fei-Huang; Yang, Jin-Dong; Li, Xiu-Tao; Chen, Qiang; Lei, Nan-Ping; Meng, Fan-Kun; Zhao, Xiao-Peng; Han, Su-Hui; Hao, Er-Jun; Mu, Yuan-Yuan
2013-09-28
A classical but new kinetic equation to estimate activation energies of various hydride transfer reactions was developed according to transition state theory using the Morse-type free energy curves of hydride donors to release a hydride anion and hydride acceptors to capture a hydride anion and by which the activation energies of 187 typical hydride self-exchange reactions and more than thirty thousand hydride cross transfer reactions in acetonitrile were safely estimated in this work. Since the development of the kinetic equation is only on the basis of the related chemical bond changes of the hydride transfer reactants, the kinetic equation should be also suitable for proton transfer reactions, hydrogen atom transfer reactions and all the other chemical reactions involved with breaking and formation of chemical bonds. One of the most important contributions of this work is to have achieved the perfect unity of the kinetic equation and thermodynamic equation for hydride transfer reactions.
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.
Figaro, S; Avril, J P; Brouers, F; Ouensanga, A; Gaspard, S
2009-01-30
Adsorption kinetic of molasses wastewaters after anaerobic digestion (MSWD) and melanoidin respectively on activated carbon was studied at different pH. The kinetic parameters could be determined using classical kinetic equations and a recently published fractal kinetic equation. A linear form of this equation can also be used to fit adsorption data. Even with lower correlation coefficients the fractal kinetic equation gives lower normalized standard deviation values than the pseudo-second order model generally used to fit adsorption kinetic data, indicating that the fractal kinetic model is much more accurate for describing the kinetic adsorption data than the pseudo-second order kinetic model.
Hamiltonian formalism of two-dimensional Vlasov kinetic equation
Pavlov, Maxim V.
2014-01-01
In this paper, the two-dimensional Benney system describing long wave propagation of a finite depth fluid motion and the multi-dimensional Russo–Smereka kinetic equation describing a bubbly flow are considered. The Hamiltonian approach established by J. Gibbons for the one-dimensional Vlasov kinetic equation is extended to a multi-dimensional case. A local Hamiltonian structure associated with the hydrodynamic lattice of moments derived by D. J. Benney is constructed. A relationship between this hydrodynamic lattice of moments and the two-dimensional Vlasov kinetic equation is found. In the two-dimensional case, a Hamiltonian hydrodynamic lattice for the Russo–Smereka kinetic model is constructed. Simple hydrodynamic reductions are presented. PMID:25484603
Hamiltonian formalism of two-dimensional Vlasov kinetic equation.
Pavlov, Maxim V
2014-12-08
In this paper, the two-dimensional Benney system describing long wave propagation of a finite depth fluid motion and the multi-dimensional Russo-Smereka kinetic equation describing a bubbly flow are considered. The Hamiltonian approach established by J. Gibbons for the one-dimensional Vlasov kinetic equation is extended to a multi-dimensional case. A local Hamiltonian structure associated with the hydrodynamic lattice of moments derived by D. J. Benney is constructed. A relationship between this hydrodynamic lattice of moments and the two-dimensional Vlasov kinetic equation is found. In the two-dimensional case, a Hamiltonian hydrodynamic lattice for the Russo-Smereka kinetic model is constructed. Simple hydrodynamic reductions are presented.
Covariant chiral kinetic equation in the Wigner function approach
NASA Astrophysics Data System (ADS)
Gao, Jian-hua; Pu, Shi; Wang, Qun
2017-07-01
The covariant chiral kinetic equation (CCKE) is derived from the four-dimensional Wigner function by an improved perturbative method under the static equilibrium conditions. The chiral kinetic equation in three dimensions can be obtained by integration over the time component of the four-momentum. There is freedom to add more terms to the CCKE allowed by conservation laws. In the derivation of the three-dimensional equation, there is also freedom to choose coefficients of some terms in d x0/d τ and d x /d τ [τ is a parameter along the worldline, and (x0,x ) denotes the time-space position of a particle] whose three-momentum integrals are vanishing. So the three-dimensional chiral kinetic equation derived from the CCKE is not uniquely determined in the current approach. The key assumption of our approach is the perturbation in powers of space-time derivative and constant electromagnetic field strength tensor under the static equilibrium conditions. To go beyond the current approach and overcome these problems one needs a new way of building up the three-dimensional chiral kinetic equation from the CCKE or directly from covariant Wigner equations.
Turbulent kinetic energy equation and free mixing
NASA Technical Reports Server (NTRS)
Morel, T.; Torda, T. P.; Bradshaw, P.
1973-01-01
Calculation of free shear flows was carried out to investigate the usefulness of several concepts which were previously successfully applied to wall flows. The method belongs to the class of differential approaches. The turbulence is taken into account by the introduction of one additional partial differential equation, the transport equation for the turbulent shear stress. The structure of turbulence is modeled after Bradshaw et al. This model was used successfully in boundary layers and its applicability to other flows is demonstrated. The work reported differs substantially from that of an earlier attempt to use this approach for calculation of free flows. The most important difference is that the region around the center line is treated by invoking the interaction hypothesis (concerning the structure of turbulence in the regions separated by the velocity extrema). The compressibility effects on shear layer spreading at low and moderate Mach numbers were investigated. In the absence of detailed experiments in free flows, the evidence from boundary layers that at low Mach numbers the structure of turbulence is unaffected by the compressibility was relied on. The present model was tested over a range of self-preserving and developing flows including pressure gradients using identical empirical input. The dependence of the structure of turbulence on the spreading rate of the shear layer was established.
The Linearized Kinetic Equation -- A Functional Analytic Approach
NASA Astrophysics Data System (ADS)
Brinkmann, Ralf Peter
2009-10-01
Kinetic models of plasma phenomena are difficult to address for two reasons. They i) are given as systems of nonlinear coupled integro-differential equations, and ii) involve generally six-dimensional distribution functions f(r,v,t). In situations which can be addressed in a linear regime, the first difficulty disappears, but the second one still poses considerable practical problems. This contribution presents an abstract approach to linearized kinetic theory which employs the methods of functional analysis. A kinetic electron equation with elastic electron-neutral interaction is studied in the electrostatic approximation. Under certain boundary conditions, a nonlinear functional, the kinetic free energy, exists which has the properties of a Lyapunov functional. In the linear regime, the functional becomes a quadratic form which motivates the definition of a bilinear scalar product, turning the space of all distribution functions into a Hilbert space. The linearized kinetic equation can then be described in terms of dynamical operators with well-defined properties. Abstract solutions can be constructed which have mathematically plausible properties. As an example, the formalism is applied to the example of the multipole resonance probe (MRP). Under the assumption of a Maxwellian background distribution, the kinetic model of that diagnostics device is compared to a previously investigated fluid model.
Controllability in Hybrid Kinetic Equations Modeling Nonequilibrium Multicellular Systems
Bianca, Carlo
2013-01-01
This paper is concerned with the derivation of hybrid kinetic partial integrodifferential equations that can be proposed for the mathematical modeling of multicellular systems subjected to external force fields and characterized by nonconservative interactions. In order to prevent an uncontrolled time evolution of the moments of the solution, a control operator is introduced which is based on the Gaussian thermostat. Specifically, the analysis shows that the moments are solution of a Riccati-type differential equation. PMID:24191137
Kinetic equation for classical particles obeying an exclusion principle
NASA Astrophysics Data System (ADS)
Kaniadakis, G.; Quarati, P.
1993-12-01
In this paper we analyze the kinetics of classical particles which obey an exclusion principle (EP) in the only-individual-transitions (OIT) approximation, and separately in the more rigorous contemporary-transitions (CT) description. In order to be able to include the EP into the kinetics equations we consider a discrete, one-dimensional, heterogeneous and anisotropic phase space and, after defining the reduced transition probabilities, we write a master equation. As a limit to the continuum of this master equation we obtain a generalized Fokker-Planck (FP) equation. This last is a nonlinear partial differential equation and reduces to the standard FP equation if the nonlinear term, which takes into account the EP, is neglected. The steady states of this equation, both in the OIT approximation and CT description, are considered. In the particularly interesting case of Brownian particles as a steady state in the OIT approximation we obtain the Fermi-Dirac (FD) distribution, while in the CT description we obtain another distribution which differs slightly from that of the FD. Moreover, our approach permits us to treat in an alternative and efficient way the problem of the determination of an effective potential to simulate the exclusion principle in classical many-body equations of motion.
Enzyme Kinetics and the Michaelis-Menten Equation
ERIC Educational Resources Information Center
Biaglow, Andrew; Erickson, Keith; McMurran, Shawnee
2010-01-01
The concepts presented in this article represent the cornerstone of classical mathematical biology. The central problem of the article relates to enzyme kinetics, which is a biochemical system. However, the theoretical underpinnings that lead to the formation of systems of time-dependent ordinary differential equations have been applied widely to…
Enzyme Kinetics and the Michaelis-Menten Equation
ERIC Educational Resources Information Center
Biaglow, Andrew; Erickson, Keith; McMurran, Shawnee
2010-01-01
The concepts presented in this article represent the cornerstone of classical mathematical biology. The central problem of the article relates to enzyme kinetics, which is a biochemical system. However, the theoretical underpinnings that lead to the formation of systems of time-dependent ordinary differential equations have been applied widely to…
Spectral function and kinetic equation for a normal Fermi liquid
Arshad, M.; Siddique, I.; Kondratyev, A. S.
2007-08-01
On the basis of the Kadanoff-Baym (KB) version of the time-dependent Green's function method, an Ansatz for the approximation of a spectral function is offered. The Ansatz possesses all the advantages of quasiparticle and extended quasiparticle approximations and satisfies the KB equation for a spectral function in the case of slightly nonequilibrium system when disturbances in space and time are taken into consideration in the gradient approximation. This feature opens opportunities for the microscopic derivation of the Landau kinetic equation for the quasiparticle distribution function of the normal Fermi liquid and provides the widening of these equations' temperature range of validity.
Modeling the turbulent kinetic energy equation for compressible, homogeneous turbulence
NASA Technical Reports Server (NTRS)
Aupoix, B.; Blaisdell, G. A.; Reynolds, William C.; Zeman, Otto
1990-01-01
The turbulent kinetic energy transport equation, which is the basis of turbulence models, is investigated for homogeneous, compressible turbulence using direct numerical simulations performed at CTR. It is shown that the partition between dilatational and solenoidal modes is very sensitive to initial conditions for isotropic decaying turbulence but not for sheared flows. The importance of the dilatational dissipation and of the pressure-dilatation term is evidenced from simulations and a transport equation is proposed to evaluate the pressure-dilatation term evolution. This transport equation seems to work well for sheared flows but does not account for initial condition sensitivity in isotropic decay. An improved model is proposed.
Bounce-averaged Kinetic Equations and Neoclassical Polarization Density
First Author = B.H. Fong; T.S. Hahm
1998-07-01
The rigorous formulation of the bounce-averaged equations is presented based upon the Poincare-Cartan one-form andLie perturbation methods. The resulting bounce-averaged Vlasov equation is Hamiltonian, thus suitable for theself-consistent simulation of low-frequency electrostatic turbulence in the trapped ion mode regime. In the bounce-kineticPoisson equation, the "neoclassical polarization density" arises from the difference between bounce-averaged banana centerand real trapped particle densities across a field line. This representation of the neoclassical polarization drift as ashielding term provides a systematic way to study the long-term behavior of the turbulence-driven E x B flow.
Exact Markovian kinetic equation for a quantum Brownian oscillator
NASA Astrophysics Data System (ADS)
Tay, B. A.; Ordonez, G.
2006-01-01
We derive an exact Markovian kinetic equation for an oscillator linearly coupled to a heat bath, describing quantum Brownian motion. Our work is based on the subdynamics formulation developed by Prigogine and collaborators. The space of distribution functions is decomposed into independent subspaces that remain invariant under Liouville dynamics. For integrable systems in Poincaré’s sense the invariant subspaces follow the dynamics of uncoupled, renormalized particles. In contrast, for nonintegrable systems, the invariant subspaces follow a dynamics with broken time symmetry, involving generalized functions. This result indicates that irreversibility and stochasticity are exact properties of dynamics in generalized function spaces. We comment on the relation between our Markovian kinetic equation and the Hu-Paz-Zhang equation.
A kinetic equation with kinetic entropy functions for scalar conservation laws
NASA Technical Reports Server (NTRS)
Perthame, Benoit; Tadmor, Eitan
1990-01-01
A nonlinear kinetic equation is constructed and proved to be well-adapted to describe general multidimensional scalar conservation laws. In particular, it is proved to be well-posed uniformly in epsilon - the microscopic scale. It is also shown that the proposed kinetic equation is equipped with a family of kinetic entropy functions - analogous to Boltzmann's microscopic H-function, such that they recover Krushkov-type entropy inequality on the macroscopic scale. Finally, it is proved by both - BV compactness arguments in the one-dimensional case, that the local density of kinetic particles admits a continuum limit, as it converges strongly with epsilon below 0 to the unique entropy solution of the corresponding conservation law.
Bifurcation in kinetic equation for interacting Fermi systems
NASA Astrophysics Data System (ADS)
Morawetz, Klaus
2003-06-01
The recently derived nonlocal quantum kinetic equation for dense interacting Fermi systems combines time derivatives with finite time stepping known from the logistic mapping. This continuous delay differential equation is a consequence of the microscopic delay time representing the dynamics of the deterministic chaotic system. The responsible delay time is explicitly calculated and discussed for short-range correlations. As a novel feature oscillations in the time evolution of the distribution function itself appear and bifurcations up to chaotic behavior occur. The temperature and density conditions are presented where such oscillations and bifurcations arise indicating an onset of phase transition.
Computer models for kinetic equations of magnetically confined plasmas
Killeen, J.; Kerbel, G.D.; McCoy, M.G.; Mirin, A.A.; Horowitz, E.J.; Shumaker, D.E.
1987-01-01
This paper presents four working computer models developed by the computational physics group of the National Magnetic Fusion Energy Computer Center. All of the models employ a kinetic description of plasma species. Three of the models are collisional, i.e., they include the solution of the Fokker-Planck equation in velocity space. The fourth model is collisionless and treats the plasma ions by a fully three-dimensional particle-in-cell method.
Kinetic theory of flocking: Derivation of hydrodynamic equations
NASA Astrophysics Data System (ADS)
Ihle, Thomas
2011-03-01
It is shown how to explicitly coarse-grain the microscopic dynamics of the rule-based Vicsek model for self-propelled agents. The hydrodynamic equations are derived by means of an Enskog-type kinetic theory. Expressions for all transport coefficients are given. The transition from a disordered to a flocking state, which at large particle speeds appears to be a fluctuation-induced first-order phase transition, is studied numerically and analytically.
On Some Properties of the Landau Kinetic Equation
NASA Astrophysics Data System (ADS)
Bobylev, Alexander; Gamba, Irene; Potapenko, Irina
2015-12-01
We discuss some general properties of the Landau kinetic equation. In particular, the difference between the "true" Landau equation, which formally follows from classical mechanics, and the "generalized" Landau equation, which is just an interesting mathematical object, is stressed. We show how to approximate solutions to the Landau equation by the Wild sums. It is the so-called quasi-Maxwellian approximation related to Monte Carlo methods. This approximation can be also useful for mathematical problems. A model equation which can be reduced to a local nonlinear parabolic equation is also constructed in connection with existence of the strong solution to the initial value problem. A self-similar asymptotic solution to the Landau equation for large v and t is discussed in detail. The solution, earlier confirmed by numerical experiments, describes a formation of Maxwellian tails for a wide class of initial data concentrated in the thermal domain. It is shown that the corresponding rate of relaxation (fractional exponential function) is in exact agreement with recent mathematically rigorous estimates.
The H-theorem for the chemical kinetic equations with discrete time and for their generalizations
NASA Astrophysics Data System (ADS)
Adzhiev, S.; Melikhov, I.; Vedenyapin, V.
2017-01-01
In this paper the generalizations of equations of chemical kinetics, including classical and quantum chemical kinetics, is considered. We make time discrete in these equations and prove the H-theorem.
Balescu, R.; Wang, H. ); Misguich, J.H. )
1994-12-01
The running diffusion coefficient [ital D]([ital t]) is evaluated for a system of charged particles undergoing the effect of a fluctuating magnetic field and of their mutual collisions. The latter coefficient can be expressed either in terms of the mean square displacement (MSD) of a test particle, or in terms of a correlation between a fluctuating distribution function and the magnetic field fluctuation. In the first case a stochastic differential equation of Langevin type for the position of a test particle must be solved; the second problem requires the determination of the distribution function from a kinetic equation. Using suitable simplifications, both problems are amenable to exact analytic solution. The conclusion is that the equivalence of the two approaches is by no means automatically guaranteed. A new type of object, the hybrid kinetic equation'' is constructed: it automatically ensures the equivalence with the Langevin results. The same conclusion holds for the generalized Fokker--Planck equation. The (Bhatnagar--Gross--Krook) (BGK) model for the collisions yields a completely wrong result. A linear approximation to the hybrid kinetic equation yields an inexact behavior, but represents an acceptable approximation in the strongly collisional limit.
Analytical Derivation of Moment Equations in Stochastic Chemical Kinetics
Sotiropoulos, Vassilios; Kaznessis, Yiannis N.
2011-01-01
The master probability equation captures the dynamic behavior of a variety of stochastic phenomena that can be modeled as Markov processes. Analytical solutions to the master equation are hard to come by though because they require the enumeration of all possible states and the determination of the transition probabilities between any two states. These two tasks quickly become intractable for all but the simplest of systems. Instead of determining how the probability distribution changes in time, we can express the master probability distribution as a function of its moments, and, we can then write transient equations for the probability distribution moments. In 1949, Moyal defined the derivative, or jump, moments of the master probability distribution. These are measures of the rate of change in the probability distribution moment values, i.e. what the impact is of any given transition between states on the moment values. In this paper we present a general scheme for deriving analytical moment equations for any N-dimensional Markov process as a function of the jump moments. Importantly, we propose a scheme to derive analytical expressions for the jump moments for any N-dimensional Markov process. To better illustrate the concepts, we focus on stochastic chemical kinetics models for which we derive analytical relations for jump moments of arbitrary order. Chemical kinetics models are widely used to capture the dynamic behavior of biological systems. The elements in the jump moment expressions are a function of the stoichiometric matrix and the reaction propensities, i.e the probabilistic reaction rates. We use two toy examples, a linear and a non-linear set of reactions, to demonstrate the applicability and limitations of the scheme. Finally, we provide an estimate on the minimum number of moments necessary to obtain statistical significant data that would uniquely determine the dynamics of the underlying stochastic chemical kinetic system. The first two moments
Analytical Derivation of Moment Equations in Stochastic Chemical Kinetics.
Sotiropoulos, Vassilios; Kaznessis, Yiannis N
2011-02-01
The master probability equation captures the dynamic behavior of a variety of stochastic phenomena that can be modeled as Markov processes. Analytical solutions to the master equation are hard to come by though because they require the enumeration of all possible states and the determination of the transition probabilities between any two states. These two tasks quickly become intractable for all but the simplest of systems. Instead of determining how the probability distribution changes in time, we can express the master probability distribution as a function of its moments, and, we can then write transient equations for the probability distribution moments. In 1949, Moyal defined the derivative, or jump, moments of the master probability distribution. These are measures of the rate of change in the probability distribution moment values, i.e. what the impact is of any given transition between states on the moment values. In this paper we present a general scheme for deriving analytical moment equations for any N-dimensional Markov process as a function of the jump moments. Importantly, we propose a scheme to derive analytical expressions for the jump moments for any N-dimensional Markov process. To better illustrate the concepts, we focus on stochastic chemical kinetics models for which we derive analytical relations for jump moments of arbitrary order. Chemical kinetics models are widely used to capture the dynamic behavior of biological systems. The elements in the jump moment expressions are a function of the stoichiometric matrix and the reaction propensities, i.e the probabilistic reaction rates. We use two toy examples, a linear and a non-linear set of reactions, to demonstrate the applicability and limitations of the scheme. Finally, we provide an estimate on the minimum number of moments necessary to obtain statistical significant data that would uniquely determine the dynamics of the underlying stochastic chemical kinetic system. The first two moments
Conformational Nonequilibrium Enzyme Kinetics: Generalized Michaelis-Menten Equation.
Piephoff, D Evan; Wu, Jianlan; Cao, Jianshu
2017-08-03
In a conformational nonequilibrium steady state (cNESS), enzyme turnover is modulated by the underlying conformational dynamics. On the basis of a discrete kinetic network model, we use an integrated probability flux balance method to derive the cNESS turnover rate for a conformation-modulated enzymatic reaction. The traditional Michaelis-Menten (MM) rate equation is extended to a generalized form, which includes non-MM corrections induced by conformational population currents within combined cyclic kinetic loops. When conformational detailed balance is satisfied, the turnover rate reduces to the MM functional form, explaining its general validity. For the first time, a one-to-one correspondence is established between non-MM terms and combined cyclic loops with unbalanced conformational currents. Cooperativity resulting from nonequilibrium conformational dynamics can be achieved in enzymatic reactions, and we provide a novel, rigorous means of predicting and characterizing such behavior. Our generalized MM equation affords a systematic approach for exploring cNESS enzyme kinetics.
Solving the Fokker-Planck kinetic equation on a lattice
NASA Astrophysics Data System (ADS)
Moroni, Daniele; Rotenberg, Benjamin; Hansen, Jean-Pierre; Succi, Sauro; Melchionna, Simone
2006-06-01
We propose a discrete lattice version of the Fokker-Planck kinetic equation in close analogy with the lattice-Boltzmann scheme. Our work extends an earlier one-dimensional formulation to arbitrary spatial dimension D . A generalized Hermite-Gauss procedure is used to construct a discretized kinetic equation and a Chapman-Enskog expansion is applied to adapt the scheme so as to correctly reproduce the macroscopic continuum equations. The linear stability of the algorithm with respect to the finite time step Δt is characterized by the eigenvalues of the collision matrix. A heuristic second-order algorithm in Δt is applied to investigate the time evolution of the distribution function of simple model systems, and compared to known analytical solutions. Preliminary investigations of sedimenting Brownian particles subjected to an orthogonal centrifugal force illustrate the numerical efficiency of the Lattice-Fokker-Planck algorithm to simulate nontrivial situations. Interactions between Brownian particles may be accounted for by adding a standard Bhatnagar-Gross-Krook collision operator to the discretized Fokker-Planck kernel.
Kinetic equations modelling wealth redistribution: a comparison of approaches.
Düring, Bertram; Matthes, Daniel; Toscani, Giuseppe
2008-11-01
Kinetic equations modelling the redistribution of wealth in simple market economies is one of the major topics in the field of econophysics. We present a unifying approach to the qualitative study for a large variety of such models, which is based on a moment analysis in the related homogeneous Boltzmann equation, and on the use of suitable metrics for probability measures. In consequence, we are able to classify the most important feature of the steady wealth distribution, namely the fatness of the Pareto tail, and the dynamical stability of the latter in terms of the model parameters. Our results apply, e.g., to the market model with risky investments [S. Cordier, L. Pareschi, and G. Toscani, J. Stat. Phys. 120, 253 (2005)], and to the model with quenched saving propensities [A. Chatterjee, B. K. Chakrabarti, and S. S. Manna, Physica A 335, 155 (2004)]. Also, we present results from numerical experiments that confirm the theoretical predictions.
Kinetic equations modelling wealth redistribution: A comparison of approaches
NASA Astrophysics Data System (ADS)
Düring, Bertram; Matthes, Daniel; Toscani, Giuseppe
2008-11-01
Kinetic equations modelling the redistribution of wealth in simple market economies is one of the major topics in the field of econophysics. We present a unifying approach to the qualitative study for a large variety of such models, which is based on a moment analysis in the related homogeneous Boltzmann equation, and on the use of suitable metrics for probability measures. In consequence, we are able to classify the most important feature of the steady wealth distribution, namely the fatness of the Pareto tail, and the dynamical stability of the latter in terms of the model parameters. Our results apply, e.g., to the market model with risky investments [S. Cordier, L. Pareschi, and G. Toscani, J. Stat. Phys. 120, 253 (2005)], and to the model with quenched saving propensities [A. Chatterjee, B. K. Chakrabarti, and S. S. Manna, Physica A 335, 155 (2004)]. Also, we present results from numerical experiments that confirm the theoretical predictions.
Model Reduction of Kinetic Equations by Operator Projection
NASA Astrophysics Data System (ADS)
Fan, Yuwei; Koellermeier, Julian; Li, Jun; Li, Ruo; Torrilhon, Manuel
2016-01-01
By a further study of the mechanism of the hyperbolic regularization of the moment system for the Boltzmann equation proposed in Cai et al. (Commun Math Sci 11(2):547-571, 2013), we point out that the key point is treating the time and space derivative in the same way. Based on this understanding, a uniform framework to derive globally hyperbolic moment systems from kinetic equations using an operator projection method is proposed. The framework is so concise and clear that it can be treated as an algorithm with four inputs to derive hyperbolic moment systems by routine calculations. Almost all existing globally hyperbolic moment systems can be included in the framework, as well as some new moment systems including globally hyperbolic regularized versions of Grad's ordered moment systems and a multi-dimensional extension of the quadrature-based moment system.
Coarse-grained kinetic equations for quantum systems
NASA Astrophysics Data System (ADS)
Petrov, E. G.
2013-01-01
The nonequilibrium density matrix method is employed to derive a master equation for the averaged state populations of an open quantum system subjected to an external high frequency stochastic field. It is shown that if the characteristic time τstoch of the stochastic process is much lower than the characteristic time τsteady of the establishment of the system steady state populations, then on the time scale Δ t ˜ τsteady, the evolution of the system populations can be described by the coarse-grained kinetic equations with the averaged transition rates. As an example, the exact averaging is carried out for the dichotomous Markov process of the kangaroo type.
Exact solutions of kinetic equations in an autocatalytic growth model.
Jędrak, Jakub
2013-02-01
Kinetic equations are introduced for the transition-metal nanocluster nucleation and growth mechanism, as proposed by Watzky and Finke [J. Am. Chem. Soc. 119, 10382 (1997)]. Equations of this type take the form of Smoluchowski coagulation equations supplemented with the terms responsible for the chemical reactions. In the absence of coagulation, we find complete analytical solutions of the model equations for the autocatalytic rate constant both proportional to the cluster mass, and the mass-independent one. In the former case, ξ(k)=s(k)(ξ(1))[proportionality]ξ(1)(k)/k was obtained, while in the latter, the functional form of s(k)(ξ(1)) is more complicated. In both cases, ξ(1)(t)=h(μ)(M(μ)(t)) is a function of the moments of the mass distribution. Both functions, s(k)(ξ(1)) and h(μ)(M(μ)), depend on the assumed mechanism of autocatalytic growth and monomer production, and not on other chemical reactions present in a system.
Causal kinetic equation of non-equilibrium plasmas
NASA Astrophysics Data System (ADS)
Treumann, Rudolf A.; Baumjohann, Wolfgang
2017-05-01
Statistical plasma theory far from thermal equilibrium is subject to Liouville's equation, which is at the base of the BBGKY hierarchical approach to plasma kinetic theory, from which, in the absence of collisions, Vlasov's equation follows. It is also at the base of Klimontovich's approach which includes single-particle effects like spontaneous emission. All these theories have been applied to plasmas with admirable success even though they suffer from a fundamental omission in their use of the electrodynamic equations in the description of the highly dynamic interactions in many-particle conglomerations. In the following we extend this theory to taking into account that the interaction between particles separated from each other at a distance requires the transport of information. Action needs to be transported and thus, in the spirit of the direct-interaction theory as developed by Wheeler and Feynman (1945), requires time. This is done by reference to the retarded potentials. We derive the fundamental causal Liouville equation for the phase space density of a system composed of a very large number of charged particles. Applying the approach of Klimontovich (1967), we obtain the retarded time evolution equation of the one-particle distribution function in plasmas, which replaces Klimontovich's equation in cases when the direct-interaction effects have to be taken into account. This becomes important in all systems where the distance between two points |Δq| ˜ ct is comparable to the product of observation time and light velocity, a situation which is typical in cosmic physics and astrophysics.
Herschlag, Gregory J; Mitran, Sorin; Lin, Guang
2015-06-21
We develop a hierarchy of approximations to the master equation for systems that exhibit translational invariance and finite-range spatial correlation. Each approximation within the hierarchy is a set of ordinary differential equations that considers spatial correlations of varying lattice distance; the assumption is that the full system will have finite spatial correlations and thus the behavior of the models within the hierarchy will approach that of the full system. We provide evidence of this convergence in the context of one- and two-dimensional numerical examples. Lower levels within the hierarchy that consider shorter spatial correlations are shown to be up to three orders of magnitude faster than traditional kinetic Monte Carlo methods (KMC) for one-dimensional systems, while predicting similar system dynamics and steady states as KMC methods. We then test the hierarchy on a two-dimensional model for the oxidation of CO on RuO2(110), showing that low-order truncations of the hierarchy efficiently capture the essential system dynamics. By considering sequences of models in the hierarchy that account for longer spatial correlations, successive model predictions may be used to establish empirical approximation of error estimates. The hierarchy may be thought of as a class of generalized phenomenological kinetic models since each element of the hierarchy approximates the master equation and the lowest level in the hierarchy is identical to a simple existing phenomenological kinetic models.
NASA Astrophysics Data System (ADS)
Adzhiev, S. Z.; Melikhov, I. V.; Vedenyapin, V. V.
2017-08-01
In this paper the generalizations of the physico-chemical kinetics equations, including classical and quantum physico-chemical kinetics, coagulation-fragmentation equations, discrete velocity models of classical and quantum kinetic equation and, generally speaking, all discrete physico-chemical kinetics is considered. We make time discrete in these equations and prove the H-theorem.
Trajectory simulation of kinetic equations for classical systems
NASA Astrophysics Data System (ADS)
Holland, M.; Williams, J.; Coakley, K.; Cooper, J.
1996-06-01
We formulate a linear theory of physical kinetics describing the relaxation of atoms from a non-equilibrium distribution. The evolution of the single-particle distribution function is decomposed into trajectories, each corresponding to a different realization of a sequence of collisions. Accumulating all possible trajectories gives the dynamics described by the classical Boltzmann equation. The significance of our method is that the required computation time scales linearly with the number of points used to sample the distribution function. This leads to the interesting possibility of extending our method to consider quantum coherences and the growth of long-range order in Bose - Einstein condensation where a large set of basis states may be required.
Accurate spectral numerical schemes for kinetic equations with energy diffusion
NASA Astrophysics Data System (ADS)
Wilkening, Jon; Cerfon, Antoine J.; Landreman, Matt
2015-08-01
We examine the merits of using a family of polynomials that are orthogonal with respect to a non-classical weight function to discretize the speed variable in continuum kinetic calculations. We consider a model one-dimensional partial differential equation describing energy diffusion in velocity space due to Fokker-Planck collisions. This relatively simple case allows us to compare the results of the projected dynamics with an expensive but highly accurate spectral transform approach. It also allows us to integrate in time exactly, and to focus entirely on the effectiveness of the discretization of the speed variable. We show that for a fixed number of modes or grid points, the non-classical polynomials can be many orders of magnitude more accurate than classical Hermite polynomials or finite-difference solvers for kinetic equations in plasma physics. We provide a detailed analysis of the difference in behavior and accuracy of the two families of polynomials. For the non-classical polynomials, if the initial condition is not smooth at the origin when interpreted as a three-dimensional radial function, the exact solution leaves the polynomial subspace for a time, but returns (up to roundoff accuracy) to the same point evolved to by the projected dynamics in that time. By contrast, using classical polynomials, the exact solution differs significantly from the projected dynamics solution when it returns to the subspace. We also explore the connection between eigenfunctions of the projected evolution operator and (non-normalizable) eigenfunctions of the full evolution operator, as well as the effect of truncating the computational domain.
Kinetic equations for baryogenesis via sterile neutrino oscillation
Asaka, Takehiko; Eijima, Sintaro; Ishida, Hiroyuki E-mail: eijima@muse.sc.niigata-u.ac.jp
2012-02-01
We investigate baryogenesis in the νMSM (neutrino Minimal Standard Model), which is the SM extended by three right-handed neutrinos with masses below the electroweak scale. The baryon asymmetry of the universe can be generated by the mechanism via flavor oscillation of right-handed (sterile) neutrinos which are responsible to masses of active neutrinos confirmed by various experiments. We present the kinetic equations for the matrix of densities of leptons which describe the generation of asymmetries. Especially, the momentum dependence of the matrix of densities is taken into account. By solving these equations numerically, it is found that the momentum distribution is significantly distorted from the equilibrium one, since the production for the modes with lower momenta k << T (T is the temperature of the universe) is enhanced, while suppressed for higher modes. As a result, the most important mode for the yields of sterile neutrinos as well as the baryon asymmetry is k ≅ 2T, which is smaller than (k) inferred from the thermal average. The comparison with the previous works is also discussed.
Ocean swell within the kinetic equation for water waves
NASA Astrophysics Data System (ADS)
Badulin, Sergei I.; Zakharov, Vladimir E.
2017-06-01
Results of extensive simulations of swell evolution within the duration-limited setup for the kinetic Hasselmann equation for long durations of up to 2 × 106 s are presented. Basic solutions of the theory of weak turbulence, the so-called Kolmogorov-Zakharov solutions, are shown to be relevant to the results of the simulations. Features of self-similarity of wave spectra are detailed and their impact on methods of ocean swell monitoring is discussed. Essential drop in wave energy (wave height) due to wave-wave interactions is found at the initial stages of swell evolution (on the order of 1000 km for typical parameters of the ocean swell). At longer times, wave-wave interactions are responsible for a universal angular distribution of wave spectra in a wide range of initial conditions. Weak power-law attenuation of swell within the Hasselmann equation is not consistent with results of ocean swell tracking from satellite altimetry and SAR (synthetic aperture radar) data. At the same time, the relatively fast weakening of wave-wave interactions makes the swell evolution sensitive to other effects. In particular, as shown, coupling with locally generated wind waves can force the swell to grow in relatively light winds.
Re-interpretation of the logistic equation for batch microbial growth in relation to Monod kinetics.
Kargi, F
2009-04-01
To determine the underlying substrate utilization mechanism in the logistic equation for batch microbial growth by revealing the relationship between the logistic and Monod kinetics. Also, to determine the logistic rate constant in terms of Monod kinetic constants. The logistic equation used to describe batch microbial growth was related to the Monod kinetics and found to be first-order in terms of the substrate and biomass concentrations. The logistic equation constant was also related to the Monod kinetic constants. Similarly, the substrate utilization kinetic equations were derived by using the logistic growth equation and related to the Monod kinetics. It is revaled that the logistic growth equation is a special form of the Monod growth kinetics when substrate limitation is first-order with respect to the substrate concentration. The logistic rate constant (k) is directly proportional to the maximum specific growth rate constant (mu(m)) and initial substrate concentration (S(0)) and also inversely related to the saturation constant (K(s)). The semi-empirical logistic equation can be used instead of Monod kinetics at low substrate concentrations to describe batch microbial growth using the relationship between the logistic rate constant and the Monod kinetic constants.
Verification of continuum drift kinetic equation solvers in NIMROD
NASA Astrophysics Data System (ADS)
Held, E. D.; Kruger, S. E.; Ji, J.-Y.; Belli, E. A.; Lyons, B. C.
2015-03-01
Verification of continuum solutions to the electron and ion drift kinetic equations (DKEs) in NIMROD [C. R. Sovinec et al., J. Comp. Phys. 195, 355 (2004)] is demonstrated through comparison with several neoclassical transport codes, most notably NEO [E. A. Belli and J. Candy, Plasma Phys. Controlled Fusion 54, 015015 (2012)]. The DKE solutions use NIMROD's spatial representation, 2D finite-elements in the poloidal plane and a 1D Fourier expansion in toroidal angle. For 2D velocity space, a novel 1D expansion in finite elements is applied for the pitch angle dependence and a collocation grid is used for the normalized speed coordinate. The full, linearized Coulomb collision operator is kept and shown to be important for obtaining quantitative results. Bootstrap currents, parallel ion flows, and radial particle and heat fluxes show quantitative agreement between NIMROD and NEO for a variety of tokamak equilibria. In addition, velocity space distribution function contours for ions and electrons show nearly identical detailed structure and agree quantitatively. A Θ-centered, implicit time discretization and a block-preconditioned, iterative linear algebra solver provide efficient electron and ion DKE solutions that ultimately will be used to obtain closures for NIMROD's evolving fluid model.
Verification of continuum drift kinetic equation solvers in NIMROD
Held, E. D.; Ji, J.-Y.; Kruger, S. E.; Belli, E. A.; Lyons, B. C.
2015-03-15
Verification of continuum solutions to the electron and ion drift kinetic equations (DKEs) in NIMROD [C. R. Sovinec et al., J. Comp. Phys. 195, 355 (2004)] is demonstrated through comparison with several neoclassical transport codes, most notably NEO [E. A. Belli and J. Candy, Plasma Phys. Controlled Fusion 54, 015015 (2012)]. The DKE solutions use NIMROD's spatial representation, 2D finite-elements in the poloidal plane and a 1D Fourier expansion in toroidal angle. For 2D velocity space, a novel 1D expansion in finite elements is applied for the pitch angle dependence and a collocation grid is used for the normalized speed coordinate. The full, linearized Coulomb collision operator is kept and shown to be important for obtaining quantitative results. Bootstrap currents, parallel ion flows, and radial particle and heat fluxes show quantitative agreement between NIMROD and NEO for a variety of tokamak equilibria. In addition, velocity space distribution function contours for ions and electrons show nearly identical detailed structure and agree quantitatively. A Θ-centered, implicit time discretization and a block-preconditioned, iterative linear algebra solver provide efficient electron and ion DKE solutions that ultimately will be used to obtain closures for NIMROD's evolving fluid model.
A bin integral method for solving the kinetic collection equation
NASA Astrophysics Data System (ADS)
Wang, Lian-Ping; Xue, Yan; Grabowski, Wojciech W.
2007-09-01
A new numerical method for solving the kinetic collection equation (KCE) is proposed, and its accuracy and convergence are investigated. The method, herein referred to as the bin integral method with Gauss quadrature (BIMGQ), makes use of two binwise moments, namely, the number and mass concentration in each bin. These two degrees of freedom define an extended linear representation of the number density distribution for each bin following Enukashvily (1980). Unlike previous moment-based methods in which the gain and loss integrals are evaluated for a target bin, the concept of source-bin pair interactions is used to transfer bin moments from source bins to target bins. Collection kernels are treated by bilinear interpolations. All binwise interaction integrals are then handled exactly by Gauss quadrature of various orders. In essence the method combines favorable features in previous spectral moment-based and bin-based pair-interaction (or flux) methods to greatly enhance the logic, consistency, and simplicity in the numerical method and its implementation. Quantitative measures are developed to rigorously examine the accuracy and convergence properties of BIMGQ for both the Golovin kernel and hydrodynamic kernels. It is shown that BIMGQ has a superior accuracy for the Golovin kernel and a monotonic convergence behavior for hydrodynamic kernels. Direct comparisons are also made with the method of Berry and Reinhardt (1974), the linear flux method of Bott (1998), and the linear discrete method of Simmel et al. (2002).
A comparison of the efficiency of numerical methods for integrating chemical kinetic rate equations
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1984-01-01
The efficiency of several algorithms used for numerical integration of stiff ordinary differential equations was compared. The methods examined included two general purpose codes EPISODE and LSODE and three codes (CHEMEQ, CREK1D and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes were applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code available for the integration of combustion kinetic rate equations. It is shown that an iterative solution of the algebraic energy conservation equation to compute the temperature can be more efficient then evaluating the temperature by integrating its time-derivative.
Wang, Lijin
2016-06-08
The stochastic protein kinetic equations can be stiff for certain parameters, which makes their numerical simulation rely on very small time step sizes, resulting in large computational cost and accumulated round-off errors. For such situation, we provide a method of reducing stiffness of the stochastic protein kinetic equation by means of a kind of variable transformation. Theoretical and numerical analysis show effectiveness of this method. Its generalization to a more general class of stochastic differential equation models is also discussed.
The kinetic equations for rotating and gravitating spheroidal body
NASA Astrophysics Data System (ADS)
Krot, A.
2003-04-01
In papers [1],[2] it has been proposed a statistical model of the gravitational interaction of particles.In the framework of this model bodies have fuzzy outlines and are represented by means of spheroidal forms. A con- sistency of the proposed statistical model the Einstein general relativity [3], [4], [5] has been shown. In work [6], which is a continuation of the paper[2], it has been investigated a slowly evolving in time process of a gravitational compression of a spheroidal body close to an unstable equilibrium state. In the paper [7] the equation of motion of particles inside the weakly gravitating spheroidal body modeled by means of an ideal liquid has been obtained. It has been derived the equations of hyperbolic type for the gravitational field of a weakly gravitating spheroidal body under observable values of velocities of particles composing it [7],[8]. This paper considers the case of gravitational compres- sion of spheroidal body with observable values of parti- cles.This means that distribution function of particles inside weakly rotating spheroidal body is a sum of an isotropic space-homogeneous stationary distribution function and its change (disturbance) under influence of dymanical gravitational field. The change of initial space-homogeneous stationary distribution function satisfyes the Boltzmann kinetic equation. This paper shows that if gravitating spheroidal body is rotating uniformly or is being at rest then distribution function of its particles satisfyes the Liouville theorem. Thus, being in unstable statistical quasiequilibrium the gravi- tating spheroidal body is rotating with constant angular velocity (or, in particular case, is being at rest). The joint distribution function of spheroidal body's particles in to coordinate space and angular velocity space is introduced. References [1] A.M.Krot, Achievements in Modern Radioelectronics, special issue "Cosmic Radiophysics",no. 8, pp.66-81, 1996 (Moscow, Russia). [2] A.M.Krot, Proc. SPIE 13
Time reversal invariance of quantum kinetic equations: Nonequilibrium Green functions formalism
NASA Astrophysics Data System (ADS)
Scharnke, Miriam; Schlünzen, Niclas; Bonitz, Michael
2017-06-01
Time reversal symmetry is a fundamental property of many quantum mechanical systems. The relation between statistical physics and time reversal is subtle and not all statistical theories conserve this particular symmetry, most notably hydrodynamic equations and kinetic equations such as the Boltzmann equation. In this article, it is shown analytically that quantum kinetic generalizations of the Boltzmann equation that are derived using the nonequilibrium Green functions formalism as well as all approximations that stem from Φ-derivable self-energies are time reversal invariant.
Generalized Klein-Gordon and Dirac Equations from Nonlocal Kinetic Approach
NASA Astrophysics Data System (ADS)
El-Nabulsi, Rami Ahmad
2016-09-01
In this note, I generalized the Klein-Gordon and the Dirac equations by using Suykens's nonlocal-in-time kinetic energy approach, which is motivated from Feynman's kinetic energy functional formalism where the position differences are shifted with respect to one another. I proved that these generalized equations are similar to those obtained in literature in the presence of minimal length based on the Quesne-Tkachuk algebra.
Entropy Analysis of Kinetic Flux Vector Splitting Schemes for the Compressible Euler Equations
NASA Technical Reports Server (NTRS)
Shiuhong, Lui; Xu, Jun
1999-01-01
Flux Vector Splitting (FVS) scheme is one group of approximate Riemann solvers for the compressible Euler equations. In this paper, the discretized entropy condition of the Kinetic Flux Vector Splitting (KFVS) scheme based on the gas-kinetic theory is proved. The proof of the entropy condition involves the entropy definition difference between the distinguishable and indistinguishable particles.
Formulation and closure of compressible turbulence equations in the light of kinetic theory
NASA Technical Reports Server (NTRS)
Tsuge, S.; Sagara, K.
1976-01-01
Fluid-dynamic moment equations, based on a kinetic hierarchy system, are derived governing the interaction between turbulent and thermal fluctuations. The kinetic theory is shown to reduce the inherent complexity of the conventional formalism of compressible turbulence theory and to minimize arbitrariness in formulating the closure condition.
Parameter Estimates in Differential Equation Models for Chemical Kinetics
ERIC Educational Resources Information Center
Winkel, Brian
2011-01-01
We discuss the need for devoting time in differential equations courses to modelling and the completion of the modelling process with efforts to estimate the parameters in the models using data. We estimate the parameters present in several differential equation models of chemical reactions of order n, where n = 0, 1, 2, and apply more general…
Parameter Estimates in Differential Equation Models for Chemical Kinetics
ERIC Educational Resources Information Center
Winkel, Brian
2011-01-01
We discuss the need for devoting time in differential equations courses to modelling and the completion of the modelling process with efforts to estimate the parameters in the models using data. We estimate the parameters present in several differential equation models of chemical reactions of order n, where n = 0, 1, 2, and apply more general…
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.
NASA Astrophysics Data System (ADS)
Salhoumi, A.; Galenko, P. K.
2017-04-01
Rapidly moving solid-liquid interface is treated analytically and numerically. Derivation and qualitative analysis of interface propagation kinetics is presented. Quantitative predictions of solutions, which follow from the Kinetic Rate Theory and the solution of Gibbs-Thomson-type equation, are compared with Molecular Dynamics simulation data (MD-data) on crystallization and melting of fcc-lattice of nickel. It is shown in the approximation of a linear behavior of the interface velocity versus undercooling that the Gibbs-Thomson-type equation and kinetic rate theory describe MD-data well enough, in the range of small growth velocity and within the range of relatively small undercooling, with a relative error for the obtained values of kinetic coefficient of the order 1.1%. Within the small-and long range of undercooling, in nonlinear behavior of the interface velocity versus undercooling, the kinetic rate theory disagrees sharply with MD-data, qualitatively and quantitatively, unlike to the Gibbs-Thomson-type equation which is in a good agreement with MD-data within the whole range of undercooling and crystal growth velocity.
Integrating chemical kinetic rate equations by selective use of stiff and nonstiff methods
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1985-01-01
The effect of switching between nonstiff and stiff methods on the efficiency of algorithms for integrating chemical kinetic rate equations was examined. Different integration methods were tested by application of the packaged code LSODE to four practical combustion kinetics problems. The problems describe adiabatic, and homogeneous gas phase combustion reactions. It is shown that selective use of nonstiff and stiff methods in different regimes of a typical batch combustion problem is faster than the use of either method for the entire problem. The implications which result in the development of fast integration techniques for combustion kinetic rate equations are discussed.
Integrating chemical kinetic rate equations by selective use of stiff and nonstiff methods
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1985-01-01
The effect of switching between nonstiff and stiff methods on the efficiency of algorithms for integrating chemical kinetic rate equations is presented. Different integration methods are tested by application of the packaged code LSODE to four practical combustion kinetics problems. The problems describe adiabatic, homogeneous gas-phase combustion reactions. It is shown that selective use of nonstiff and stiff methods in different regimes of a typical batch combustion problem is faster than the use of either method for the entire problem. The implications of this result to the development of fast integration techniques for combustion kinetic rate equations are discussed.
Cluster equations for the Glauber kinetic Ising ferromagnet: I. Existence and uniqueness
NASA Astrophysics Data System (ADS)
Kreer, Markus
The infinite set of cluster equations, proposed by Binder and Müller-Krumbhaar for a Glauber kinetic Ising ferromagnet in 1974, generalize the Becker-Döring equations used in classical nucleation theory. For positive symmetric transition rates satisfying certain growth conditions and a detailed balance condition we prove for sufficiently fast decaying initial cluster distributions the existence of a positive cluster distribution with finite density for all finite times solving the cluster equations. Uniqueness is proven under some further conditions on the transition rates. Our existence and uniqueness results apply e.g. for a Glauber kinetic Ising ferromagnet in two dimensions.
Denicol, G. S.; Koide, T.; Rischke, D. H.
2010-10-15
We rederive the equations of motion of dissipative relativistic fluid dynamics from kinetic theory. In contrast with the derivation of Israel and Stewart, which considered the second moment of the Boltzmann equation to obtain equations of motion for the dissipative currents, we directly use the latter's definition. Although the equations of motion obtained via the two approaches are formally identical, the coefficients are different. We show that, for the one-dimensional scaling expansion, our method is in better agreement with the solution obtained from the Boltzmann equation.
Belyi, V.V.; Kukharenko, Y.A.; Wallenborn, J. ||
1996-05-01
Taking into account the first non-Markovian correction to the Balescu-Lenard equation, we have derived an expression for the pair correlation function and a nonlinear kinetic equation valid for a nonideal polarized classical plasma. This last equation allows for the description of the correlational energy evolution and shows the global conservation of energy with dynamical polarization. {copyright} {ital 1996 The American Physical Society.}
A comparison of the efficiency of numerical methods for integrating chemical kinetic rate equations
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1984-01-01
A comparison of the efficiency of several algorithms recently developed for the efficient numerical integration of stiff ordinary differential equations is presented. The methods examined include two general-purpose codes EPISODE and LSODE and three codes (CHEMEQ, CREK1D, and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes are applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code currently available for the integration of combustion kinetic rate equations. An important finding is that an iterative solution of the algebraic energy conservation equation to compute the temperature can be more efficient than evaluating the temperature by integrating its time-derivative.
Solving Point-Reactor Kinetics Equations Using Exponential Moment Methods
2013-03-21
equations of the following form: ( ) ( ) ( ) ( ) ( )i i i dn t t n t c t S t dt (2) ( ) ( ) ( )i ii i dc t c t n...presented in the function. Exponential moment functions are orderless; that is, the value of the function is invariant under permutations of its...turned into an integral equation by ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) i i i i i i i i i i i i dn
A new code for collisional drift kinetic equation solving
Reynolds, J. M.; Velasco, J. L.; Tarancon, A.; Lopez-Bruna, D.; Guasp, J.
2008-11-02
We introduce a new code of plasma transport based on evolving the Boltzmann equation in guiding center approximation where collisions has been taken into account. The spatial geometry is discretized using high order elements in space and a moment expansion in velocity space. First calculations with non-evolving electric field agree with the particle code ISDEP.
Valencia, Pedro L; Astudillo-Castro, Carolina; Gajardo, Diego; Flores, Sebastián
2017-04-01
We provide initial rate data from enzymatic reaction experiments and tis processing to estimate the kinetic parameters from the substrate uncompetitive inhibition equation using the median method published by Eisenthal and Cornish-Bowden (Cornish-Bowden and Eisenthal, 1974; Eisenthal and Cornish-Bowden, 1974). The method was denominated the direct linear plot and consists in the calculation of the median from a dataset of kinetic parameters Vmax and Km from the Michaelis-Menten equation. In this opportunity we present the procedure to applicate the direct linear plot to the substrate uncompetitive inhibition equation; a three-parameter equation. The median method is characterized for its robustness and its insensibility to outlier. The calculations are presented in an Excel datasheet and a computational algorithm was developed in the free software Python. The kinetic parameters of the substrate uncompetitive inhibition equation Vmax , Km and Ks were calculated using three experimental points from the dataset formed by 13 experimental points. All the 286 combinations were calculated. The dataset of kinetic parameters resulting from this combinatorial was used to calculate the median which corresponds to the statistic estimator of the real kinetic parameters. A comparative statistical analyses between the median method and the least squares was published in Valencia et al. [3].
Kinetically reduced local Navier-Stokes equations: an alternative approach to hydrodynamics.
Karlin, Iliya V; Tomboulides, Ananias G; Frouzakis, Christos E; Ansumali, Santosh
2006-09-01
An alternative approach, the kinetically reduced local Navier-Stokes (KRLNS) equations for the grand potential and the momentum, is proposed for the simulation of low Mach number flows. The Taylor-Green vortex flow is considered in the KRLNS framework, and compared to the results of the direct numerical simulation of the incompressible Navier-Stokes equations. The excellent agreement between the KRLNS equations and the incompressible nonlocal Navier-Stokes equations for this nontrivial time-dependent flow indicates that the former is a viable alternative for computational fluid dynamics at low Mach numbers.
Diffusion Limit of Kinetic Equations for Multiple Species Charged Particles
NASA Astrophysics Data System (ADS)
Wu, Hao; Lin, Tai-Chia; Liu, Chun
2015-02-01
In ionic solutions, there are multi-species charged particles (ions) with different properties like mass, charge etc. Macroscopic continuum models like the Poisson-Nernst-Planck (PNP) systems have been extensively used to describe the transport and distribution of ionic species in the solvent. Starting from the kinetic theory for the ion transport, we study a Vlasov-Poisson-Fokker-Planck (VPFP) system in a bounded domain with reflection boundary conditions for charge distributions and prove that the global renormalized solutions of the VPFP system converge to the global weak solutions of the PNP system, as the small parameter related to the scaled thermal velocity and mean free path tends to zero. Our results may justify the PNP system as a macroscopic model for the transport of multi-species ions in dilute solutions.
Bai, Shirong; Skodje, Rex T
2017-08-17
A new approach is presented for simulating the time-evolution of chemically reactive systems. This method provides an alternative to conventional modeling of mass-action kinetics that involves solving differential equations for the species concentrations. The method presented here avoids the need to solve the rate equations by switching to a representation based on chemical pathways. In the Sum Over Histories Representation (or SOHR) method, any time-dependent kinetic observable, such as concentration, is written as a linear combination of probabilities for chemical pathways leading to a desired outcome. In this work, an iterative method is introduced that allows the time-dependent pathway probabilities to be generated from a knowledge of the elementary rate coefficients, thus avoiding the pitfalls involved in solving the differential equations of kinetics. The method is successfully applied to the model Lotka-Volterra system and to a realistic H2 combustion model.
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.
Critical dynamics in systems controlled by fractional kinetic equations
NASA Astrophysics Data System (ADS)
Batalov, Lev; Batalova, Anastasia
2013-02-01
The field theory renormalization group is used for analyzing the fractional Langevin equation with the order of the temporal derivative 0<α<1, fractional Laplacian of the order σ, and Gaussian noise correlator. The case of non-linearity φm with odd m≥3 is considered. It is proved that the model is multiplicatively renormalizable. Propagators were found in the momentum and coordinate representation, expressed in terms of Fox’s H functions. Existence of the dissipative scaling regime in the framework of the ε expansion for σ=2, α=1/l, l=1,2,… is proved. Requirement of the continuous dependence of the critical exponents on α imposes the condition m=3. The main quantitative result is the calculation of the dynamical critical exponent z for α=1/2 up to ε2. We have obtained for it the expression z=4+0.1555ε2+O(ε3).
Phu, Jack; Al-Saleem, Noha; Kalloniatis, Michael; Khuu, Sieu K
2016-11-01
In the present study, we measured the extent of statokinetic dissociation (SKD) in normal observers and then equated the psychophysical tasks into a two-interval forced choice (2IFC) procedure. In Experiment 1, we used the Humphrey visual field analyzer in static perimetry and automated kinetic perimetry modes to measure contrast sensitivity thresholds and the Goldmann manual kinetic perimeter to measure isopters. This was carried out using a Goldmann size II target. Goldmann kinetic perimetry was performed manually with both inward (peripheral to center) and outward (center to periphery) directions of movement to deduce an "average" isopter. This revealed the presence of SKD when superimposed upon the map of static contrast threshold results. There was no evidence of any contribution of examiner technique or instrument-specific differences to SKD. In Experiment 2, we determined the psychometric curves plotting proportion seen as a function of stimulus eccentricity with static and kinetic stimuli with a 2IFC procedure and method of constant stimuli. In an additional experiment, we also showed that subjects were able to reliably discriminate whether a stimulus was static, moving inward, or moving outward, and hence, comparisons could be made between static and kinetic perimetry tasks. Overall, by making the task objective and reducing criterion bias, eccentricity thresholds were equated across static and kinetic perimetry methods; hence, no evidence of SKD was seen. We suggest SKD is inherent to the differences in methodology of threshold measurement in conventional static and kinetic perimetry and individual criterion bias.
NASA Astrophysics Data System (ADS)
Yano, Ryosuke; Suzuki, Kojiro; Kuroda, Hisayasu
2007-01-01
The direct description of chemical reactions by the Boltzmann equation still involves some difficulties in the kinetic theory. In this paper, we describe diatomic molecular dissociation due to transitions of vibrational quantum states resulting from inelastic collisions. These can be described by the Wang Chang-Uhlenbeck (WCU) equation. To avoid direct evaluation of the strong nonlinear collision kernel of the WCU equation, we used a kinetic equation. For accurate description of the dissociation process, we describe improvements we made to the conventional inelastic collision model (the so-called Morse model). Combining this inelastic collision model with the gas mixture model by Oguchi, we formulated a model for representing diatomic molecular dissociations. We validated this model by simulating a hypersonic shock layer with diatomic molecular dissociation.
New integration techniques for chemical kinetic rate equations. 2: Accuracy comparison
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1985-01-01
A comparison of the accuracy of several techniques recently developed for solving stiff differential equations is presented. The techniques examined include two general purpose codes EEPISODE and LSODE developed for an arbitrary system of ordinary differential equations, and three specialized codes CHEMEQ, CREKID, and GCKP84 developed specifically to solve chemical kinetic rate equations. The accuracy comparisons are made by applying these solution procedures to two practical combustion kinetics problems. Both problems describe adiabatic, homogeneous, gas phase chemical reactions at constant pressure, and include all three combustion regimes: induction, heat release, and equilibration. The comparisons show that LSODE is the most efficient code - in the sense that it requires the least computational work to attain a specified accuracy level. An important finding is that an iterative solution of the algebraic enthalpy conservation equation for the temperature can be more accurate and efficient than computing the temperature by integrating its time derivative.
New integration techniques for chemical kinetic rate equations. II - Accuracy comparison
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1986-01-01
A comparison of the accuracy of several techniques recently developed for solving stiff differential equations is presented. The techniques examined include two general purpose codes EEPISODE and LSODE developed for an arbitrary system of ordinary differential equations, and three specialized codes CHEMEQ, CREKID, and GCKP84 developed specifically to solve chemical kinetic rate equations. The accuracy comparisons are made by applying these solution procedures to two practical combustion kinetics problems. Both problems describe adiabatic, homogeneous, gas phase chemical reactions at constant pressure, and include all three combustion regimes: induction heat release, and equilibration. The comparisons show that LSODE is the most efficient code - in the sense that it requires the least computational work to attain a specified accuracy level. An important finding is that an iterative solution of the algebraic enthalpy conservation equation for the temperature can be more accurate and efficient than computing the temperature by integrating its time derivative.
New integration techniques for chemical kinetic rate equations. II - Accuracy comparison
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1986-01-01
A comparison of the accuracy of several techniques recently developed for solving stiff differential equations is presented. The techniques examined include two general purpose codes EEPISODE and LSODE developed for an arbitrary system of ordinary differential equations, and three specialized codes CHEMEQ, CREKID, and GCKP84 developed specifically to solve chemical kinetic rate equations. The accuracy comparisons are made by applying these solution procedures to two practical combustion kinetics problems. Both problems describe adiabatic, homogeneous, gas phase chemical reactions at constant pressure, and include all three combustion regimes: induction heat release, and equilibration. The comparisons show that LSODE is the most efficient code - in the sense that it requires the least computational work to attain a specified accuracy level. An important finding is that an iterative solution of the algebraic enthalpy conservation equation for the temperature can be more accurate and efficient than computing the temperature by integrating its time derivative.
Kinetic equations for a density matrix describing nonlinear effects in spectral line wings
Parkhomenko, A. I. Shalagin, A. M.
2011-11-15
Kinetic quantum equations are derived for a density matrix with collision integrals describing nonlinear effects in spectra line wings. These equations take into account the earlier established inequality of the spectral densities of Einstein coefficients for absorption and stimulated radiation emission by a two-level quantum system in the far wing of a spectral line in the case of frequent collisions. The relationship of the absorption and stimulated emission probabilities with the characteristics of radiation and an elementary scattering event is found.
Analytical solution of the kinetic equation for a uniform plasma in a magnetic field
Ji, Jeong-Young; Held, Eric D.
2010-07-15
The kinetic equation for a single-component uniform plasma in a magnetic field is analytically solved by the moment method. The linear system of ordinary differential equations for the moments is decomposed into subsystems of lower dimensions by a geometric method. The eigensystem of each subsystem shows that parallel moments decay monotonically, but perpendicular lth harmonic moments decay while oscillating with the l,l-2,...,-th harmonics of gyrofrequency. A generalization to a multicomponent plasma is discussed.
Analytical solution of the kinetic equation for a uniform plasma in a magnetic field.
Ji, Jeong-Young; Held, Eric D
2010-07-01
The kinetic equation for a single-component uniform plasma in a magnetic field is analytically solved by the moment method. The linear system of ordinary differential equations for the moments is decomposed into subsystems of lower dimensions by a geometric method. The eigensystem of each subsystem shows that parallel moments decay monotonically, but perpendicular lth harmonic moments decay while oscillating with the l,l-2,… ,-th harmonics of gyrofrequency. A generalization to a multicomponent plasma is discussed.
NASA Technical Reports Server (NTRS)
Pai, S. I.
1973-01-01
The fundamental equations of a mixture of a gas and pseudofluid of small spherical solid particles are derived from the Boltzmann equation of two-fluid theory. The distribution function of the gas molecules is defined in the same manner as in the ordinary kinetic theory of gases, but the distribution function for the solid particles is different from that of the gas molecules, because it is necessary to take into account the different size and physical properties of solid particles. In the proposed simple kinetic theory, two additional parameters are introduced: one is the radius of the spheres and the other is the instantaneous temperature of the solid particles in the distribution of the solid particles. The Boltzmann equation for each species of the mixture is formally written, and the transfer equations of these Boltzmann equations are derived and compared to the well-known fundamental equations of the mixture of a gas and small solid particles from continuum theory. The equations obtained reveal some insight into various terms in the fundamental equations. For instance, the partial pressure of the pseudofluid of solid particles is not negligible if the volume fraction of solid particles is not negligible as in the case of lunar ash flow.
A generalized Fisher equation and its utility in chemical kinetics.
Ross, John; Fernández Villaverde, Alejandro; Banga, Julio R; Vázquez, Sara; Morán, Federico
2010-07-20
A generalized Fisher equation (GFE) relates the time derivative of the average of the intrinsic rate of growth to its variance. The GFE is an exact mathematical result that has been widely used in population dynamics and genetics, where it originated. Here we demonstrate that the GFE can also be useful in other fields, specifically in chemistry, with models of two chemical reaction systems for which the mechanisms and rate coefficients correspond reasonably well to experiments. A bad fit of the GFE can be a sign of high levels of measurement noise; for low or moderate levels of noise, fulfillment of the GFE is not degraded. Hence, the GFE presents a noise threshold that may be used to test the validity of experimental measurements without requiring any additional information. In a different approach information about the system (model) is included in the calculations. In that case, the discrepancy with the GFE can be used as an optimization criterion for the determination of rate coefficients in a given reaction mechanism.
A generalized Fisher equation and its utility in chemical kinetics
Ross, John; Villaverde, Alejandro Fernández; Banga, Julio R.; Vázquez, Sara; Morán, Federico
2010-01-01
A generalized Fisher equation (GFE) relates the time derivative of the average of the intrinsic rate of growth to its variance. The GFE is an exact mathematical result that has been widely used in population dynamics and genetics, where it originated. Here we demonstrate that the GFE can also be useful in other fields, specifically in chemistry, with models of two chemical reaction systems for which the mechanisms and rate coefficients correspond reasonably well to experiments. A bad fit of the GFE can be a sign of high levels of measurement noise; for low or moderate levels of noise, fulfillment of the GFE is not degraded. Hence, the GFE presents a noise threshold that may be used to test the validity of experimental measurements without requiring any additional information. In a different approach information about the system (model) is included in the calculations. In that case, the discrepancy with the GFE can be used as an optimization criterion for the determination of rate coefficients in a given reaction mechanism. PMID:20615992
New integration techniques for chemical kinetic rate equations. I - Efficiency comparison
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1986-01-01
A comparison of the efficiency of several recently developed numerical techniques for solving chemical kinetic rate equations is presented. The solution procedures examined include two general-purpose codes, EPISODE and LSODE, developed as multipurpose differential equation solvers, and three specialzed codes, CHEMEQ, CREK1D, and GCKP84, developed specifically for chemical kinetics. The efficiency comparison is made by applying these codes to two practical combustion kinetics problems. Both problems describe adiabatic, constant-pressure, gas-phase chemical reactions and include all three combustion regimes: induction, heat release, and equilibration. The comparison shows that LSODE is the fastest routine currently available for solving chemical kinetic rate equations. An important finding is that an iterative solution of the algebraic enthalpy conservation equation for temperature can be significantly faster than evaluation of the temperature by integration of its time derivative. Significant increases in computational speed are realized by updating the reaction rate constants only when the temperature change exceeds an amount Delta-T that is problem dependent. An approximate expression for the automatic evaluation of Delta-T is presented and is shown to result in increased computational speed.
New integration techniques for chemical kinetic rate equations. I - Efficiency comparison
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1986-01-01
A comparison of the efficiency of several recently developed numerical techniques for solving chemical kinetic rate equations is presented. The solution procedures examined include two general-purpose codes, EPISODE and LSODE, developed as multipurpose differential equation solvers, and three specialzed codes, CHEMEQ, CREK1D, and GCKP84, developed specifically for chemical kinetics. The efficiency comparison is made by applying these codes to two practical combustion kinetics problems. Both problems describe adiabatic, constant-pressure, gas-phase chemical reactions and include all three combustion regimes: induction, heat release, and equilibration. The comparison shows that LSODE is the fastest routine currently available for solving chemical kinetic rate equations. An important finding is that an iterative solution of the algebraic enthalpy conservation equation for temperature can be significantly faster than evaluation of the temperature by integration of its time derivative. Significant increases in computational speed are realized by updating the reaction rate constants only when the temperature change exceeds an amount Delta-T that is problem dependent. An approximate expression for the automatic evaluation of Delta-T is presented and is shown to result in increased computational speed.
Solvability of certain inverse problems for the nonstationary kinetic transport equation
NASA Astrophysics Data System (ADS)
Volkov, N. P.
2016-09-01
Linear and nonlinear inverse problems for the nonstationary multispeed anisotropic kinetic transport equation are studied. Sufficient conditions for the existence and uniqueness of weak solutions to these problems in various function spaces are found. The proofs of the corresponding theorems imply that solutions of the inverse problems under study can be obtained by applying the method of successive approximations.
General theory of multistage geminate reactions of isolated pairs of reactants. I. Kinetic equations
NASA Astrophysics Data System (ADS)
Doktorov, Alexander B.; Kipriyanov, Alexey A.
2014-05-01
General matrix approach to the consideration of multistage geminate reactions of isolated pairs of reactants depending on reactant mobility is formulated on the basis of the concept of "effective" particles. Various elementary reactions (stages of multistage reaction including physicochemical processes of internal quantum state changes) proceeding with the participation of isolated pairs of reactants (or isolated reactants) are taken into account. Investigation has been made in terms of kinetic approach implying the derivation of general (matrix) kinetic equations for local and mean probabilities of finding any of the reaction species in the sample under study (or for local and mean concentrations). The recipes for the calculation of kinetic coefficients of the equations for mean quantities in terms of relative coordinates of reactants have been formulated in the general case of inhomogeneous reacting systems. Important specific case of homogeneous reacting systems is considered.
A multi-dimensional kinetic-based upwind solver for the Euler equations
NASA Technical Reports Server (NTRS)
Eppard, W. M.; Grossman, B.
1993-01-01
A multidimensional kinetic fluctuation-splitting scheme has been developed for the Euler equations. The scheme is based on an N-scheme discretization of the Boltzmann equation at the kinetic level for triangulated Cartesian meshes with a diagonal-adaptive strategy. The resulting Euler scheme is a cell-vertex fluctuation-splitting scheme where fluctuations in the conserved-variable vector Q are obtained as moments of the fluctuation in the Maxwellian velocity distribution function at the kinetic level. Encouraging preliminary results have been obtained for perfect gases on Cartesian meshes with first-order spatial accuracy. The present approach represents an improvement to the well-established dimensionally-split upwind schemes.
General theory of multistage geminate reactions of isolated pairs of reactants. I. Kinetic equations
Doktorov, Alexander B.; Kipriyanov, Alexey A.
2014-05-14
General matrix approach to the consideration of multistage geminate reactions of isolated pairs of reactants depending on reactant mobility is formulated on the basis of the concept of “effective” particles. Various elementary reactions (stages of multistage reaction including physicochemical processes of internal quantum state changes) proceeding with the participation of isolated pairs of reactants (or isolated reactants) are taken into account. Investigation has been made in terms of kinetic approach implying the derivation of general (matrix) kinetic equations for local and mean probabilities of finding any of the reaction species in the sample under study (or for local and mean concentrations). The recipes for the calculation of kinetic coefficients of the equations for mean quantities in terms of relative coordinates of reactants have been formulated in the general case of inhomogeneous reacting systems. Important specific case of homogeneous reacting systems is considered.
Geodesic acoustic mode in anisotropic plasmas using double adiabatic model and gyro-kinetic equation
Ren, Haijun; Cao, Jintao
2014-12-15
Geodesic acoustic mode in anisotropic tokamak plasmas is theoretically analyzed by using double adiabatic model and gyro-kinetic equation. The bi-Maxwellian distribution function for guiding-center ions is assumed to obtain a self-consistent form, yielding pressures satisfying the magnetohydrodynamic (MHD) anisotropic equilibrium condition. The double adiabatic model gives the dispersion relation of geodesic acoustic mode (GAM), which agrees well with the one derived from gyro-kinetic equation. The GAM frequency increases with the ratio of pressures, p{sub ⊥}/p{sub ∥}, and the Landau damping rate is dramatically decreased by p{sub ⊥}/p{sub ∥}. MHD result shows a low-frequency zonal flow existing for all p{sub ⊥}/p{sub ∥}, while according to the kinetic dispersion relation, no low-frequency branch exists for p{sub ⊥}/p{sub ∥}≳ 2.
A multi-dimensional kinetic-based upwind solver for the Euler equations
NASA Technical Reports Server (NTRS)
Eppard, W. M.; Grossman, B.
1993-01-01
A multidimensional kinetic fluctuation-splitting scheme has been developed for the Euler equations. The scheme is based on an N-scheme discretization of the Boltzmann equation at the kinetic level for triangulated Cartesian meshes with a diagonal-adaptive strategy. The resulting Euler scheme is a cell-vertex fluctuation-splitting scheme where fluctuations in the conserved-variable vector Q are obtained as moments of the fluctuation in the Maxwellian velocity distribution function at the kinetic level. Encouraging preliminary results have been obtained for perfect gases on Cartesian meshes with first-order spatial accuracy. The present approach represents an improvement to the well-established dimensionally-split upwind schemes.
Kinetic equations for systems with long-range interactions: a unified description
NASA Astrophysics Data System (ADS)
Chavanis, Pierre-Henri
2010-05-01
We complete the existing literature on the kinetic theory of systems with long-range interactions. Starting from the BBGKY hierarchy, or using projection operator technics or a quasilinear theory, a general kinetic equation can be derived when collective effects are neglected. This equation (which is not well known) applies to possibly spatially inhomogeneous systems, which is specific to systems with long-range interactions. Interestingly, the structure of this kinetic equation bears a clear physical meaning in terms of generalized Kubo relations. Furthermore, this equation takes a very similar form for stellar systems and two-dimensional point vortices, providing therefore a unified description of the kinetic theory of these systems. If we assume that the system is spatially homogeneous (or axisymmetric for point vortices), this equation can be simplified and reduces to the Landau equation (or its counterpart for point vortices). Our formalism thus offers a simple derivation of Landau-type equations. We also use this general formalism to derive a kinetic equation, written in angle-action variables, describing spatially inhomogeneous systems with long-range interactions. This new derivation solves the shortcomings of our previous derivation (Chavanis 2007 Physica A 377 469). Finally, we consider a test particle approach and derive general expressions for the diffusion and friction (or drift) coefficients of a test particle evolving in a bath of field particles. We make contact with the expressions previously obtained in the literature. As an application of the kinetic theory, we argue that, for one-dimensional systems and two-dimensional point vortices, the relaxation time is shorter for inhomogeneous (or non-axisymmetric) distributions than for homogeneous (or axisymmetric) distributions because there are potentially more resonances. We compare this prediction with existing numerical results. For the HMF model, we argue that the relaxation time scales like N for
Rudzinski, Władysław; Plazinski, Wojciech
2008-05-20
It is shown that the modified pseudo-first-order (MPFO) kinetic equation proposed recently by Yang and Al-Duri simulates well the behavior of the kinetics governed by the rate of surface reaction and described by our general kinetic equation, based on the statistical rate theory. The linear representation with respect to time, offered by the MPFO equation seems to be a convenient tool for distinguishing between the surface reaction and the diffusional kinetics. Also, a method of distinguishing between the surface reaction and the intraparticle diffusion model based on analyzing the initial kinetic isotherms of sorption is proposed. The applicability of these procedures is demonstrated by the analysis of adsorption kinetics of the reactive yellow dye onto an activated carbon.
Functional integral derivation of the kinetic equation of two-dimensional point vortices
NASA Astrophysics Data System (ADS)
Fouvry, Jean-Baptiste; Chavanis, Pierre-Henri; Pichon, Christophe
2016-08-01
We present a brief derivation of the kinetic equation describing the secular evolution of point vortices in two-dimensional hydrodynamics, by relying on a functional integral formalism. We start from Liouville's equation which describes the exact dynamics of a two-dimensional system of point vortices. At the order 1 / N, the evolution of the system is characterised by the first two equations of the BBGKY hierarchy involving the system's 1-body distribution function and its 2-body correlation function. Thanks to the introduction of auxiliary fields, these two evolution constraints may be rewritten as a functional integral. When functionally integrated over the 2-body correlation function, this rewriting leads to a new constraint coupling the 1-body distribution function and the two auxiliary fields. Once inverted, this constraint provides, through a new route, the closed non-linear kinetic equation satisfied by the 1-body distribution function. Such a method sheds new lights on the origin of these kinetic equations complementing the traditional derivation methods.
The H-theorem for the physico-chemical kinetic equations with explicit time discretization
NASA Astrophysics Data System (ADS)
Adzhiev, S. Z.; Melikhov, I. V.; Vedenyapin, V. V.
2017-09-01
There is demonstrated in the present paper, that the H-theorem in the case of explicit time discretization of the physico-chemical kinetic equations, generally speaking, is not valid. We prove the H-theorem, when the system of the physico-chemical kinetic equations with explicit time discretization has the form of non-linear analogue of the Markov process with doubly stochastic matrix, and for more general cases. In these cases the proof is reduced to the proof of the H-theorem for Markov chains. The simplest discrete velocity models of the Boltzmann equation with explicit time discretization -the Carleman and Broadwell models are discussed and the H-theorem for them in the case of discrete time is proved.
Properties-preserving high order numerical methods for a kinetic eikonal equation
NASA Astrophysics Data System (ADS)
Luo, Songting; Payne, Nicholas
2017-02-01
For the BGK (Bhatnagar-Gross-Krook) equation in the large scale hyperbolic limit, the density of particles can be transformed as the Hopf-Cole transformation, where the phase function converges uniformly to the viscosity solution of an effective Hamilton-Jacobi equation, referred to as the kinetic eikonal equation. In this work, we present efficient high order finite difference methods for numerically solving the kinetic eikonal equation. The methods are based on monotone schemes such as the Godunov scheme. High order weighted essentially non-oscillatory techniques and Runge-Kutta procedures are used to obtain high order accuracy in both space and time. The effective Hamiltonian is determined implicitly by a nonlinear equation given as integrals with respect to the velocity variable. Newton's method is applied to solve the nonlinear equation, where integrals with respect to the velocity variable are evaluated either by a Gauss quadrature formula or as expansions with respect to moments of the Maxwellian. The methods are designed such that several key properties such as the positivity of the viscosity solution and the positivity of the effective Hamiltonian are preserved. Numerical experiments are presented to demonstrate the effectiveness of the methods.
A Gas-kinetic Scheme for the Two-Fluid MHD Equations with Resistivity
NASA Astrophysics Data System (ADS)
Anderson, Steven; Girimaji, Sharath; da Silva, Eduardo; Siebert, Diogo; Salazar, Juan
2016-11-01
The two-fluid MHD equations are a simplified model of plasma flow wherein a mixture of two species (electrons and ions) is considered. In this model, unlike single-fluid MHD, quasi-neutrality is not enforced, Ohm's Law is not used, and the fluids are not in thermal equilibrium - thus both fluids assume their own density, velocity, and temperature. Here we present a numerical scheme to solve the two-fluid MHD equations based on an extension of the gas-kinetic method. In contrast to previous implementations of the gas-kinetic scheme for MHD, the solution of the non-equilibrium distribution function for each gas at the cell interface is extended to include the effect of the electromagnetic forces as well as the inter-species collisions (resistivity). Closure of the fluid equations with the electromagnetic fields is obtained through Maxwell's equations, and physically correct divergences are enforced via correction potentials. Maxwell's equations are integrated via a simple Lax-Friedrichs type flux-splitting. To separate integration of the source and flux terms in the governing equations we use Strang splitting. Some numerical results are presented to demonstrate accuracy of the scheme and we discuss advantages and potential applications of the scheme. This research was supported by National Science Foundation Grant Number DGE-1252521 and the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) of Brazil.
Valencia, Pedro L; Astudillo-Castro, Carolina; Gajardo, Diego; Flores, Sebastián
2017-04-07
In 1974, Eisenthal and Cornish-Bowden published the direct linear plot method, which used the median to estimate the Vmax and Km from a set of initial rates as a function of substrate concentrations. The robustness of this non-parametric method was clearly demonstrated by comparing it with the least-squares method. The authors commented that the method cannot readily be generalized to equations of more than two parameters. Unfortunately, this comment has been misread by other authors. Comments such as "this method cannot be extended directly to equations with more than two parameters" were found in some publications. In addition, recently, the most drastic comment was published: "this method cannot be applied for the analysis of substrate inhibition." Given all of these presumptions, we have been motivated to publish a demonstration of the contrary: the median method can be applied to more than two-parameter equations, using as an example, the substrate uncompetitive inhibition equation. A computer algorithm was written to evaluate the effect of simulated experimental error of the initial rates on the estimation of Vmax, Km and KS. The error was assigned to different points of the experimental design. Four different KS/Km ratios were analyzed with the values 10, 100, 1000 and 10,000. The results indicated that the least-squares method was slightly better than the median method in terms of accuracy and variance. However, the presence of outliers affected the estimation of kinetic constants using the least-squares method more severely than the median method. The estimation of KS using the median method to estimate 1/KS was much better than the direct estimation of KS, causing a negative effect of non-linearity of KS in the kinetic equation. Considering that the median method is free from the assumptions of the least-squares method and the arbitrary assumptions implicit in the linearization methods to estimate the kinetic constants Vmax, Km and KS from the substrate
Cabrales, Luis E Bergues; Nava, Juan J Godina; Aguilera, Andrés Ramírez; Joa, Javier A González; Ciria, Héctor M Camué; González, Maraelys Morales; Salas, Miriam Fariñas; Jarque, Manuel Verdecia; González, Tamara Rubio; Mateus, Miguel A O'Farril; Brooks, Soraida C Acosta; Palencia, Fabiola Suárez; Zamora, Lisset Ortiz; Quevedo, María C Céspedes; Seringe, Sarah Edward; Cuitié, Vladimir Crombet; Cabrales, Idelisa Bergues; González, Gustavo Sierra
2010-10-28
Electrotherapy effectiveness at different doses has been demonstrated in preclinical and clinical studies; however, several aspects that occur in the tumor growth kinetics before and after treatment have not yet been revealed. Mathematical modeling is a useful instrument that can reveal some of these aspects. The aim of this paper is to describe the complete growth kinetics of unperturbed and perturbed tumors through use of the modified Gompertz equation in order to generate useful insight into the mechanisms that underpin this devastating disease. The complete tumor growth kinetics for control and treated groups are obtained by interpolation and extrapolation methods with different time steps, using experimental data of fibrosarcoma Sa-37. In the modified Gompertz equation, a delay time is introduced to describe the tumor's natural history before treatment. Different graphical strategies are used in order to reveal new information in the complete kinetics of this tumor type. The first stage of complete tumor growth kinetics is highly non linear. The model, at this stage, shows different aspects that agree with those reported theoretically and experimentally. Tumor reversibility and the proportionality between regions before and after electrotherapy are demonstrated. In tumors that reach partial remission, two antagonistic post-treatment processes are induced, whereas in complete remission, two unknown antitumor mechanisms are induced. The modified Gompertz equation is likely to lead to insights within cancer research. Such insights hold promise for increasing our understanding of tumors as self-organizing systems and, the possible existence of phase transitions in tumor growth kinetics, which, in turn, may have significant impacts both on cancer research and on clinical practice.
2010-01-01
Background Electrotherapy effectiveness at different doses has been demonstrated in preclinical and clinical studies; however, several aspects that occur in the tumor growth kinetics before and after treatment have not yet been revealed. Mathematical modeling is a useful instrument that can reveal some of these aspects. The aim of this paper is to describe the complete growth kinetics of unperturbed and perturbed tumors through use of the modified Gompertz equation in order to generate useful insight into the mechanisms that underpin this devastating disease. Methods The complete tumor growth kinetics for control and treated groups are obtained by interpolation and extrapolation methods with different time steps, using experimental data of fibrosarcoma Sa-37. In the modified Gompertz equation, a delay time is introduced to describe the tumor's natural history before treatment. Different graphical strategies are used in order to reveal new information in the complete kinetics of this tumor type. Results The first stage of complete tumor growth kinetics is highly non linear. The model, at this stage, shows different aspects that agree with those reported theoretically and experimentally. Tumor reversibility and the proportionality between regions before and after electrotherapy are demonstrated. In tumors that reach partial remission, two antagonistic post-treatment processes are induced, whereas in complete remission, two unknown antitumor mechanisms are induced. Conclusion The modified Gompertz equation is likely to lead to insights within cancer research. Such insights hold promise for increasing our understanding of tumors as self-organizing systems and, the possible existence of phase transitions in tumor growth kinetics, which, in turn, may have significant impacts both on cancer research and on clinical practice. PMID:21029411
NASA Astrophysics Data System (ADS)
Tessarotto, Massimo; Asci, Claudio
2017-05-01
In this paper the problem is posed of determining the physically-meaningful asymptotic orderings holding for the statistical description of a large N-body system of hard spheres, i.e., formed by N ≡1/ε ≫ 1 particles, which are allowed to undergo instantaneous and purely elastic unary, binary or multiple collisions. Starting point is the axiomatic treatment recently developed [Tessarotto et al., 2013-2016] and the related discovery of an exact kinetic equation realized by Master equation which advances in time the 1-body probability density function (PDF) for such a system. As shown in the paper the task involves introducing appropriate asymptotic orderings in terms of ε for all the physically-relevant parameters. The goal is that of identifying the relevant physically-meaningful asymptotic approximations applicable for the Master kinetic equation, together with their possible relationships with the Boltzmann and Enskog kinetic equations, and holding in appropriate asymptotic regimes. These correspond either to dilute or dense systems and are formed either by small-size or finite-size identical hard spheres, the distinction between the various cases depending on suitable asymptotic orderings in terms of ε.
Towards an ultra efficient kinetic scheme. Part I: Basics on the BGK equation
NASA Astrophysics Data System (ADS)
Dimarco, Giacomo; Loubere, Raphaël
2013-12-01
In this paper we present a new ultra efficient numerical method for solving kinetic equations. In this preliminary work, we present the scheme in the case of the BGK relaxation operator. The scheme, being based on a splitting technique between transport and collision, can be easily extended to other collisional operators as the Boltzmann collision integral or to other kinetic equations such as the Vlasov equation. The key idea, on which the method relies, is to solve the collision part on a grid and then to solve exactly the transport linear part by following the characteristics backward in time. The main difference between the method proposed and semi-Lagrangian methods is that here we do not need to reconstruct the distribution function at each time step. This allows to tremendously reduce the computational cost of the method and it permits for the first time, to the author's knowledge, to compute solutions of full six dimensional kinetic equations on a single processor laptop machine. Numerical examples, up to the full three dimensional case, are presented which validate the method and assess its efficiency in 1D, 2D and 3D.
Equations for the kinetic modeling of supersonically flowing electrically excited lasers
NASA Technical Reports Server (NTRS)
Lind, R. C.
1973-01-01
The equations for the kinetic modeling of a supersonically flowing electrically excited laser system are presented. The work focuses on the use of diatomic gases, in particular carbon monoxide mixtures. The equations presented include the vibrational rate equation which describes the vibrational population distribution, the electron, ion and electronic level rate equations, the gasdynamic equations for an ionized gas in the presence of an applied electric field, and the free electron Boltzmann equation including flow and gradient coupling terms. The model developed accounts for vibration-vibration collisions, vibration-translation collisions, electron-molecule inelastic excitation and superelastic de-excitation collisions, charge particle collisions, ionization and three body recombination collisions, elastic collisions, and radiative decay, all of which take place in such a system. A simplified form of the free electron Boltzmann equation is developed and discussed with emphasis placed on its coupling with the supersonic flow. A brief description of a possible solution procedure for the set of coupled equations is then discussed.
2009-01-01
Background Classical descriptions of enzyme kinetics ignore the physical nature of the intracellular environment. Main implicit assumptions behind such approaches are that reactions occur in compartment volumes which are large enough so that molecular discreteness can be ignored and that molecular transport occurs via diffusion. Though these conditions are frequently met in laboratory conditions, they are not characteristic of the intracellular environment, which is compartmentalized at the micron and submicron scales and in which active means of transport play a significant role. Results Starting from a master equation description of enzyme reaction kinetics and assuming metabolic steady-state conditions, we derive novel mesoscopic rate equations which take into account (i) the intrinsic molecular noise due to the low copy number of molecules in intracellular compartments (ii) the physical nature of the substrate transport process, i.e. diffusion or vesicle-mediated transport. These equations replace the conventional macroscopic and deterministic equations in the context of intracellular kinetics. The latter are recovered in the limit of infinite compartment volumes. We find that deviations from the predictions of classical kinetics are pronounced (hundreds of percent in the estimate for the reaction velocity) for enzyme reactions occurring in compartments which are smaller than approximately 200 nm, for the case of substrate transport to the compartment being mediated principally by vesicle or granule transport and in the presence of competitive enzyme inhibitors. Conclusion The derived mesoscopic rate equations describe subcellular enzyme reaction kinetics, taking into account, for the first time, the simultaneous influence of both intrinsic noise and the mode of transport. They clearly show the range of applicability of the conventional deterministic equation models, namely intracellular conditions compatible with diffusive transport and simple enzyme
An equation of state for purely kinetic k-essence inspired by cosmic topological defects
NASA Astrophysics Data System (ADS)
Cordero, Rubén; González, Eduardo L.; Queijeiro, Alfonso
2017-06-01
We investigate the physical properties of a purely kinetic k-essence model with an equation of state motivated in superconducting membranes. We compute the equation of state parameter w and discuss its physical evolution via a nonlinear equation of state. Using the adiabatic speed of sound and energy density, we restrict the range of parameters of the model in order to have an acceptable physical behavior. We study the evolution of the scale factor and address the question of the possible existence of finite-time future singularities. Furthermore, we analyze the evolution of the luminosity distance dL with redshift z by comparing (normalizing) it with the Λ CDM model. Since the equation of state parameter is z-dependent the evolution of the luminosity distance is also analyzed using the Alcock-Paczyński test.
A gas-kinetic BGK scheme for the compressible Navier-Stokes equations
NASA Technical Reports Server (NTRS)
Xu, Kun
2000-01-01
This paper presents an improved gas-kinetic scheme based on the Bhatnagar-Gross-Krook (BGK) model for the compressible Navier-Stokes equations. The current method extends the previous gas-kinetic Navier-Stokes solver developed by Xu and Prendergast by implementing a general nonequilibrium state to represent the gas distribution function at the beginning of each time step. As a result, the requirement in the previous scheme, such as the particle collision time being less than the time step for the validity of the BGK Navier-Stokes solution, is removed. Therefore, the applicable regime of the current method is much enlarged and the Navier-Stokes solution can be obtained accurately regardless of the ratio between the collision time and the time step. The gas-kinetic Navier-Stokes solver developed by Chou and Baganoff is the limiting case of the current method, and it is valid only under such a limiting condition. Also, in this paper, the appropriate implementation of boundary condition for the kinetic scheme, different kinetic limiting cases, and the Prandtl number fix are presented. The connection among artificial dissipative central schemes, Godunov-type schemes, and the gas-kinetic BGK method is discussed. Many numerical tests are included to validate the current method.
New forms of two-particle and one-particle kinetic equations
NASA Astrophysics Data System (ADS)
Saveliev, V. L.; Yonemura, S.
2012-11-01
Pair collisions are the main interaction process in the Boltzmann gas dynamics. By making use of exactly the same physical assumptions as was done by Ludwig Boltzmann we wrote the kinetic equation for two-particle distribution function of molecules in gas mixtures. Instead of the collision integral, there are the linear scattering operator and the chaos projector in the right part of this equation. We developed a new technique for factorization of the scattering operator on the bases of right inverses to the Casimir operator of the group of rotations. We exactly transformed the Boltzmann collision integral to the Landau-Fokker-Planck like form.
Gelß, Patrick Matera, Sebastian Schütte, Christof
2016-06-01
In multiscale modeling of heterogeneous catalytic processes, one crucial point is the solution of a Markovian master equation describing the stochastic reaction kinetics. Usually, this is too high-dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic Monte Carlo method. In this study we break the curse of dimensionality for the direct solution of the Markovian master equation by exploiting the Tensor Train Format for this purpose. The performance of the approach is demonstrated on a first principles based, reduced model for the CO oxidation on the RuO{sub 2}(110) surface. We investigate the complexity for increasing system size and for various reaction conditions. The advantage over the stochastic simulation approach is illustrated by a problem with increased stiffness.
Klinman, Judith P
2014-01-01
The final arbiter of enzyme mechanism is the ability to establish and test a kinetic mechanism. Isotope effects play a major role in expanding the scope and insight derived from the Michaelis-Menten equation. The integration of isotope effects into the formalism of the Michaelis-Menten equation began in the 1970s and has continued until the present. This review discusses a family of eukaryotic copper proteins, including dopamine β-monooxygenase, tyramine β-monooxygenase and peptidylglycine α-amidating enzyme, which are responsible for the synthesis of neuroactive compounds, norepinephrine, octopamine and C-terminally carboxamidated peptides, respectively. The review highlights the results of studies showing how combining kinetic isotope effects with initial rate parameters permits the evaluation of: (a) the order of substrate binding to multisubstrate enzymes; (b) the magnitude of individual rate constants in complex, multistep reactions; (c) the identification of chemical intermediates; and (d) the role of nonclassical (tunnelling) behaviour in C-H activation.
NASA Astrophysics Data System (ADS)
Gelß, Patrick; Matera, Sebastian; Schütte, Christof
2016-06-01
In multiscale modeling of heterogeneous catalytic processes, one crucial point is the solution of a Markovian master equation describing the stochastic reaction kinetics. Usually, this is too high-dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic Monte Carlo method. In this study we break the curse of dimensionality for the direct solution of the Markovian master equation by exploiting the Tensor Train Format for this purpose. The performance of the approach is demonstrated on a first principles based, reduced model for the CO oxidation on the RuO2(110) surface. We investigate the complexity for increasing system size and for various reaction conditions. The advantage over the stochastic simulation approach is illustrated by a problem with increased stiffness.
A kinetic equation for linear stable fractional motion with applications to space plasma physics
Watkins, Nicholas W; Credgington, Daniel; Sanchez, Raul; Rosenberg, SJ; Chapman, Sandra C
2009-01-01
Levy flights and fractional Brownian motion have become exemplars of the heavy-tailed jumps and long-ranged memory widely seen in physics. Natural time series frequently combine both effects, and linear fractional stable motion (lfsm) is a model process of this type, combining {alpha}-stable jumps with a memory kernel. In contrast complex physical spatiotemporal diffusion processes where both the above effects compete have for many years been modeled using the fully fractional kinetic equation for the continuous-time random walk (CTRW), with power laws in the probability density functions of both jump size and waiting time. We derive the analogous kinetic equation for lfsm and show that it has a diffusion coefficient with a power law in time rather than having a fractional time derivative like the CTRW. We discuss some preliminary results on the scaling of burst 'sizes' and 'durations' in lfsm time series, with applications to modeling existing observations in space physics and elsewhere.
An asymptotic-preserving scheme for linear kinetic equation with fractional diffusion limit
NASA Astrophysics Data System (ADS)
Wang, Li; Yan, Bokai
2016-05-01
We present a new asymptotic-preserving scheme for the linear Boltzmann equation which, under appropriate scaling, leads to a fractional diffusion limit. Our scheme rests on novel micro-macro decomposition to the distribution function, which splits the original kinetic equation following a reshuffled Hilbert expansion. As opposed to classical diffusion limit, a major difficulty comes from the fat tail in the equilibrium which makes the truncation in velocity space depending on the small parameter. Our idea is, while solving the macro-micro part in a truncated velocity domain (truncation only depends on numerical accuracy), to incorporate an integrated tail over the velocity space that is beyond the truncation, and its major component can be precomputed once with any accuracy. Such an addition is essential to drive the solution to the correct asymptotic limit. Numerical experiments validate its efficiency in both kinetic and fractional diffusive regimes.
Nguyen, Thanh Lam; Stanton, John F
2015-07-16
In the field of chemical kinetics, the solution of a two-dimensional master equation that depends explicitly on both total internal energy (E) and total angular momentum (J) is a challenging problem. In this work, a weak-E/fixed-J collisional model (i.e., weak-collisional internal energy relaxation/free-collisional angular momentum relaxation) is used along with the steady-state approach to solve the resulting (simplified) two-dimensional (E,J)-grained master equation. The corresponding solutions give thermal rate constants and product branching ratios as functions of both temperature and pressure. We also have developed a program that can be used to predict and analyze experimental chemical kinetics results. This expedient technique, when combined with highly accurate potential energy surfaces, is cable of providing results that may be meaningfully compared to experiments. The reaction of singlet oxygen with methane proceeding through vibrationally excited methanol is used as an illustrative example.
NASA Astrophysics Data System (ADS)
Maulidah, Rifa'atul; Purqon, Acep
2016-08-01
Mendong (Fimbristylis globulosa) has a potentially industrial application. We investigate a predictive model for heat and mass transfer in drying kinetics during drying a Mendong. We experimentally dry the Mendong by using a microwave oven. In this study, we analyze three mathematical equations and feed forward neural network (FNN) with back propagation to describe the drying behavior of Mendong. Our results show that the experimental data and the artificial neural network model has a good agreement and better than a mathematical equation approach. The best FNN for the prediction is 3-20-1-1 structure with Levenberg- Marquardt training function. This drying kinetics modeling is potentially applied to determine the optimal parameters during mendong drying and to estimate and control of drying process.
NASA Astrophysics Data System (ADS)
Tassi, E.
2015-11-01
We present an infinite family of Hamiltonian electromagnetic fluid models for plasmas, derived from drift-kinetic equations. An infinite hierarchy of fluid equations is obtained from a Hamiltonian drift-kinetic system by taking moments of a generalized distribution function and using Hermite polynomials as weight functions of the velocity coordinate along the magnetic guide field. Each fluid model is then obtained by truncating the hierarchy to a finite number N + 1 of equations by means of a closure relation. We show that, for any positive N, a linear closure relation between the moment of order N + 1 and the moment of order N guarantees that the resulting fluid model possesses a Hamiltonian structure, thus respecting the Hamiltonian character of the parent drift-kinetic model. An orthogonal transformation is identified which maps the fluid moments to a new set of dynamical variables in terms of which the Poisson brackets of the fluid models become a direct sum and which unveils remarkable dynamical properties of the models in the two-dimensional (2D) limit. Indeed, when imposing translational symmetry with respect to the direction of the magnetic guide field, all models belonging to the infinite family can be reformulated as systems of advection equations for Lagrangian invariants transported by incompressible generalized velocities. These are reminiscent of the advection properties of the parent drift-kinetic model in the 2D limit and are related to the Casimirs of the Poisson brackets of the fluid models. The Hamiltonian structure of the generic fluid model belonging to the infinite family is illustrated treating a specific example of a fluid model retaining five moments in the electron dynamics and two in the ion dynamics. We also clarify the connection existing between the fluid models of this infinite family and some fluid models already present in the literature.
A Kinetic Equation for the Distribution of Interaction Clusters in Rarefied Gases
NASA Astrophysics Data System (ADS)
Patterson, Robert I. A.; Simonella, Sergio; Wagner, Wolfgang
2017-08-01
We consider a stochastic particle model governed by an arbitrary binary interaction kernel. A kinetic equation for the distribution of interaction clusters is established. Under some additional assumptions a recursive representation of the solution is found. For particular choices of the interaction kernel (including the Boltzmann case) several explicit formulas are obtained. These formulas are confirmed by numerical experiments. The experiments are also used to illustrate various conjectures and open problems.
A Kinetic Equation for the Distribution of Interaction Clusters in Rarefied Gases
NASA Astrophysics Data System (ADS)
Patterson, Robert I. A.; Simonella, Sergio; Wagner, Wolfgang
2017-10-01
We consider a stochastic particle model governed by an arbitrary binary interaction kernel. A kinetic equation for the distribution of interaction clusters is established. Under some additional assumptions a recursive representation of the solution is found. For particular choices of the interaction kernel (including the Boltzmann case) several explicit formulas are obtained. These formulas are confirmed by numerical experiments. The experiments are also used to illustrate various conjectures and open problems.
NASA Technical Reports Server (NTRS)
Radhadrishnan, Krishnan
1993-01-01
A detailed analysis of the accuracy of several techniques recently developed for integrating stiff ordinary differential equations is presented. The techniques include two general-purpose codes EPISODE and LSODE developed for an arbitrary system of ordinary differential equations, and three specialized codes CHEMEQ, CREK1D, and GCKP4 developed specifically to solve chemical kinetic rate equations. The accuracy study is made by application of these codes to two practical combustion kinetics problems. Both problems describe adiabatic, homogeneous, gas-phase chemical reactions at constant pressure, and include all three combustion regimes: induction, heat release, and equilibration. To illustrate the error variation in the different combustion regimes the species are divided into three types (reactants, intermediates, and products), and error versus time plots are presented for each species type and the temperature. These plots show that CHEMEQ is the most accurate code during induction and early heat release. During late heat release and equilibration, however, the other codes are more accurate. A single global quantity, a mean integrated root-mean-square error, that measures the average error incurred in solving the complete problem is used to compare the accuracy of the codes. Among the codes examined, LSODE is the most accurate for solving chemical kinetics problems. It is also the most efficient code, in the sense that it requires the least computational work to attain a specified accuracy level. An important finding is that use of the algebraic enthalpy conservation equation to compute the temperature can be more accurate and efficient than integrating the temperature differential equation.
Implementation and verification of Chapman-Enskog-like drift kinetic equations in NIMROD
NASA Astrophysics Data System (ADS)
Held, Eric; Jepson, Joseph; Ji, Jeong-Young; Nimrod Team
2016-10-01
Rigorous closure of the extended magnetohydrodynamic equations used in plasma fluid codes incorporates important effects for tokamak plasmas such as perturbed bootstrap current physics and generalized viscosity at low collisionality. In this work we discuss continuum numerical solutions of the Chapman-Enskog-like electron and ion drift kinetic equations which have been implemented recently in the NIMROD code. Among other things, these solutions supply the CGL electron stress closure for Ohms Law and CGL ion stress closure for the plasma flow evolution equation. Such closures are paramount to understanding the macroscopic stability properties of high-performance tokamak plasmas. Verification of the orthogonal nature of the Maxwellian and non-Maxwellian parts of the distribution function inherent in the adopted Chapman-Enskog-like approach is provided along with simulation results of neoclassical transport and the Spitzer thermalization and conduction problems.
A coarse-grained kinetic equation for neutral particles in turbulent fusion plasmas
Mekkaoui, A.; Marandet, Y.; Genesio, P.; Rosato, J.; Stamm, R.; Capes, H.; Koubiti, M.; Godbert-Mouret, L.; Catoire, F.
2012-06-15
A coarse-grained kinetic equation for neutral particles (atoms, molecules) in magnetized fusion plasmas, valid on time scales large compared to the turbulence correlation time, is presented. This equation includes the effects of plasma density fluctuations, described by gamma statistics, on the transport of neutral particles. These effects have so far been neglected in plasma edge modeling, in spite of the fact that the amplitude of fluctuations can be of order unity. Density fluctuations are shown to have a marked effect on the screening of neutrals and on the spatial localization of the ionization source, in particular at high density. The coarse-grained equations obtained in this work are readily implemented in edge code suites currently used for fusion plasma analysis and future divertor design (ITER, DEMO).
Proton-pumping mechanism of cytochrome c oxidase: a kinetic master-equation approach.
Kim, Young C; Hummer, Gerhard
2012-04-01
Cytochrome c oxidase is an efficient energy transducer that reduces oxygen to water and converts the released chemical energy into an electrochemical membrane potential. As a true proton pump, cytochrome c oxidase translocates protons across the membrane against this potential. Based on a wealth of experiments and calculations, an increasingly detailed picture of the reaction intermediates in the redox cycle has emerged. However, the fundamental mechanism of proton pumping coupled to redox chemistry remains largely unresolved. Here we examine and extend a kinetic master-equation approach to gain insight into redox-coupled proton pumping in cytochrome c oxidase. Basic principles of the cytochrome c oxidase proton pump emerge from an analysis of the simplest kinetic models that retain essential elements of the experimentally determined structure, energetics, and kinetics, and that satisfy fundamental physical principles. The master-equation models allow us to address the question of how pumping can be achieved in a system in which all reaction steps are reversible. Whereas proton pumping does not require the direct modulation of microscopic reaction barriers, such kinetic gating greatly increases the pumping efficiency. Further efficiency gains can be achieved by partially decoupling the proton uptake pathway from the active-site region. Such a mechanism is consistent with the proposed Glu valve, in which the side chain of a key glutamic acid shuttles between the D channel and the active-site region. We also show that the models predict only small proton leaks even in the absence of turnover. The design principles identified here for cytochrome c oxidase provide a blueprint for novel biology-inspired fuel cells, and the master-equation formulation should prove useful also for other molecular machines. .
Proton-pumping mechanism of cytochrome c oxidase: A kinetic master-equation approach
Kim, Young C.; Hummer, Gerhard
2011-01-01
Cytochrome c oxidase (CcO) is an efficient energy transducer that reduces oxygen to water and converts the released chemical energy into an electrochemical membrane potential. As a true proton pump, CcO translocates protons across the membrane against this potential. Based on a wealth of experiments and calculations, an increasingly detailed picture of the reaction intermediates in the redox cycle has emerged. However, the fundamental mechanism of proton pumping coupled to redox chemistry remains largely unresolved. Here we examine and extend a kinetic master-equation approach to gain insight into redox-coupled proton pumping in CcO. Basic principles of the CcO proton pump emerge from an analysis of the simplest kinetic models that retain essential elements of the experimentally determined structure, energetics, and kinetics, and that satisfy fundamental physical principles. The master-equation models allow us to address the question of how pumping can be achieved in a system in which all reaction steps are reversible. Whereas proton pumping does not require the direct modulation of microscopic reaction barriers, such kinetic gating greatly increases the pumping efficiency. Further efficiency gains can be achieved by partially decoupling the proton uptake pathway from the ative-site region. Such a mechanism is consistent with the proposed Glu valve, in which the side chain of a key glutamic acid shuttles between the D channel and the active-site region. We also show that the models predict only small proton leaks even in the absence of turnover. The design principles identified here for CcO provide a blueprint for novel biology-inspired fuel cells, and the master-equation formulation should prove useful also for other molecular machines. PMID:21946020
NASA Astrophysics Data System (ADS)
Enukashvily, Isaac M.
1980-11-01
An extension of Bleck' method and of the method of moments is developed for the numerical integration of the kinetic equation of coalescence and breakup of cloud droplets. The number density function nk(x,t) in each separate cloud droplet packet between droplet mass grid points (xk,xk+1) is represented by an expansion in orthogonal polynomials with a given weighting function wk(x,k). The expansion coefficients describe the deviations of nk(x,t) from wk(x,k). In this way droplet number concentrations, liquid water contents and other moments in each droplet packet are conserved, and the problem of solving the kinetic equation is replaced by one of solving a set of coupled differential equations for the moments of the number density function nk(x,t). Equations for these moments in each droplet packet are derived. The method is tested against existing solutions of the coalescence equation. Numerical results are obtained when Bleck's uniform distribution hypothesis for nk(x,t) and Golovin's asymptotic solution of the coalescence equation is chosen for the, weighting function wk(x, k). A comparison between numerical results computed by Bleck's method and by the method of this study is made. It is shown that for the correct computation of the coalescence and breakup interactions between cloud droplet packets it is very important that the, approximation of the nk(x,t) between grid points (xk,xk+1) satisfies the conservation conditions for the number concentration, liquid water content and other moments of the cloud droplet packets. If these conservation conditions are provided, even the quasi-linear approximation of the nk(x,t) in comparison with Berry's six-point interpolation will give reasonable results which are very close to the existing analytic solutions.
Kolobov, Vladimir; Arslanbekov, Robert; Frolova, Anna
2014-12-09
The paper describes an Adaptive Mesh in Phase Space (AMPS) technique for solving kinetic equations with deterministic mesh-based methods. The AMPS technique allows automatic generation of adaptive Cartesian mesh in both physical and velocity spaces using a Tree-of-Trees data structure. We illustrate advantages of AMPS for simulations of rarefied gas dynamics and electron kinetics on low temperature plasmas. In particular, we consider formation of the velocity distribution functions in hypersonic flows, particle kinetics near oscillating boundaries, and electron kinetics in a radio-frequency sheath. AMPS provide substantial savings in computational cost and increased efficiency of the mesh-based kinetic solvers.
Kinetic Thomas-Fermi solutions of the Gross-Pitaevskii equation
NASA Astrophysics Data System (ADS)
Ölschläger, M.; Wirth, G.; Smith, C. Morais; Hemmerich, A.
2009-04-01
Approximate solutions of the Gross-Pitaevskii (GP) equation, obtained upon neglection of the kinetic energy, are well known as Thomas-Fermi solutions. They are characterized by the compensation of the local potential by the collisional energy. In this article we consider exact solutions of the GP-equation with this property and definite values of the kinetic energy, which suggests the term "kinetic Thomas-Fermi" (KTF) solutions. Despite their formal simplicity, KTF-solutions can possess complex current density fields with unconventional topology. We point out that a large class of light-shift potentials gives rise to KTF-solutions. As elementary examples, we consider one-dimensional and two-dimensional optical lattice scenarios, obtained by means of the superposition of two, three and four laser beams, and discuss the stability properties of the corresponding KTF-solutions. A general method is proposed to excite two-dimensional KTF-solutions in experiments by means of time-modulated light-shift potentials.
Utilization of integrated Michaelis-Menten equation to determine kinetic constants.
Bezerra, Rui M F; Dias, Albino A
2007-03-01
Students of biochemistry and related biosciences are urged to solve problems where kinetic parameters are calculated from initial rates obtained at different substrate concentrations. Troubles begin when they go to the laboratory to perform kinetic experiments and realize that usual laboratory instruments do not measure initial rates but only substrate or product concentrations as a function of reaction time. To overcome this problem we present a methodology which uses the integrated form of Michaelis-Menten equation. The method presented has a theoretical and pedagogic basis which is not as arbitrary as other approaches. Here we present and describe the methodology for analyzing time course data together with some examples of the essential computer procedures to implement these analyses. To simplify the understanding of this methodology the experimental examples are confined to linear inhibitions and experimental points utilized are the same from which the initial rates are determined.
Garnier, Alain; Gaillet, Bruno
2015-12-01
Not so many fermentation mathematical models allow analytical solutions of batch process dynamics. The most widely used is the combination of the logistic microbial growth kinetics with Luedeking-Piret bioproduct synthesis relation. However, the logistic equation is principally based on formalistic similarities and only fits a limited range of fermentation types. In this article, we have developed an analytical solution for the combination of Monod growth kinetics with Luedeking-Piret relation, which can be identified by linear regression and used to simulate batch fermentation evolution. Two classical examples are used to show the quality of fit and the simplicity of the method proposed. A solution for the combination of Haldane substrate-limited growth model combined with Luedeking-Piret relation is also provided. These models could prove useful for the analysis of fermentation data in industry as well as academia.
An asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations
Sun, Wenjun; Jiang, Song; Xu, Kun
2015-03-15
The solutions of radiative transport equations can cover both optical thin and optical thick regimes due to the large variation of photon's mean-free path and its interaction with the material. In the small mean free path limit, the nonlinear time-dependent radiative transfer equations can converge to an equilibrium diffusion equation due to the intensive interaction between radiation and material. In the optical thin limit, the photon free transport mechanism will emerge. In this paper, we are going to develop an accurate and robust asymptotic preserving unified gas kinetic scheme (AP-UGKS) for the gray radiative transfer equations, where the radiation transport equation is coupled with the material thermal energy equation. The current work is based on the UGKS framework for the rarefied gas dynamics [14], and is an extension of a recent work [12] from a one-dimensional linear radiation transport equation to a nonlinear two-dimensional gray radiative system. The newly developed scheme has the asymptotic preserving (AP) property in the optically thick regime in the capturing of diffusive solution without using a cell size being smaller than the photon's mean free path and time step being less than the photon collision time. Besides the diffusion limit, the scheme can capture the exact solution in the optical thin regime as well. The current scheme is a finite volume method. Due to the direct modeling for the time evolution solution of the interface radiative intensity, a smooth transition of the transport physics from optical thin to optical thick can be accurately recovered. Many numerical examples are included to validate the current approach.
Convective kinetic energy equation under the mass-flux subgrid-scale parameterization
NASA Astrophysics Data System (ADS)
Yano, Jun-Ichi
2015-03-01
The present paper originally derives the convective kinetic energy equation under mass-flux subgrid-scale parameterization in a formal manner based on the segmentally-constant approximation (SCA). Though this equation is long since presented by Arakawa and Schubert (1974), a formal derivation is not known in the literature. The derivation of this formulation is of increasing interests in recent years due to the fact that it can explain basic aspects of the convective dynamics such as discharge-recharge and transition from shallow to deep convection. The derivation is presented in two manners: (i) for the case that only the vertical component of the velocity is considered and (ii) the case that both the horizontal and vertical components are considered. The equation reduces to the same form as originally presented by Arakwa and Schubert in both cases, but with the energy dissipation term defined differently. In both cases, nevertheless, the energy "dissipation" (loss) term consists of the three principal contributions: (i) entrainment-detrainment, (ii) outflow from top of convection, and (iii) pressure effects. Additionally, inflow from the bottom of convection contributing to a growth of convection is also formally counted as a part of the dissipation term. The eddy dissipation is also included for a completeness. The order-of-magnitude analysis shows that the convective kinetic energy "dissipation" is dominated by the pressure effects, and it may be approximately described by Rayleigh damping with a constant time scale of the order of 102-103 s. The conclusion is also supported by a supplementary analysis of a cloud-resolving model (CRM) simulation. The Appendix discusses how the loss term ("dissipation") of the convective kinetic energy is qualitatively different from the conventional eddy-dissipation process found in turbulent flows.
Plasma drift-kinetic equation calculations in three-dimensional magnetic geometries
Reynolds, J. M.; Lopez-Bruna, D.
2010-07-15
A new code to simulate three-dimensional plasmas in complex toroidal geometries is presented. It solves drift-kinetic equations for the one-particle distribution function f based on their projection onto a functional basis consisting of an arbitrary number of Legendre-Laguerre polynomials. In this paper, these theoretical aspects of the code are exposed together with their relation with the standard formalism. Comparisons with neoclassical theory for the large aspect ratio case and first calculations in the geometry of the TJ-II Heliac are also presented.
Theory of warm ionized gases: equation of state and kinetic Schottky anomaly.
Capolupo, A; Giampaolo, S M; Illuminati, F
2013-10-01
Based on accurate Lennard-Jones-type interaction potentials, we derive a closed set of state equations for the description of warm atomic gases in the presence of ionization processes. The specific heat is predicted to exhibit peaks in correspondence to single and multiple ionizations. Such kinetic analog in atomic gases of the Schottky anomaly in solids is enhanced at intermediate and low atomic densities. The case of adiabatic compression of noble gases is analyzed in detail and the implications on sonoluminescence are discussed. In particular, the predicted plasma electron density in a sonoluminescent bubble turns out to be in good agreement with the value measured in recent experiments.
NASA Technical Reports Server (NTRS)
Kaiser, T. B.; Jones, F. C.; Birmingham, T. J.
1972-01-01
The problem of deriving a kinetic equation for the cosmic ray distribution function in a random magnetic field is considered. A model is adopted which is mathematically simple but which contains the essential physics. The perturbation expansion upon which the quasi-linear treatment is based is investigated. The existence of resonant particles causes the breakdown of the adiabatic approximation frequently used in this theory. Resonant particles cause a general secular growth of higher order terms in the expansion which invalidates the entire perturbative approach.
NASA Astrophysics Data System (ADS)
Kierkels, A. H. M.; Velázquez, J. J. L.
2016-06-01
We construct a family of self-similar solutions with fat tails to a quadratic kinetic equation. This equation describes the long time behaviour of weak solutions with finite mass to the weak turbulence equation associated to the nonlinear Schrödinger equation. The solutions that we construct have finite mass, but infinite energy. In Kierkels and Velázquez (J Stat Phys 159:668-712, 2015) self-similar solutions with finite mass and energy were constructed. Here we prove upper and lower exponential bounds on the tails of these solutions.
Kinetically reduced local Navier-Stokes equations for simulation of incompressible viscous flows.
Borok, S; Ansumali, S; Karlin, I V
2007-12-01
Recently, another approach to study incompressible fluid flow was suggested [S. Ansumali, I. Karlin, and H. Ottinger, Phys. Rev. Lett. 94, 080602 (2005)]-the kinetically reduced local Navier-Stokes (KRLNS) equations. We consider a simplified two-dimensional KRLNS system and compare it with Chorin's artificial compressibility method. A comparison of the two methods for steady state computation of the flow in a lid-driven cavity at various Reynolds numbers shows that the results from both methods are in good agreement with each other. However, in the transient flow, it is demonstrated that the KRLNS equations correctly describe the time evolution of the velocity and of the pressure, unlike the artificial compressibility method.
NASA Astrophysics Data System (ADS)
Dubrovskii, V. G.; Hervieu, Yu. Yu.
2014-09-01
In this work, we present a theoretical analysis of the diffusion-induced growth of "vapor-liquid-solid" nanowires, based on the stationary equations with generalized boundary conditions. We discuss why and how the earlier results are modified when the adatom chemical potential is discontinuous at the nanowire base. Several simplified models for the adatom diffusion flux are discussed, yielding the 1 /Rp radius dependence of the length, with p ranging from 0.5 to 2. The self-consistent approach is used to couple the diffusion transport with the kinetics of 2D nucleation under the droplet. This leads to a new growth equation that contains only two dimensional parameters and the power exponents p and q, where q=1 or 2 depends on the nucleus position. We show that this equation describes the size-dependent depression of the growth rate of narrow nanowires much better than the Gibbs-Thomson correction in several important cases. Overall, our equation fits very well the experimental data on the length-radius correlations of III-V and group IV nanowires obtained by different epitaxy techniques.
Doktorov, A B
2014-09-14
In the framework of unified many-particle approach the familiar problem of fluorescence concentration quenching in the presence of pumping (light pulse) of arbitrary intensity is considered. This process is a vivid and the simplest example of multistage bulk reaction including bimolecular irreversible quenching reaction and reversible monomolecular transformation as elementary stages. General relation between the kinetics of multistage bulk reaction and that of the elementary stage of quenching has been established. This allows one to derive general kinetic equations (of two types) for the multistage reaction in question on the basis of general kinetic equations (differential and integro-differential) of elementary stage of quenching. Relying on the same unified many-particle approach we have developed binary approximations with the use of two (frequently employed in the literature) many-particle methods (such as simple superposition approximation and the method of extracting pair channels in three-particle correlation evolution) to the derivation of non-Markovian binary kinetic equations. The possibility of reducing the obtained binary equations to the Markovian equations of formal chemical kinetics has been considered. As an example the exact solution of the problem (for the specific case) is examined, and the applicability of two many particle methods of derivation of binary equations is analyzed.
Doktorov, A. B.
2014-09-14
In the framework of unified many-particle approach the familiar problem of fluorescence concentration quenching in the presence of pumping (light pulse) of arbitrary intensity is considered. This process is a vivid and the simplest example of multistage bulk reaction including bimolecular irreversible quenching reaction and reversible monomolecular transformation as elementary stages. General relation between the kinetics of multistage bulk reaction and that of the elementary stage of quenching has been established. This allows one to derive general kinetic equations (of two types) for the multistage reaction in question on the basis of general kinetic equations (differential and integro-differential) of elementary stage of quenching. Relying on the same unified many-particle approach we have developed binary approximations with the use of two (frequently employed in the literature) many-particle methods (such as simple superposition approximation and the method of extracting pair channels in three-particle correlation evolution) to the derivation of non-Markovian binary kinetic equations. The possibility of reducing the obtained binary equations to the Markovian equations of formal chemical kinetics has been considered. As an example the exact solution of the problem (for the specific case) is examined, and the applicability of two many particle methods of derivation of binary equations is analyzed.
Bayesian inference for kinetic models of biotransformation using a generalized rate equation.
Ying, Shanshan; Zhang, Jiangjiang; Zeng, Lingzao; Shi, Jiachun; Wu, Laosheng
2017-03-06
Selecting proper rate equations for the kinetic models is essential to quantify biotransformation processes in the environment. Bayesian model selection method can be used to evaluate the candidate models. However, comparisons of all plausible models can result in high computational cost, while limiting the number of candidate models may lead to biased results. In this work, we developed an integrated Bayesian method to simultaneously perform model selection and parameter estimation by using a generalized rate equation. In the approach, the model hypotheses were represented by discrete parameters and the rate constants were represented by continuous parameters. Then Bayesian inference of the kinetic models was solved by implementing Markov Chain Monte Carlo simulation for parameter estimation with the mixed (i.e., discrete and continuous) priors. The validity of this approach was illustrated through a synthetic case and a nitrogen transformation experimental study. It showed that our method can successfully identify the plausible models and parameters, as well as uncertainties therein. Thus this method can provide a powerful tool to reveal more insightful information for the complex biotransformation processes.
The solution of the point kinetics equations via converged accelerated Taylor series (CATS)
Ganapol, B.; Picca, P.; Previti, A.; Mostacci, D.
2012-07-01
This paper deals with finding accurate solutions of the point kinetics equations including non-linear feedback, in a fast, efficient and straightforward way. A truncated Taylor series is coupled to continuous analytical continuation to provide the recurrence relations to solve the ordinary differential equations of point kinetics. Non-linear (Wynn-epsilon) and linear (Romberg) convergence accelerations are employed to provide highly accurate results for the evaluation of Taylor series expansions and extrapolated values of neutron and precursor densities at desired edits. The proposed Converged Accelerated Taylor Series, or CATS, algorithm automatically performs successive mesh refinements until the desired accuracy is obtained, making use of the intermediate results for converged initial values at each interval. Numerical performance is evaluated using case studies available from the literature. Nearly perfect agreement is found with the literature results generally considered most accurate. Benchmark quality results are reported for several cases of interest including step, ramp, zigzag and sinusoidal prescribed insertions and insertions with adiabatic Doppler feedback. A larger than usual (9) number of digits is included to encourage honest benchmarking. The benchmark is then applied to the enhanced piecewise constant algorithm (EPCA) currently being developed by the second author. (authors)
Widder, M.E.; Titulaer, U.M. )
1993-03-01
The authors consider a mixture of heavy vapor molecules and a light carrier gas surrounding a liquid droplet. The vapor is described by a variant of the Klein-Kramers equation; the gas is described by the Navier-Stokes equations; the droplet acts as a heat source due to the released heat of condensation. The exchange of momentum and energy between the constituents of the mixture is taken into account by force terms in the kinetic equation and source terms in the Navier-Stokes equations. These are chosen to obtain maximal agreement with the irreversible thermodynamics of a gas mixture. The structure of the kinetic boundary layer around the sphere is determined from the self-consistent solution of this set of coupled equations with appropriate boundary conditions at the surface of the sphere. The kinetic equation is rewritten as a set of coupled moment equations. A complete set of solutions of these moment equations is constructed by numerical integration inward from the region far away from the droplet, where the background inhomogeneities are small. A technique developed earlier is used to deal with the numerical instability of the moment equations. The solutions obtained for given temperature and pressure profiles in the gas are then combined linearly such that they obey the boundary conditions at the droplet surface; from this solution source terms for the Navier-Stokes equation of the gas are constructed and used to determine improved temperature and pressure profiles for the background gas. For not too large temperature differneces between the droplet and the gas at infinity, self-consistency is reached after a few iterations. The method is applied to the condensation of droplets from a supersaturated vapor as well as to strong evaporation of droplets under the influence of an external heat source, where corrections of up to 40% are obtained.
R. Skartlien; E. Sollum; A. Akselsen; P. Meakin; B. Grimes; J. Sjoblom
2012-12-01
Lattice Boltzmann simulations were used to study the coalescence kinetics in emulsions with amphiphilic surfactant, under neutrally buoyant conditions, and with a significant kinematic viscosity contrast between the phases (emulating water in oil emulsions). The 3D simulation domain was large enough (256 3rd power -- 10 7th power grid points) to obtain good statistics with droplet numbers ranging from a few thousand at early times to a few hundred near equilibrium. Increased surfactant contents slowed down the coalescence rate between droplets due to the Gibbs-Marangoni effect, and the coalescence was driven by a quasi-turbulent velocity field. The kinetic energy decayed at a relatively slow rate at early times, due to conversion of interfacial energy to kinetic energy in the flow during coalescence. Phenomenological, coupled differential equations for the mean droplet diameter D(t) and the number density nd(t) were obtained from the simulation data and from film draining theories. Local (in time) power law exponents for the growth of the mean diameter (and for the concomitant decrease of nd) were established in terms of the instantaneous values of the kinetic energy, coalescence probability, Gibbs elasticity, and interfacial area. The model studies indicated that true power laws for the growth of the droplet size and decrease of the number of droplets with time may not be justified, since the exponents derived using the phenomenological model were time dependent. In contrast to earlier simulation results for symmetric blends with surfactant, we found no evidence for stretched logarithmic scaling of the formD -- [ln (ct)]a for the morphology length, or exponential scalings associated with arrested growth, on the basis of the phenomenological model.
On the Rate of Relaxation for the Landau Kinetic Equation and Related Models
NASA Astrophysics Data System (ADS)
Bobylev, Alexander; Gamba, Irene M.; Zhang, Chenglong
2017-08-01
We study the rate of relaxation to equilibrium for Landau kinetic equation and some related models by considering the relatively simple case of radial solutions of the linear Landau-type equations. The well-known difficulty is that the evolution operator has no spectral gap, i.e. its spectrum is not separated from zero. Hence we do not expect purely exponential relaxation for large values of time t>0. One of the main goals of our work is to numerically identify the large time asymptotics for the relaxation to equilibrium. We recall the work of Strain and Guo (Arch Rat Mech Anal 187:287-339 2008, Commun Partial Differ Equ 31:17-429 2006), who rigorously show that the expected law of relaxation is \\exp (-ct^{2/3}) with some c > 0. In this manuscript, we find an heuristic way, performed by asymptotic methods, that finds this "law of two thirds", and then study this question numerically. More specifically, the linear Landau equation is approximated by a set of ODEs based on expansions in generalized Laguerre polynomials. We analyze the corresponding quadratic form and the solution of these ODEs in detail. It is shown that the solution has two different asymptotic stages for large values of time t and maximal order of polynomials N: the first one focus on intermediate asymptotics which agrees with the "law of two thirds" for moderately large values of time t and then the second one on absolute, purely exponential asymptotics for very large t, as expected for linear ODEs. We believe that appearance of intermediate asymptotics in finite dimensional approximations must be a generic behavior for different classes of equations in functional spaces (some PDEs, Boltzmann equations for soft potentials, etc.) and that our methods can be applied to related problems.
About and beyond the Henri-Michaelis-Menten rate equation for single-substrate enzyme kinetics.
Bajzer, Zeljko; Strehler, Emanuel E
2012-01-20
For more than a century the simple single-substrate enzyme kinetics model and related Henri-Michaelis-Menten (HMM) rate equation have been thoroughly explored in various directions. In the present paper we are concerned with a possible generalization of this rate equation recently proposed by F. Kargi (BBRC 382 (2009) 157-159), which is assumed to be valid both in the case that the total substrate or enzyme is in excess and the quasi-steady-state is achieved. We demonstrate that this generalization is grossly inadequate and propose another generalization based on application of the quasi-steady-state condition and conservation equations for both enzyme and substrate. The standard HMM equation is derived by (a) assuming the quasi-steady-state condition, (b) applying the conservation equation only for the enzyme, and (c) assuming that the substrate concentration at quasi-steady-state can be approximated by the total substrate concentration [S](0). In our formula the rate is already expressed through [S](0), and we only assume that when quasi-steady-state is achieved the amount of product formed is negligible compared to [S](0). Numerical simulations show that our formula is generally more accurate than the HMM formula and also can provide a good approximation when the enzyme is in excess, which is not the case for the HMM formula. We show that the HMM formula can be derived from our expression by further assuming that the total enzyme concentration is negligible compared to [S](0). Copyright © 2011 Elsevier Inc. All rights reserved.
Solution of fractional kinetic equation by a class of integral transform of pathway type
NASA Astrophysics Data System (ADS)
Kumar, Dilip
2013-04-01
Solutions of fractional kinetic equations are obtained through an integral transform named Pα-transform introduced in this paper. The Pα-transform is a binomial type transform containing many class of transforms including the well known Laplace transform. The paper is motivated by the idea of pathway model introduced by Mathai [Linear Algebra Appl. 396, 317-328 (2005), 10.1016/j.laa.2004.09.022]. The composition of the transform with differential and integral operators are proved along with convolution theorem. As an illustration of applications to the general theory of differential equations, a simple differential equation is solved by the new transform. Being a new transform, the Pα-transform of some elementary functions as well as some generalized special functions such as H-function, G-function, Wright generalized hypergeometric function, generalized hypergeometric function, and Mittag-Leffler function are also obtained. The results for the classical Laplace transform is retrieved by letting α → 1.
Sub-Alfvénic reduced full-f Kinetic MHD equations to study flute like instabilities
NASA Astrophysics Data System (ADS)
Sengupta, W.; Hassam, A.; Antonsen, T. M., Jr.
2016-10-01
We develop a set of reduced sub-Alfvénic fluid as well as kinetic MHD equations which are suitable for studying flute like instabilities in MHD ordering. The full-f kinetic equations are obtained by reducing Kulsrud's complete set of kinetic MHD system and includes trapped ion dynamics in a toroidal geometry. The nonlinear equations show the presence of Mercier modes, electromagnetic effects, GAMs and Rosenbluth-Hinton zero frequency zonal flows. Linear stability based on our equations shall be compared to the well known Kruskal-Oberman Kinetic MHD stability criteria. In the supersonic limit, for large q, our system can be shown to be equivalent to CGL double adiabatic theory. In the marginal stability limit, we shall discuss trapped particle stabilization of interchange modes. Comparison will also be made to the sub-Alfvénic reduced MHD fluid equations in a large aspect ratio tokamak. We shall show that the trapped particle effects in kinetic theory can be treated as a boundary layer of width the square root of the inverse aspect ratio in phase space. Work supported by DOE.
Elementary solutions of coupled model equations in the kinetic theory of gases
NASA Technical Reports Server (NTRS)
Kriese, J. T.; Siewert, C. E.; Chang, T. S.
1974-01-01
The method of elementary solutions is employed to solve two coupled integrodifferential equations sufficient for determining temperature-density effects in a linearized BGK model in the kinetic theory of gases. Full-range completeness and orthogonality theorems are proved for the developed normal modes and the infinite-medium Green's function is constructed as an illustration of the full-range formalism. The appropriate homogeneous matrix Riemann problem is discussed, and half-range completeness and orthogonality theorems are proved for a certain subset of the normal modes. The required existence and uniqueness theorems relevant to the H matrix, basic to the half-range analysis, are proved, and an accurate and efficient computational method is discussed. The half-space temperature-slip problem is solved analytically, and a highly accurate value of the temperature-slip coefficient is reported.
Monte Carlo implementation of a guiding-center Fokker-Planck kinetic equation
Hirvijoki, E.; Snicker, A.; Kurki-Suonio, T.; Brizard, A.
2013-09-15
A Monte Carlo method for the collisional guiding-center Fokker-Planck kinetic equation is derived in the five-dimensional guiding-center phase space, where the effects of magnetic drifts due to the background magnetic field nonuniformity are included. It is shown that, in the limit of a homogeneous magnetic field, our guiding-center Monte Carlo collision operator reduces to the guiding-center Monte Carlo Coulomb operator previously derived by Xu and Rosenbluth [Phys. Fluids B 3, 627 (1991)]. Applications of the present work will focus on the collisional transport of energetic ions in complex nonuniform magnetized plasmas in the large mean-free-path (collisionless) limit, where magnetic drifts must be retained.
Liao, Fei; Tian, Kao-Cong; Yang, Xiao; Zhou, Qi-Xin; Zeng, Zhao-Chun; Zuo, Yu-Ping
2003-03-01
The reliability of kinetic substrate quantification by nonlinear fitting of the enzyme reaction curve to the integrated Michaelis-Menten equation was investigated by both simulation and preliminary experimentation. For simulation, product absorptivity epsilon was 3.00 mmol(-1) L cm(-1) and K(m) was 0.10 mmol L(-1), and uniform absorbance error sigma was randomly inserted into the error-free reaction curve of product absorbance A(i) versus reaction time t(i) calculated according to the integrated Michaelis-Menten equation. The experimental reaction curve of arylesterase acting on phenyl acetate was monitored by phenol absorbance at 270 nm. Maximal product absorbance A(m) was predicted by nonlinear fitting of the reaction curve to Eq. (1) with K(m) as constant. There were unique A(m) for best fitting of both the simulated and experimental reaction curves. Neither the error in reaction origin nor the variation of enzyme activity changed the background-corrected value of A(m). But the range of data under analysis, the background absorbance, and absorbance error sigma had an effect. By simulation, A(m) from 0.150 to 3.600 was predicted with reliability and linear response to substrate concentration when there was 80% consumption of substrate at sigma of 0.001. Restriction of absorbance to 0.700 enabled A(m) up to 1.800 to be predicted at sigma of 0.001. Detection limit reached A(m) of 0.090 at sigma of 0.001. By experimentation, the reproducibility was 4.6% at substrate concentration twice the K(m), and A(m) linearly responded to phenyl acetate with consistent absorptivity for phenol, and upper limit about twice the maximum of experimental absorbance. These results supported the reliability of this new kinetic method for enzymatic analysis with enhanced upper limit and precision.
Wu, Fuke; Tian, Tianhai; Rawlings, James B; Yin, George
2016-05-07
The frequently used reduction technique is based on the chemical master equation for stochastic chemical kinetics with two-time scales, which yields the modified stochastic simulation algorithm (SSA). For the chemical reaction processes involving a large number of molecular species and reactions, the collection of slow reactions may still include a large number of molecular species and reactions. Consequently, the SSA is still computationally expensive. Because the chemical Langevin equations (CLEs) can effectively work for a large number of molecular species and reactions, this paper develops a reduction method based on the CLE by the stochastic averaging principle developed in the work of Khasminskii and Yin [SIAM J. Appl. Math. 56, 1766-1793 (1996); ibid. 56, 1794-1819 (1996)] to average out the fast-reacting variables. This reduction method leads to a limit averaging system, which is an approximation of the slow reactions. Because in the stochastic chemical kinetics, the CLE is seen as the approximation of the SSA, the limit averaging system can be treated as the approximation of the slow reactions. As an application, we examine the reduction of computation complexity for the gene regulatory networks with two-time scales driven by intrinsic noise. For linear and nonlinear protein production functions, the simulations show that the sample average (expectation) of the limit averaging system is close to that of the slow-reaction process based on the SSA. It demonstrates that the limit averaging system is an efficient approximation of the slow-reaction process in the sense of the weak convergence.
NASA Astrophysics Data System (ADS)
Wu, Fuke; Tian, Tianhai; Rawlings, James B.; Yin, George
2016-05-01
The frequently used reduction technique is based on the chemical master equation for stochastic chemical kinetics with two-time scales, which yields the modified stochastic simulation algorithm (SSA). For the chemical reaction processes involving a large number of molecular species and reactions, the collection of slow reactions may still include a large number of molecular species and reactions. Consequently, the SSA is still computationally expensive. Because the chemical Langevin equations (CLEs) can effectively work for a large number of molecular species and reactions, this paper develops a reduction method based on the CLE by the stochastic averaging principle developed in the work of Khasminskii and Yin [SIAM J. Appl. Math. 56, 1766-1793 (1996); ibid. 56, 1794-1819 (1996)] to average out the fast-reacting variables. This reduction method leads to a limit averaging system, which is an approximation of the slow reactions. Because in the stochastic chemical kinetics, the CLE is seen as the approximation of the SSA, the limit averaging system can be treated as the approximation of the slow reactions. As an application, we examine the reduction of computation complexity for the gene regulatory networks with two-time scales driven by intrinsic noise. For linear and nonlinear protein production functions, the simulations show that the sample average (expectation) of the limit averaging system is close to that of the slow-reaction process based on the SSA. It demonstrates that the limit averaging system is an efficient approximation of the slow-reaction process in the sense of the weak convergence.
Conditions for critical effects in the mass action kinetics equations for water radiolysis.
Wittman, Richard S; Buck, Edgar C; Mausolf, Edward J; McNamara, Bruce K; Smith, Frances N; Soderquist, Chuck Z
2014-12-26
We report on a subtle global feature of the mass action kinetics equations for water radiolysis that results in predictions of a critical behavior in H2O2 and associated radical concentrations. While radiolysis kinetics have been studied extensively in the past, it is only in recent years that high-speed computing has allowed the rapid exploration of the solution over widely varying dose and compositional conditions. We explore the radiolytic production of H2O2 under various externally fixed conditions of molecular H2 and O2 that have been regarded as problematic in the literature-specifically, "jumps" in predicted concentrations, and inconsistencies between predictions and experiments have been reported for α radiolysis. We computationally map-out a critical concentration behavior for α radiolysis kinetics using a comprehensive set of reactions. We then show that all features of interest are accurately reproduced with 15 reactions. An analytical solution for steady-state concentrations of the 15 reactions reveals regions in [H2] and [O2] where the H2O2 concentration is not unique-both stable and unstable concentrations exist. The boundary of this region can be characterized analytically as a function of G-values and rate constants independent of dose rate. Physically, the boundary can be understood as separating a region where a steady-state H2O2 concentration exists from one where it does not exist without a direct decomposition reaction. We show that this behavior is consistent with reported α radiolysis data and that no such behavior should occur for γ radiolysis. We suggest experiments that could verify or discredit a critical concentration behavior for α radiolysis and could place more restrictive ranges on G-values from derived relationships between them.
Conditions for critical effects in the mass action kinetics equations for water radiolysis
Wittman, Richard S.; Buck, Edgar C.; Mausolf, Edward J.; McNamara, Bruce K.; Smith, Frances N.; Soderquist, Chuck Z.
2014-11-25
We report on a subtle global feature of the mass action kinetics equations for water radiolysis that results in predictions of a critical behavior in H2O2 and associated radical concentrations. While radiolysis kinetics has been studied extensively in the past, it is only in recent years that high speed computing has allowed the rapid exploration of the solution over widely varying dose and compositional conditions. We explore the radiolytic production of H2O2 under various externally fixed conditions of molecular H2 and O2 that have been regarded as problematic in the literature – specifically, “jumps” in predicted concentrations, and inconsistencies between predictions and experiments have been reported for alpha radiolysis. We computationally map-out a critical concentration behavior for alpha radiolysis kinetics using a comprehensive set of reactions. We then show that all features of interest are accurately reproduced with 15 reactions. An analytical solution for steady-state concentrations of the 15 reactions reveals regions in [H2] and [O2] where the H2O2 concentration is not unique – both stable and unstable concentrations exist. The boundary of this region can be characterized analytically as a function of G-values and rate constants independent of dose rate. Physically, the boundary can be understood as separating a region where a steady-state H2O2 concentration exists, from one where it does not exist without a direct decomposition reaction. We show that this behavior is consistent with reported alpha radiolysis data and that no such behavior should occur for gamma radiolysis. We suggest experiments that could verify or discredit a critical concentration behavior for alpha radiolysis and could place more restrictive ranges on G-values from derived relationships between them.
Conditions for critical effects in the mass action kinetics equations for water radiolysis
Wittman, Richard S.; Buck, Edgar C.; Mausolf, Edward J.; McNamara, Bruce K.; Smith, Frances N.; Soderquist, Chuck Z.
2014-12-26
We report on a subtle global feature of the mass action kinetics equations for water radiolysis that results in predictions of a critical behavior in H2O2 and associated radical concentrations. While radiolysis kinetics has been studied extensively in the past, it is only in recent years that high speed computing has allowed the rapid exploration of the solution over widely varying dose and compositional conditions. We explore the radiolytic production of H2O2 under various externally fixed conditions of molecular H2 and O2 that have been regarded as problematic in the literature – specifically, “jumps” in predicted concentrations, and inconsistencies between predictions and experiments have been reported for alpha radiolysis. We computationally map-out a critical concentration behavior for alpha radiolysis kinetics using a comprehensive set of reactions. We then show that all features of interest are accurately reproduced with 15 reactions. An analytical solution for steady-state concentrations of the 15 reactions reveals regions in [H2] and [O2] where the H2O2 concentration is not unique – both stable and unstable concentrations exist. The boundary of this region can be characterized analytically as a function of G-values and rate constants independent of dose rate. Physically, the boundary can be understood as separating a region where a steady-state H2O2 concentration exists, from one where it does not exist without a direct decomposition reaction. We show that this behavior is consistent with reported alpha radiolysis data and that no such behavior should occur for gamma radiolysis. We suggest experiments that could verify or discredit a critical concentration behavior for alpha radiolysis and could place more restrictive ranges on G-values from derived relationships between them.
Verification of particle-in-cell simulations against exact solutions of kinetic equations
NASA Astrophysics Data System (ADS)
Turner, Miles
2015-09-01
Demonstrating correctness of computer simulations (or verification) has become a matter of increasing concern in recent years. The strongest type of verification is a demonstration that the simulation converges to an exact solution of the mathematical model that is supposed to be solved. Of course, this is possible only if such an exact solution is available. In this paper, we are interested in kinetic simulation using the particle-in-cell method, and consequently a relevant exact solution must be a solution of a kinetic equation. While we know of no such solutions that exercise all the features of a typical particle-in-cell simulation, in this paper we show that the mathematical literature contains several such solutions that involve a large fraction of the functionality of such a code, and which collectively exercise essentially all of the code functionality. These solutions include the plane diode, the neutron criticality problem, and the calculation of ion energy distribution functions in oscillating fields. In each of theses cases, we can show the the particle-in-cell simulation converges to the exact solution in the expected way. These demonstrations are strong evidence of correct implementation. Work supported by Science Foundation Ireland under grant 08/SRC/I1411.
PADÉ APPROXIMANTS FOR THE EQUATION OF STATE FOR RELATIVISTIC HYDRODYNAMICS BY KINETIC THEORY
Tsai, Shang-Hsi; Yang, Jaw-Yen
2015-07-20
A two-point Padé approximant (TPPA) algorithm is developed for the equation of state (EOS) for relativistic hydrodynamic systems, which are described by the classical Maxwell–Boltzmann statistics and the semiclassical Fermi–Dirac statistics with complete degeneracy. The underlying rational function is determined by the ratios of the macroscopic state variables with various orders of accuracy taken at the extreme relativistic limits. The nonunique TPPAs are validated by Taub's inequality for the consistency of the kinetic theory and the special theory of relativity. The proposed TPPA is utilized in deriving the EOS of the dilute gas and in calculating the specific heat capacity, the adiabatic index function, and the isentropic sound speed of the ideal gas. Some general guidelines are provided for the application of an arbitrary accuracy requirement. The superiority of the proposed TPPA is manifested in manipulating the constituent polynomials of the approximants, which avoids the arithmetic complexity of struggling with the modified Bessel functions and the hyperbolic trigonometric functions arising from the relativistic kinetic theory.
Padé Approximants for the Equation of State for Relativistic Hydrodynamics by Kinetic Theory
NASA Astrophysics Data System (ADS)
Tsai, Shang-Hsi; Yang, Jaw-Yen
2015-07-01
A two-point Padé approximant (TPPA) algorithm is developed for the equation of state (EOS) for relativistic hydrodynamic systems, which are described by the classical Maxwell-Boltzmann statistics and the semiclassical Fermi-Dirac statistics with complete degeneracy. The underlying rational function is determined by the ratios of the macroscopic state variables with various orders of accuracy taken at the extreme relativistic limits. The nonunique TPPAs are validated by Taub's inequality for the consistency of the kinetic theory and the special theory of relativity. The proposed TPPA is utilized in deriving the EOS of the dilute gas and in calculating the specific heat capacity, the adiabatic index function, and the isentropic sound speed of the ideal gas. Some general guidelines are provided for the application of an arbitrary accuracy requirement. The superiority of the proposed TPPA is manifested in manipulating the constituent polynomials of the approximants, which avoids the arithmetic complexity of struggling with the modified Bessel functions and the hyperbolic trigonometric functions arising from the relativistic kinetic theory.
NASA Technical Reports Server (NTRS)
Catto, P. J.
1979-01-01
A simpler technique than those introduced by Lenard and Bernstein (1958), and Dougherty (1964) is employed to obtain the perturbed species density from a specified kinetic equation for a plasma in a given uniform magnetic field. The technique is a generalization of the velocity-Fourier transform method employed by Karpman (1967) for B sub 0 identical to zero, and relies on the fact that in transform space the model collision operator, used to obtain the kinetic equation for waves in a magnetized plasma, contains only first derivatives. The technique is illustrated by evaluating the perturbed density of an arbitrary species.
Relativistic kinetic equation for spin-1/2 particles in the long-scale-length approximation
NASA Astrophysics Data System (ADS)
Ekman, R.; Asenjo, F. A.; Zamanian, J.
2017-08-01
In this paper, we derive a fully relativistic kinetic theory for spin-1/2 particles and its coupling to Maxwell's equations, valid in the long-scale-length limit, where the fields vary on a scale much longer than the localization of the particles; we work to first order in ℏ . Our starting point is a Foldy-Wouthuysen (FW) transformation, applicable to this regime, of the Dirac Hamiltonian. We derive the corresponding evolution equation for the Wigner quasidistribution in an external electromagnetic field. Using a Lagrangian method we find expressions for the charge and current densities, expressed as free and bound parts. It is furthermore found that the velocity is nontrivially related to the momentum variable, with the difference depending on the spin and the external electromagnetic fields. This fact that has previously been discussed as "hidden momentum" and is due to that the FW transformation maps pointlike particles to particle clouds for which the prescription of minimal coupling is incorrect, as they have multipole moments. We express energy and momentum conservation for the system of particles and the electromagnetic field, and discuss our results in the context of the Abraham-Minkowski dilemma.
Chang, Z.; Callen, J.D. )
1992-05-01
Unified fluid/kinetic equations for the plasma perturbed density ({ital {tilde n}}), parallel flow velocity ({ital {tilde u}}{sub {parallel}}) and temperature ({ital {tilde T}}) are developed in a sheared slab geometry by calculating the fluid moment closure relations kinetically. At first, a set of (unclosed) nonlinear perturbed fluid equations for {ital {tilde n}}, {ital {tilde u}}{sub {parallel}} and {ital {tilde T}} is developed using a drift ordering analysis and a new gyroviscous force ((spec. char. missing){center dot}{Pi}{sub {ital g}}). Thereafter, to develop linear closure relations for {bold b}{center dot}{del}{center dot}{tilde {Pi}}{sub {parallel}} and {ital {tilde q}}{sub {parallel}}, a drift-kinetic version of a new Chapman--Enskog-like (CEL) equation is developed and solved by using a moment approach and a physically realistic collision operator (Lorentz scattering operator plus the momentum restoring terms). The resultant closure relations for {bold b}{center dot}(spec. char. missing){center dot}{tilde {Pi}}{sub {parallel}} and {ital {tilde q}}{sub {parallel}} unify the fluid and kinetic approaches. In the collisional fluid limit the equations reduce to the well-known Braginskii equations. In the adiabatic limit they reproduce the usual kinetic results, including Landau damping. It is shown that this new CEL approach is more compatible with a fluidlike description of plasmas than the usual drift/gyrokinetic approach. Remarkably simplified forms of the closure relations are presented. The results are compared with other Landau damping models and shown to be more accurate, complete, and physically realistic. Applications of this set of equations to various microinstabilities in tokamak plasmas are presented in a separate paper (Part II) (Phys. Fluids B {bold 4}, 1182 (1992)).
Pinto, Leticia N.; Dos Santos, Adimir
2015-07-01
Multiplying Subcritical Systems were for a long time poorly studied and its theoretical description remains with plenty open questions. Great interest on such systems arose partly due to the improvement of hybrid concepts, such as the Accelerator-Driven Systems (ADS). Along with the need for new technologies to be developed, further study and understanding of subcritical systems are essential also in more practical situations, such as in the case of a PWR criticalization in their physical startup tests. Point kinetics equations are fundamental to continuously monitor the reactivity behavior to a possible variation of external sources intensity. In this case, quickly and accurately predicting power transients and reactivity becomes crucial. It is known that conventional Reactivity Meters cannot operate in subcritical levels nor describe the dynamics of multiplying systems in these conditions, by the very structure of the classical kinetic equations. Several theoretical models have been proposed to characterize the kinetics of such systems with special regard to the reactivity, as the one developed by Gandini and Salvatores among others. This work presents a discussion about the derivation of point kinetics equations for subcritical systems and the importance of considering the external source. From the point of view of the Gandini and Salvatores' point kinetics model and based on the experimental results provided by Lee and dos Santos, it was possible to develop an innovative approach. This article proposes an algorithm that describes the subcritical reactivity with external source, contributing to the advancement of studies in the field. (authors)
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.
Applications of nonlinear science and kinetic equations to the spread of epidemics
NASA Astrophysics Data System (ADS)
Macinnis, David Robert
The study of the spread of epidemics is currently growing into a successful subfield of a combination of nonlinear science and statistical mechanics. Topics studied in this field include kinetic and mean field levels of epidemiological models. This thesis consists of the analysis of such topics and specifically directed at the Hantavirus, West Nile virus, and the Bubonic Plague. A successful reaction-diffusion equation approach developed recently by Abramson and Kenkre was able to describe spatiotemporal patterns of the Hantavirus model. From measurements of the parameters of their model it was found that the mice, the carriers of the infection, must be regarded as moving diffusively within attractive potentials representative of home ranges. Several attempts have been made to incorporate home ranges into their model. Two of these attempts are discussed within this thesis. A model to explain the transmission of the West Nile virus within bird and mosquito populations was recently developed by Kenkre, Parmenter, Peixoto, and Sadasiv who showed how spatially resolved issues could be discussed but restricted their analysis to mean field considerations. This thesis extends that study by investigating spatial resolution of the infected populations. Traveling waves of the bird and mosquito populations are found in the West Nile context. Infection control of various epidemics has become increasingly important to limit the potential force of infection into the human population. This thesis contains a quantitative attempt at a theory of such control (for the West Nile virus) via spraying of the mosquito population. Mean field and kinetic level models are proposed in this thesis to describe the transmission of the Bubonic Plague which involves flea and mammal populations. The various populations are found to undergo a variety of bifurcations as well as hysteresis in their steady state regime. Spatially resolved analysis of the populations is also presented.
NASA Astrophysics Data System (ADS)
Luo, Songting; Payne, Nicholas
2017-07-01
We present an effective asymptotic method for approximating the density of particles for kinetic equations with a Bhatnagar-Gross-Krook (BGK) relaxation operator in the large scale hyperbolic limit. The density of particles is transformed via a Hopf-Cole transformation, where the phase function is expanded as a power series with respect to the Knudsen number. The expansion terms can be determined by solving a sequence of equations. In particular, it has been proved in [3] that the leading order term is the viscosity solution of an effective Hamilton-Jacobi equation, and we show that the higher order terms can be formally determined by solving a sequence of transport equations. Both the effective Hamilton-Jacobi equation and the transport equations are independent of the Knudsen number, and are formulated in the physical space, where the effective Hamiltonian is obtained as the solution of a nonlinear equation that is given as an integral in the velocity variable, and the coefficients of the transport equations are given as integrals in the velocity variable. With appropriate Gauss quadrature rules for evaluating these integrals effectively, the effective Hamilton-Jacobi equation and the transport equations can be solved efficiently to obtain the expansion terms for approximating the density function. In this work, the zeroth, first and second order terms in the expansion are used to obtain second order accuracy with respect to the Knudsen number. The proposed method balances efficiency and accuracy, and has the potential to deal with kinetic equations with more general BGK models. Numerical experiments verify the effectiveness of the proposed method.
Kiernan, D; Malone, A; O'Brien, T; Simms, C K
2015-01-01
Regression equations based on pelvic anatomy are routinely used to estimate the hip joint centre during gait analysis. While the associated errors have been well documented, the clinical significance of these errors has not been reported. This study investigated the clinical agreement of three commonly used regression equation sets (Bell et al., Davis et al. and Orthotrak software) against the equations of Harrington et al. Full 3-dimensional gait analysis was performed on 18 healthy paediatric subjects. Kinematic and kinetic data were calculated using each set of regression equations and compared to Harrington et al. In addition, the Gait Profile Score and GDI-Kinetic were used to assess clinical significance. Bell et al. was the best performing set with differences in Gait Profile Score (0.13°) and GDI-Kinetic (0.84 points) falling below the clinical significance threshold. Small deviations were present for the Orthotrak set for hip abduction moment (0.1 Nm/kg), however differences in Gait Profile Score (0.27°) and GDI-Kinetic (2.26 points) remained below the clinical threshold. Davis et al. showed least agreement with a clinically significant difference in GDI-Kinetic score (4.36 points). It is proposed that Harrington et al. or Bell et al. regression equation sets are used during gait analysis especially where inverse dynamic data are calculated. Orthotrak is a clinically acceptable alternative however clinicians must be aware of the effects of error on hip abduction moment. The Davis et al. set should be used with caution for inverse dynamic analysis as error could be considered clinically meaningful.
Numerical solutions of the semiclassical Boltzmann ellipsoidal-statistical kinetic model equation
Yang, Jaw-Yen; Yan, Chin-Yuan; Huang, Juan-Chen; Li, Zhihui
2014-01-01
Computations of rarefied gas dynamical flows governed by the semiclassical Boltzmann ellipsoidal-statistical (ES) kinetic model equation using an accurate numerical method are presented. The semiclassical ES model was derived through the maximum entropy principle and conserves not only the mass, momentum and energy, but also contains additional higher order moments that differ from the standard quantum distributions. A different decoding procedure to obtain the necessary parameters for determining the ES distribution is also devised. The numerical method in phase space combines the discrete-ordinate method in momentum space and the high-resolution shock capturing method in physical space. Numerical solutions of two-dimensional Riemann problems for two configurations covering various degrees of rarefaction are presented and various contours of the quantities unique to this new model are illustrated. When the relaxation time becomes very small, the main flow features a display similar to that of ideal quantum gas dynamics, and the present solutions are found to be consistent with existing calculations for classical gas. The effect of a parameter that permits an adjustable Prandtl number in the flow is also studied. PMID:25104904
Study of carbon dioxide gas treatment based on equations of kinetics in plasma discharge reactor
NASA Astrophysics Data System (ADS)
Abedi-Varaki, Mehdi
2017-08-01
Carbon dioxide (CO2) as the primary greenhouse gas, is the main pollutant that is warming earth. CO2 is widely emitted through the cars, planes, power plants and other human activities that involve the burning of fossil fuels (coal, natural gas and oil). Thus, there is a need to develop some method to reduce CO2 emission. To this end, this study investigates the behavior of CO2 in dielectric barrier discharge (DBD) plasma reactor. The behavior of different species and their reaction rates are studied using a zero-dimensional model based on equations of kinetics inside plasma reactor. The results show that the plasma reactor has an effective reduction on the CO2 density inside the reactor. As a result of reduction in the temporal variations of reaction rate, the speed of chemical reactions for CO2 decreases and very low concentration of CO2 molecules inside the plasma reactor is generated. The obtained results are compared with the existing experimental and simulation findings in the literature.
Rudzinski, Wladyslaw; Plazinski, Wojciech
2006-08-24
For practical applications of solid/solution adsorption processes, the kinetics of these processes is at least as much essential as their features at equilibrium. Meanwhile, the general understanding of this kinetics and its corresponding theoretical description are far behind the understanding and the level of theoretical interpretation of adsorption equilibria in these systems. The Lagergren empirical equation proposed at the end of 19th century to describe the kinetics of solute sorption at the solid/solution interfaces has been the most widely used kinetic equation until now. This equation has also been called the pseudo-first order kinetic equation because it was intuitively associated with the model of one-site occupancy adsorption kinetics governed by the rate of surface reaction. More recently, its generalization for the two-sites-occupancy adsorption was proposed and called the pseudo-second-order kinetic equation. However, the general use and the wide applicability of these empirical equations during more than one century have not resulted in a corresponding fundamental search for their theoretical origin. Here the first theoretical development of these equations is proposed, based on applying the new fundamental approach to kinetics of interfacial transport called the Statistical Rate Theory. It is shown that these empirical equations are simplified forms of a more general equation developed here, for the case when the adsorption kinetics is governed by the rate of surface reactions. The features of that general equation are shown by presenting exhaustive model investigations, and the applicability of that equation is tested by presenting a quantitative analysis of some experimental data reported in the literature.
Sun, Wenjun; Jiang, Song; Xu, Kun; Li, Shu
2015-12-01
This paper presents an extension of previous work (Sun et al., 2015 [22]) of the unified gas kinetic scheme (UGKS) for the gray radiative transfer equations to the frequency-dependent (multi-group) radiative transfer system. Different from the gray radiative transfer equations, where the optical opacity is only a function of local material temperature, the simulation of frequency-dependent radiative transfer is associated with additional difficulties from the frequency-dependent opacity. For the multiple frequency radiation, the opacity depends on both the spatial location and the frequency. For example, the opacity is typically a decreasing function of frequency. At the same spatial region the transport physics can be optically thick for the low frequency photons, and optically thin for high frequency ones. Therefore, the optical thickness is not a simple function of space location. In this paper, the UGKS for frequency-dependent radiative system is developed. The UGKS is a finite volume method and the transport physics is modeled according to the ratio of the cell size to the photon's frequency-dependent mean free path. When the cell size is much larger than the photon's mean free path, a diffusion solution for such a frequency radiation will be obtained. On the other hand, when the cell size is much smaller than the photon's mean free path, a free transport mechanism will be recovered. In the regime between the above two limits, with the variation of the ratio between the local cell size and photon's mean free path, the UGKS provides a smooth transition in the physical and frequency space to capture the corresponding transport physics accurately. The seemingly straightforward extension of the UGKS from the gray to multiple frequency radiation system is due to its intrinsic consistent multiple scale transport modeling, but it still involves lots of work to properly discretize the multiple groups in order to design an asymptotic preserving (AP) scheme in all
Use of a generalized fisher equation for global optimization in chemical kinetics.
Villaverde, Alejandro F; Ross, John; Morán, Federico; Balsa-Canto, Eva; Banga, Julio R
2011-08-04
A new approach for parameter estimation in chemical kinetics has been recently proposed (Ross et al. Proc. Natl. Acad. Sci. U.S.A. 2010, 107, 12777). It makes use of an optimization criterion based on a Generalized Fisher Equation (GFE). Its utility has been demonstrated with two reaction mechanisms, the chlorite-iodide and Oregonator, which are computationally stiff systems. In this Article, the performance of the GFE-based algorithm is compared to that obtained from minimization of the squared distances between the observed and predicted concentrations obtained by solving the corresponding initial value problem (we call this latter approach "traditional" for simplicity). Comparison of the proposed GFE-based optimization method with the "traditional" one has revealed their differences in performance. This difference can be seen as a trade-off between speed (which favors GFE) and accuracy (which favors the traditional method). The chlorite-iodide and Oregonator systems are again chosen as case studies. An identifiability analysis is performed for both of them, followed by an optimal experimental design based on the Fisher Information Matrix (FIM). This allows to identify and overcome most of the previously encountered identifiability issues, improving the estimation accuracy. With the new data, obtained from optimally designed experiments, it is now possible to estimate effectively more parameters than with the previous data. This result, which holds for both GFE-based and traditional methods, stresses the importance of an appropriate experimental design. Finally, a new hybrid method that combines advantages from the GFE and traditional approaches is presented.
NASA Astrophysics Data System (ADS)
Needham, D. J.
1992-05-01
In this paper we consider the question of the existence of solutions to an initial-boundary value problem for a singular, semilinear, parabolic equation arising from the study of autocatalytic chemical kinetics of the type A → B at rate k[A][B] P , where A is a reactant, B is the autocatalyst, k>0 the rate constant and 0< p<1 the order of the reaction, King and Needham [1]. A monotone iteration method is adopted in the spirit of Sattinger [2], However the general theory due to Sattinger [2] cannot be applied directly because of the singular nature of the source term in the equation.
Sasorov, P. V.; Fomin, I. V.
2015-06-15
The collision integral in the kinetic equation for a rarefied spin-polarized gas of fermions (electrons) is derived. The collisions between these fermions and the collisions with much heavier particles (ions) forming a randomly located stationary background (gas) are taken into account. An important new circumstance is that the particle-particle scattering amplitude is not assumed to be small, which could be obtained, for example, in the first Born approximation. The derived collision integral can be used in the kinetic equation, including that for a relatively cold rarefied spin-polarized plasma with a characteristic electron energy below α{sup 2}m{sub e}c{sup 2}, where α is the fine-structure constant.
NASA Technical Reports Server (NTRS)
Frost, W.; Harper, W. L.
1975-01-01
Flow over surface obstructions can produce significantly large wind shears such that adverse flying conditions can occur for aeronautical systems (helicopters, STOL vehicles, etc.). Atmospheric flow fields resulting from a semi-elliptical surface obstruction in an otherwise horizontally homogeneous statistically stationary flow are modelled with the boundary-layer/Boussinesq-approximation of the governing equation of fluid mechanics. The turbulence kinetic energy equation is used to determine the dissipative effects of turbulent shear on the mean flow. Iso-lines of turbulence kinetic energy and turbulence intensity are plotted in the plane of the flow and highlight regions of high turbulence intensity in the stagnation zone and sharp gradients in intensity along the transition from adverse to favourable pressure gradient. Discussion of the effects of the disturbed wind field in CTOL and STOL aircraft flight path and obstruction clearance standards is given. The results indicate that closer inspection of these presently recommended standards as influenced by wind over irregular terrains is required.
Goličnik, Marko
2012-03-01
To facilitate the determination of a reaction type and its kinetics constants for reversible inhibitors of Michaelis-Menten-type enzymes using progress-curve analysis, I present here an explicit equation for direct curve fitting to full time-course data of inhibited enzyme-catalyzed reactions. This algebraic expression involves certain elementary functions where their values are readily available using any standard nonlinear regression program. Hence this allows easy analysis of experimentally observed kinetics without any data conversion prior to fitting. Its implementation gives correct parameter estimates that are in very good agreement with results obtained using both the numerically integrated Michaelis-Menten rate equation or its exact closed-form solution which is expressed in terms of the Lambert W function.
NASA Astrophysics Data System (ADS)
Murugan, R.
2002-09-01
A composite approximate solution of Michaelis-Menten enzyme kinetic equation, which could describe both transient and slow dynamics, was obtained by ordinary perturbation methods in terms of undetermined gauge functions up to a first-order level. It was found that the zeroth-order perturbation function itself solved the paradox due to steady-state approximation and predicted well the maximum enzyme-substrate complex ([ES]max) and time tm to attain it. Extensive kinetic simulations using a chemical kinetic simulator proved the validity of these results. A comparison between simulated and predicted results showed that error in the prediction of tm was negligible when perturbation parameter falls in the range of (0<ɛ≪1). Apart from these, also the effect of transient dynamics on the linearity of Lineweaver-Burk plot (especially near the origin) has been explained.
Costa, Rafael S; Machado, Daniel; Rocha, Isabel; Ferreira, Eugénio C
2010-05-01
The construction of dynamic metabolic models at reaction network level requires the use of mechanistic enzymatic rate equations that comprise a large number of parameters. The lack of knowledge on these equations and the difficulty in the experimental identification of their associated parameters, represent nowadays the limiting factor in the construction of such models. In this study, we compare four alternative modeling approaches based on Michaelis-Menten kinetics for the bi-molecular reactions and different types of simplified rate equations for the remaining reactions (generalized mass action, convenience kinetics, lin-log and power-law). Using the mechanistic model for Escherichia coli central carbon metabolism as a benchmark, we investigate the alternative modeling approaches through comparative simulations analyses. The good dynamic behavior and the powerful predictive capabilities obtained using the hybrid model composed of Michaelis-Menten and the approximate lin-log kinetics indicate that this is a possible suitable approach to model complex large-scale networks where the exact rate laws are unknown.
NASA Astrophysics Data System (ADS)
Ungar, F.; Cygorek, M.; Axt, V. M.
2017-06-01
Quantum kinetic equations of motion for the description of the exciton spin dynamics in II-VI diluted magnetic semiconductor quantum wells with laser driving are derived. The model includes the magnetic as well as the nonmagnetic carrier-impurity interaction, the Coulomb interaction, Zeeman terms, and the light-matter coupling, allowing for an explicit treatment of arbitrary excitation pulses. Based on a dynamics-controlled truncation scheme, contributions to the equations of motion up to second order in the generating laser field are taken into account. The correlations between the carrier and the impurity subsystems are treated within the framework of a correlation expansion. For vanishing magnetic field, the Markov limit of the quantum kinetic equations formulated in the exciton basis agrees with existing theories based on Fermi's golden rule. For narrow quantum wells excited at the 1 s exciton resonance, numerical quantum kinetic simulations reveal pronounced deviations from the Markovian behavior. In particular, the spin decays initially with approximately half the Markovian rate and a nonmonotonic decay in the form of an overshoot of up to 10 % of the initial spin polarization is predicted.
A unified gas-kinetic scheme for continuum and rarefied flows IV: Full Boltzmann and model equations
Liu, Chang; Xu, Kun; Sun, Quanhua; Cai, Qingdong
2016-06-01
Fluid dynamic equations are valid in their respective modeling scales, such as the particle mean free path scale of the Boltzmann equation and the hydrodynamic scale of the Navier–Stokes (NS) equations. With a variation of the modeling scales, theoretically there should have a continuous spectrum of fluid dynamic equations. Even though the Boltzmann equation is claimed to be valid in all scales, many Boltzmann solvers, including direct simulation Monte Carlo method, require the cell resolution to the order of particle mean free path scale. Therefore, they are still single scale methods. In order to study multiscale flow evolution efficiently, the dynamics in the computational fluid has to be changed with the scales. A direct modeling of flow physics with a changeable scale may become an appropriate approach. The unified gas-kinetic scheme (UGKS) is a direct modeling method in the mesh size scale, and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path. The cell size of UGKS is not limited by the particle mean free path. With the variation of the ratio between the numerical cell size and local particle mean free path, the UGKS recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale. The previous UGKS is mostly constructed from the evolution solution of kinetic model equations. Even though the UGKS is very accurate and effective in the low transition and continuum flow regimes with the time step being much larger than the particle mean free time, it still has space to develop more accurate flow solver in the region, where the time step is comparable with the local particle mean free time. In such a scale, there is dynamic difference from the full Boltzmann collision term and the model equations. This work is about the further development of the UGKS with the implementation of the full Boltzmann collision term in the region
Utilization of Integrated Michaelis-Menten Equation to Determine Kinetic Constants
ERIC Educational Resources Information Center
Bezerra, Rui M. F.; Dias, Albino A.
2007-01-01
Students of biochemistry and related biosciences are urged to solve problems where kinetic parameters are calculated from initial rates obtained at different substrate concentrations. Troubles begin when they go to the laboratory to perform kinetic experiments and realize that usual laboratory instruments do not measure initial rates but only…
Utilization of Integrated Michaelis-Menten Equation to Determine Kinetic Constants
ERIC Educational Resources Information Center
Bezerra, Rui M. F.; Dias, Albino A.
2007-01-01
Students of biochemistry and related biosciences are urged to solve problems where kinetic parameters are calculated from initial rates obtained at different substrate concentrations. Troubles begin when they go to the laboratory to perform kinetic experiments and realize that usual laboratory instruments do not measure initial rates but only…
NASA Astrophysics Data System (ADS)
Shi, Feng
This dissertation is organized in two parts, the first part is about fundamental characteristics of multiple dimensional systems, the second part is about parallel KMC calculation of coarsening process. In Part I, we first study the fundamental characteristics of nucleation and growth in 3 dimensional (3D) systems using a simplified model of nucleation and growth. One of the main goals of this work is to compare with previous work on 2D nucleation and growth in order to understand the effects of dimensionality. The scaling of the average island-size, island density, monomer density, island-size distribution (ISD), capture number distribution (CND), and capture zone distribution (CZD) are studied as a function of the fraction of occupied sites (coverage) and the ratio D/F of the monomer hopping rate D to the (per site) monomer creation rate F. Our model may be viewed as a simple model of the early-stages of vacancy cluster nucleation and growth under irradiation. Good agreement is found between our mean-field (MF) rate-equation results for the average island and monomer densities and our simulation results. In addition, we find that due to the decreased influence of correlations and fluctuations in 3D as compared to 2D, the scaled CND depends only weakly on the island-size. As a result the scaled ISD is significantly sharper than obtained in 2D and diverges with increasing D/F. However, the scaled ISD obtained in kinetic Monte Carlo (KMC) simulations appears to diverge more slowly with increasing D/F than the MF prediction while the divergence occurs at a value of the scaled island-size which is somewhat beyond the MF prediction. These results are supported by an analysis of the asymptotic CND. The final goal for understanding the mechanism of nucleation and growth is to develop a theory to concisely and precisely disclose the law underlying the nucleation and growth process. From the theoretical point view, dimension can be taken as a variable to develop theory. In
Czerwonko, J.
1988-04-01
Since nonlinear effects are of the same importance as the particle-hole asymmetry (PHA) effects for normal Fermi liquids, at least for some physical situations, a formalism is presented taking both into account. Moreover, because the nonlinearity or PHA is easiest to induce by strong magnetic fields, weak polarization effects are also included. The kinetic equations for the weakly coupled density and magnetization modes are obtained under these circumstances. They lead to an additional effective mass equation in comparison to the Landau formula, joining the suitable angular average of the effective interaction of triples of quasiparticles with gradient of the two-quasiparticle interaction with PHA effects included. The equations are investigated in detail for ac magnetic field much smaller than the dc field in two cases: (1) at almost equilibrium magnetization of the sample and (2) at almost equilibrium (in the length) magnetization precessing around a dc field tipped to it by an angle Theta not equal to O.
NASA Astrophysics Data System (ADS)
Li, Zhi-Hui; Peng, Ao-Ping; Zhang, Han-Xin; Yang, Jaw-Yen
2015-04-01
This article reviews rarefied gas flow computations based on nonlinear model Boltzmann equations using deterministic high-order gas-kinetic unified algorithms (GKUA) in phase space. The nonlinear Boltzmann model equations considered include the BGK model, the Shakhov model, the Ellipsoidal Statistical model and the Morse model. Several high-order gas-kinetic unified algorithms, which combine the discrete velocity ordinate method in velocity space and the compact high-order finite-difference schemes in physical space, are developed. The parallel strategies implemented with the accompanying algorithms are of equal importance. Accurate computations of rarefied gas flow problems using various kinetic models over wide ranges of Mach numbers 1.2-20 and Knudsen numbers 0.0001-5 are reported. The effects of different high resolution schemes on the flow resolution under the same discrete velocity ordinate method are studied. A conservative discrete velocity ordinate method to ensure the kinetic compatibility condition is also implemented. The present algorithms are tested for the one-dimensional unsteady shock-tube problems with various Knudsen numbers, the steady normal shock wave structures for different Mach numbers, the two-dimensional flows past a circular cylinder and a NACA 0012 airfoil to verify the present methodology and to simulate gas transport phenomena covering various flow regimes. Illustrations of large scale parallel computations of three-dimensional hypersonic rarefied flows over the reusable sphere-cone satellite and the re-entry spacecraft using almost the largest computer systems available in China are also reported. The present computed results are compared with the theoretical prediction from gas dynamics, related DSMC results, slip N-S solutions and experimental data, and good agreement can be found. The numerical experience indicates that although the direct model Boltzmann equation solver in phase space can be computationally expensive
Steady-state benchmarks of DK4D: A time-dependent, axisymmetric drift-kinetic equation solver
Lyons, B. C.; Jardin, S. C.; Ramos, J. J.
2015-05-15
The DK4D code has been written to solve a set of time-dependent, axisymmetric, finite-Larmor-radius drift-kinetic equations (DKEs) for the non-Maxwellian part of the electron and ion distribution functions using the full, linearized Fokker–Planck–Landau collision operator. The plasma is assumed to be in the low- to finite-collisionality regime, as is found in the cores of modern and future magnetic confinement fusion experiments. Each DKE is formulated such that the perturbed distribution function carries no net density, parallel momentum, or kinetic energy. Rather, these quantities are contained within the background Maxwellians and would be evolved by an appropriate set of extended magnetohydrodynamic (MHD) equations. This formulation allows for straight-forward coupling of DK4D to existing extended MHD time evolution codes. DK4D uses a mix of implicit and explicit temporal representations and finite element and spectral spatial representations. These, along with other computational methods used, are discussed extensively. Steady-state benchmarks are then presented comparing the results of DK4D to expected analytic results at low collisionality, qualitatively, and to the Sauter analytic fits for the neoclassical conductivity and bootstrap current, quantitatively. These benchmarks confirm that DK4D is capable of solving for the correct, gyroaveraged distribution function in stationary magnetic equilibria. Furthermore, the results presented demonstrate how the exact drift-kinetic solution varies with collisionality as a function of the magnetic moment and the poloidal angle.
Steady-state benchmarks of DK4D: A time-dependent, axisymmetric drift-kinetic equation solvera)
NASA Astrophysics Data System (ADS)
Lyons, B. C.; Jardin, S. C.; Ramos, J. J.
2015-05-01
The DK4D code has been written to solve a set of time-dependent, axisymmetric, finite-Larmor-radius drift-kinetic equations (DKEs) for the non-Maxwellian part of the electron and ion distribution functions using the full, linearized Fokker-Planck-Landau collision operator. The plasma is assumed to be in the low- to finite-collisionality regime, as is found in the cores of modern and future magnetic confinement fusion experiments. Each DKE is formulated such that the perturbed distribution function carries no net density, parallel momentum, or kinetic energy. Rather, these quantities are contained within the background Maxwellians and would be evolved by an appropriate set of extended magnetohydrodynamic (MHD) equations. This formulation allows for straight-forward coupling of DK4D to existing extended MHD time evolution codes. DK4D uses a mix of implicit and explicit temporal representations and finite element and spectral spatial representations. These, along with other computational methods used, are discussed extensively. Steady-state benchmarks are then presented comparing the results of DK4D to expected analytic results at low collisionality, qualitatively, and to the Sauter analytic fits for the neoclassical conductivity and bootstrap current, quantitatively. These benchmarks confirm that DK4D is capable of solving for the correct, gyroaveraged distribution function in stationary magnetic equilibria. Furthermore, the results presented demonstrate how the exact drift-kinetic solution varies with collisionality as a function of the magnetic moment and the poloidal angle.
McAuliffe, J.J. ); Perry, S.B. ); Brooks, E.E. ); Ingwall, J.S. Harvard Medical School, Boston, MA )
1991-03-12
Here the authors define the kinetics of the creatine kinase (CK) reaction in an intact mammalian heart containing the full rnage of CK isoenzymes. Previously derived kinetic constants were refit for the reaction occurring at 37C. Steady-state metabolite concentrations from {sup 31}P NMR and standard biochemical techniques were determined. {sup 31}P magnetization transfer data were obtained to determine unidirectional creatine kinase fluxes in hearts with differing total creatine contents and differing mitochondrial CK activities during KCl arrest and isovolumic work for both the forward reaction (MgATP synthesis) and reverse reaction (phosphocreatine synthesis). The NMR kinetic data and substrate concentrations data were used in conjunction with a kinetic model based on MM-CK in solution to determine the applicability of the solution-based kinetic models to the CK kinetics of the intact heart. The results indicated that no single set of rate equation constants could describe both the KCl-arrested and working hearts. Analysis of the results indicated that the CK reaction is rate limited in the direction of ATP synthesis, the size of the guanidino substrate pool drives the measured CK flux in the intact heart, and during isovolumic work, the CK reaction operates under saturating conditions; that is, the substrate concentrations are at least 2-fold greater than the K{sub m} or K{sub im} for each substrate. However, during KCl arrest the reaction does not operate under saturating conditions and the CK reaction velocity is strongly influenced by the guanidino substrate pool size.
Iyer, Ramakrishnan; Mukhopadhyay, Ayan
2010-04-15
The AdS/CFT correspondence defines a sector with universal strongly coupled dynamics in the field theory as the dual of pure gravity in AdS described by Einstein's equation with a negative cosmological constant. We explain here, from the field-theoretic viewpoint how the dynamics in this sector gets determined by the expectation value of the energy-momentum tensor alone. We first show that the Boltzmann equation has very special solutions which could be functionally completely determined in terms of the energy-momentum tensor alone. We call these solutions conservative solutions. We indicate why conservative solutions should also exist when we refine this kinetic description to go closer to the exact microscopic theory or even move away from the regime of weak coupling so that no kinetic description could be employed. We argue that these conservative solutions form the universal sector dual to pure gravity at strong coupling and large N. Based on this observation, we propose a regularity condition on the energy-momentum tensor so that the dual solution in pure gravity has a smooth future horizon. We also study if irreversibility emerges only at long time scales of observation, unlike the case of the Boltzmann equation.
NASA Technical Reports Server (NTRS)
Pratt, D. T.; Radhakrishnan, K.
1986-01-01
The design of a very fast, automatic black-box code for homogeneous, gas-phase chemical kinetics problems requires an understanding of the physical and numerical sources of computational inefficiency. Some major sources reviewed in this report are stiffness of the governing ordinary differential equations (ODE's) and its detection, choice of appropriate method (i.e., integration algorithm plus step-size control strategy), nonphysical initial conditions, and too frequent evaluation of thermochemical and kinetic properties. Specific techniques are recommended (and some advised against) for improving or overcoming the identified problem areas. It is argued that, because reactive species increase exponentially with time during induction, and all species exhibit asymptotic, exponential decay with time during equilibration, exponential-fitted integration algorithms are inherently more accurate for kinetics modeling than classical, polynomial-interpolant methods for the same computational work. But current codes using the exponential-fitted method lack the sophisticated stepsize-control logic of existing black-box ODE solver codes, such as EPISODE and LSODE. The ultimate chemical kinetics code does not exist yet, but the general characteristics of such a code are becoming apparent.
Zhao, Renjie; Evans, James W.; Oliveira, Tiago J.
2016-04-08
Here, a discrete version of deposition-diffusion equations appropriate for description of step flow on a vicinal surface is analyzed for a two-dimensional grid of adsorption sites representing the stepped surface and explicitly incorporating kinks along the step edges. Model energetics and kinetics appropriately account for binding of adatoms at steps and kinks, distinct terrace and edge diffusion rates, and possible additional barriers for attachment to steps. Analysis of adatom attachment fluxes as well as limiting values of adatom densities at step edges for nonuniform deposition scenarios allows determination of both permeability and kinetic coefficients. Behavior of these quantities is assessed as a function of key system parameters including kink density, step attachment barriers, and the step edge diffusion rate.
Zhao, Renjie; Evans, James W.; Oliveira, Tiago J.
2016-04-08
Here, a discrete version of deposition-diffusion equations appropriate for description of step flow on a vicinal surface is analyzed for a two-dimensional grid of adsorption sites representing the stepped surface and explicitly incorporating kinks along the step edges. Model energetics and kinetics appropriately account for binding of adatoms at steps and kinks, distinct terrace and edge diffusion rates, and possible additional barriers for attachment to steps. Analysis of adatom attachment fluxes as well as limiting values of adatom densities at step edges for nonuniform deposition scenarios allows determination of both permeability and kinetic coefficients. Behavior of these quantities is assessedmore » as a function of key system parameters including kink density, step attachment barriers, and the step edge diffusion rate.« less
Brautigam, Chad A.
2011-01-01
The interaction of macromolecules with themselves and with other macromolecules is fundamental to the functioning of living systems. Recent advances in the analysis of sedimentation velocity (SV) data obtained by analytical ultracentrifugation allow the experimenter to determine important features of such interactions, including the equilibrium association constant and information about the kinetic off-rate of the interaction. The determination of these parameters is made possible by the ability of modern software to fit numerical solutions of the Lamm Equation with kinetic considerations directly to SV data. Herein, the SV analytical advances implemented in the software package SEDPHAT are summarized. Detailed analyses of SV data using these strategies are presented. Finally, a few highlights of recent literature reports that feature this type of SV data analysis are surveyed. PMID:21187153
Wang, Jia X; Uribe, Francisco A; Springer, Thomas E; Zhang, Junliang; Adzic, Radoslav R
2008-01-01
According to Sergio Trasatti, "A true theory of electrocatalysis will not be available until activity can be calculated a priori from some known properties of the materials." Toward this goal, we developed intrinsic kinetic equations for the hydrogen oxidation reaction (HOR) and the oxygen reduction reaction (ORR) using as the kinetic parameters the free energies of adsorption and activation for elementary reactions. Rigorous derivation retained the intrinsic connection between the intermediates' adsorption isotherms and the kinetic equations, affording us an integrated approach for establishing the reaction mechanisms based upon various experimental and theoretical results. Using experimentally deduced free energy diagrams and activity-and-barriers plot for the ORR on Pt(111), we explained why the Tafel slope in the large overpotential region is double that in the small overpotential region. For carbon-supported Pt nanoparticles (Pt/C), the polarization curves measured with thin-film rotating disk electrodes also exhibit the double Tafel slope, albeit Pt(111) is several times more active than the Pt nanoparticles when the current is normalized by real surface area. An analytic method was presented for the polarization curves measured with H2 in proton exchange membrane fuel cells (PEMFCs). The fit to a typical iR-free polarization curve at 80 degrees C revealed that the change of the Tafel slope occurs at about 0.77 V that is the reversible potential for the transition between adsorbed O and OH on Pt/C. This is significant because it predicts that the Butler-Volmer equation can only fit the data above this potential, regardless the current density. We also predicted a decrease of the Tafel slope from 70 to 65 mV dec(-1) at 80 degrees C with increasing oxygen partial pressure, which is consistent with the observation reported in literature.
Galerkin method for unsplit 3-D Dirac equation using atomically/kinetically balanced B-spline basis
Fillion-Gourdeau, F.; Lorin, E.; Bandrauk, A.D.
2016-02-15
A Galerkin method is developed to solve the time-dependent Dirac equation in prolate spheroidal coordinates for an electron–molecular two-center system. The initial state is evaluated from a variational principle using a kinetic/atomic balanced basis, which allows for an efficient and accurate determination of the Dirac spectrum and eigenfunctions. B-spline basis functions are used to obtain high accuracy. This numerical method is used to compute the energy spectrum of the two-center problem and then the evolution of eigenstate wavefunctions in an external electromagnetic field.
Parkhomenko, A I; Shalagin, Anatolii M
2011-11-30
Using the eikonal approximation, we have calculated effective collision frequencies in density-matrix kinetic equations describing nonlinear effects in the wings of spectral lines. We have established the relation between the probabilities of absorption and stimulated emission and the characteristics of the radiation and elementary scattering event. The example of the power interaction potential shows that quantum mechanical calculation of the collision frequencies in the eikonal approximation and previously known spectral line wing theory give similar results for the probability of radiation absorption.
A Gas-Kinetic Method for Hyperbolic-Elliptic Equations and Its Application in Two-Phase Fluid Flow
NASA Technical Reports Server (NTRS)
Xu, Kun
1999-01-01
A gas-kinetic method for the hyperbolic-elliptic equations is presented in this paper. In the mixed type system, the co-existence and the phase transition between liquid and gas are described by the van der Waals-type equation of state (EOS). Due to the unstable mechanism for a fluid in the elliptic region, interface between the liquid and gas can be kept sharp through the condensation and evaporation process to remove the "averaged" numerical fluid away from the elliptic region, and the interface thickness depends on the numerical diffusion and stiffness of the phase change. A few examples are presented in this paper for both phase transition and multifluid interface problems.
Sánchez, Ana; Vázquez, José A; Quinteiro, Javier; Sotelo, Carmen G
2013-04-10
Real-time PCR is the most sensitive method for detection and precise quantification of specific DNA sequences, but it is not usually applied as a quantitative method in seafood. In general, benchmark techniques, mainly cycle threshold (Ct), are the routine method for quantitative estimations, but they are not the most precise approaches for a standard assay. In the present work, amplification data from European hake (Merluccius merluccius) DNA samples were accurately modeled by three sigmoid reparametrized equations, where the lag phase parameter (λc) from the Richards equation with four parameters was demonstrated to be the perfect substitute for Ct for PCR quantification. The concentrations of primers and probes were subsequently optimized by means of that selected kinetic parameter. Finally, the linear correlation among DNA concentration and λc was also confirmed.
Goncharov, P. R.; Kuteev, B. V.; Ozaki, T.; Sudo, S.
2010-11-15
Analytical and semianalytical solutions have been obtained using a practical dimensionless form of Boltzmann kinetic equation assuming spatial homogeneity, azimuthal symmetry, and Maxwellian distributions of target plasma species. In contrast with formerly considered simplified equations with truncated collision terms, the exact Landau-Boltzmann collision operator is used, which conserves the number of particles, nullifies the collision term at statistical equilibrium, and describes the Maxwellization process naturally observed in correct solutions. A comparison with previous stationary and time-dependent analytical solutions is given. The new semianalytical results can be used in numerical modeling, for verification of solutions in more complex models, and in experimental data analysis, especially concerning nuclear processes and advanced localized, angle-resolved suprathermal particle diagnostics.
A covariant Fokker-Planck equation for a simple gas from relativistic kinetic theory
Chacon-Acosta, Guillermo; Dagdug, Leonardo; Morales-Tecotl, Hugo A.
2010-12-14
A manifestly covariant Fokker-Planck differential equation is derived for the case of a relativistic simple gas by taking a small momentum transfer approximation within the collision integral of the relativistic Boltzmann equation. We follow closely previous work, with the main difference that we keep manifest covariance at every stage of the analysis. In addition, we use the covariant Juettner distribution function to find a relativistic generalization of the Einstein's fluctuation-dissipation relation.
NASA Astrophysics Data System (ADS)
Shiwa, Y.
1988-02-01
The equation of motion for an interface in the presence of a gravitational field is considered, when the order parameter is conserved. The kinetic drumhead model is derived directly from the Cahn-Hilliard equation without recourse to the drumhead free energy. We append a systematic derivation of the drumhead free-energy functional for the interface in a nonuniform external field.
Wu, Zhen; Zhang, Xian; Zhou, Chunjiao; Pang, Jing-Lin; Zhang, Panyue
2017-02-22
Single-molecule aluminum salt AlCl3, medium polymerized polyaluminum chloride (PAC), and high polymerized polyaluminum chloride (HPAC) were prepared in a laboratory. The characteristics and coagulation properties of these prepared aluminum salts were investigated. The Langmuir, Freundlich, and Sips adsorption isotherms were first used to describe the adsorption neutralization process in coagulation, and the Boltzmann equation was used to fit the reaction kinetics of floc growth in flocculation. It was novel to find that the experimental data fitted well with the Sips and Boltzmann equation, and the significance of parameters in the equations was discussed simultaneously. Through the Sips equation, the adsorption neutralization reaction was proved to be spontaneous and the adsorption neutralization capacity was HPAC > PAC > AlCl3. Sips equation also indicated that the zeta potential of water samples would reach a limit with the increase of coagulant dosage, and the equilibrium zeta potential values were 30.25, 30.23, and 27.25 mV for AlCl3, PAC, and HPAC, respectively. The lower equilibrium zeta potential value of HPAC might be the reason why the water sample was not easy to achieve restabilization at a high coagulant dosage. Through the Boltzmann equation modeling, the maximum average floc size formed by AlCl3, PAC, and HPAC were 196.0, 188.0, and 203.6 μm, respectively, and the halfway time of reactions were 31.23, 17.08, and 9.55 min, respectively. The HPAC showed the strongest floc formation ability and the fastest floc growth rate in the flocculation process, which might be caused by the stronger adsorption and bridging functions of Alb and Alc contained in HPAC.
NASA Astrophysics Data System (ADS)
Maslov, Lev A.; Chebotarev, Vladimir I.
2017-02-01
The generalized logistic equation is proposed to model kinetics and statistics of natural processes such as earthquakes, forest fires, floods, landslides, and many others. This equation has the form dN(A)/dA = s dot (1-N(A)) dot N(A)q dot A-α, q>0q>0 and A>0A>0 is the size of an element of a structure, and α≥0. The equation contains two exponents α and q taking into account two important properties of elements of a system: their fractal geometry, and their ability to interact either to enhance or to damp the process of aggregation. The function N(A)N(A) can be understood as an approximation to the number of elements the size of which is less than AA. The function dN(A)/dAdN(A)/dA where N(A)N(A) is the general solution of this equation for q=1 is a product of an increasing bounded function and power-law function with stretched exponential cut-off. The relation with Tsallis non-extensive statistics is demonstrated by solving the generalized logistic equation for q>0q>0. In the case 0equation models super-additive, and the case q>1q>1 it models sub-additive structures. The Gutenberg-Richter (G-R) formula results from interpretation of empirical data as a straight line in the area of stretched exponent with small α. The solution is applied for modeling distribution of foreshocks and aftershocks in the regions of Napa Valley 2014, and Sumatra 2004 earthquakes fitting the observed data well, both qualitatively and quantitatively.
NASA Astrophysics Data System (ADS)
Privalov, T.; Shalagin, A.
1999-06-01
The interaction of a plane monochromatic traveling wave with two-level particles suffering collisions with buffer-gas particles is considered. Collision rates are assumed to be velocity dependent. The collision integral is obtained on the basis of the strong-collision model, generalized to the case of velocity-dependent collision rates (the so-called ``kangaroo'' model). We obtained the exact analytical solution of the problem for arbitrary intensity of radiation, arbitrary ratio of homogeneous and Doppler widths of the absorption line, and arbitrary mass ratio between absorbing- and buffer-gas particles. The obtained analytical solutions of the quantum kinetic equations allowed us to analyze the spectral shape of the strong-field absorption line as well as the probe-field absorption line (the nonlinear part of the work done by the probe field) and the frequency dependence of the light-induced drift (LID) velocity. A comprehensive comparative analysis for the three- and one-dimensional versions of the model is given. On the basis of this analysis, we reach the conclusion that the one-dimensional quantum kinetic equation has quite a wide range of application. We also reveal the conditions for the strongest manifestation of the velocity dependence of the collision rates, which affects most strongly the anomalous LID.
NASA Technical Reports Server (NTRS)
Emukashvily, I. M.
1982-01-01
An extension of the method of moments is developed for the numerical integration of the kinetic equations of droplet spectra evolution by condensation/evaporation and by coalescence/breakup processes. The number density function n sub k (x,t) in each separate droplet packet between droplet mass grid points (x sub k, x sub k+1) is represented by an expansion in orthogonal polynomials with a given weighting function. In this way droplet number concentrations, liquid water contents and other moments in each droplet packet are conserved and the problem of solving the kinetic equations is replaced by one of solving a set of coupled differential equations for the number density function moments. The method is tested against analytic solutions of the corresponding kinetic equations. Numerical results are obtained for different coalescence/breakup and condensation/evaporation kernels and for different initial droplet spectra. Also droplet mass grid intervals, weighting functions, and time steps are varied.
Feng, Xin; Ye, Xingyou; Park, Jun-Bom; Lu, Wenli; Morott, Joe; Beissner, Brad; Lian, Zhuoyang John; Pinto, Elanor; Bi, Vivian; Porter, Stu; Durig, Tom; Majumdar, Soumyajit; Repka, Michael A
2015-01-01
The recrystallization of an amorphous drug in a solid dispersion system could lead to a loss in the drug solubility and bioavailability. The primary objective of the current research was to use an improved kinetic model to evaluate the recrystallization kinetics of amorphous structures and to further understand the factors influencing the physical stability of amorphous solid dispersions. Amorphous solid dispersions of fenofibrate with different molecular weights of hydroxypropylcellulose, HPC (Klucel™ LF, EF, ELF) were prepared utilizing hot-melt extrusion technology. Differential scanning calorimetry was utilized to quantitatively analyze the extent of recrystallization in the samples stored at different temperatures and relative humidity (RH) conditions. The experimental data were fitted into the improved kinetics model of a modified Avrami equation to calculate the recrystallization rate constants. Klucel LF, the largest molecular weight among the HPCs used, demonstrated the greatest inhibition of fenofibrate recrystallization. Additionally, the recrystallization rate (k) decreased with increasing polymer content, however exponentially increased with higher temperature. Also k increased linearly rather than exponentially over the range of RH studied.
Döntgen, Malte; Leonhard, Kai
2017-03-02
Chemical activation of intermediates, such as hydrogen abstraction products, is emerging as a basis for a fully new reaction type: hot β-scission. While for thermally equilibrated intermediates chemical kinetics are typically orders of magnitude slower than relaxational kinetics, chemically activated intermediates raise the issue of inseparable chemical and relaxational kinetics. Here, this separation problem is discussed in the framework of master equation simulations, proposing three cases often encountered in chemistry: insignificant chemical activation, predominant chemical activation, and the transition between these two limits. These three cases are illustrated via three example systems: methoxy (CH3Ȯ), diazenyl (ṄNH), and methyl formate radicals (CH3OĊO). For diazenyl, it is found that hot β-scission fully replaces the sequence of hydrogen abstraction and β-scission of thermally equilibrated diazenyl. Building on the example systems, a rule of thumb is proposed that can be used to intuitively judge the significance of hot β-scission: if the reverse hydrogen abstraction barrier height is comparable to or larger than the β-scission barrier height, hot β-scission should be considered in more detail.
Fisher, Harvey F
2016-08-01
The transient-state kinetic approach has failed to reach its full potential despite its advantage over the steady-state approach in its ability to observe mechanistic events directly and in real time. This failure has been due in part to the lack of any rigorously derived and readily applicable body of theory corresponding to that which currently characterizes the steady-state approach. In order to clarify the causes of this discrepancy and to suggest a route to its solution we examine the capabilities and limitations of the various forms of transient-state kinetic approaches to the mathematical resolution of enzymatic reaction mechanisms currently available. We document a lack of validity inherent in their basic assumptions and suggest the need for a potentially more rigorous analytic approach.
Eggert, Matthew W; Byrne, Mark E; Chambers, Robert P
2012-10-01
In order to evaluate the effectiveness of aldehyde dehydrogenase (ALDH) from Saccharomyces cerevisiae as a catalyst for the conversion of acetaldehyde into its physiologically and biologically less toxic acetate, the kinetics over broad concentrations were studied to develop a suitable kinetic rate expression. Even with literature accounts of the binding complexations, the yeast ALDH currently lacks a quantitative kinetic rate expression accounting for simultaneous inhibition parameters under higher acetaldehyde concentrations. Both substrate acetaldehyde and product NADH were observed as individual sources of inhibition with the combined effect of a ternary complex of acetaldehyde and the coenzyme leading to experimental rates as little as an eighth of the expected activity. Furthermore, the onset and strength of inhibition from each component were directly affected by the concentration of the co-substrate NAD. While acetaldehyde inhibition of ALDH is initiated below concentrations of 0.05 mM in the presence of 0.5 mM NAD or less, the acetaldehyde inhibition onset shifts to 0.2 mM with as much as 1.6 mM NAD. The convenience of the statistical software package JMP allowed for effective determination of experimental kinetic constants and simplification to a suitable rate expression: v = Vmax(AB)/(K(ia)K(b) + K(b)A + K(a)B + AB + B(2)/K(I-Ald) + B(2)Q/K(I-Ald-NADH) + BQ/K(I-NADH)) where the last three terms represent the inhibition complex terms for acetaldehyde, acetaldehyde-NADH, and NADH, respectively. The corresponding values of K(I-Ald), K(I-Ald-NADH), and K(I-NADH) for yeast ALDH are 2.55, 0.0269, and 0.162 mM at 22 °C and pH 7.8.
Mohamad, Nur Ikhwan; Cronin, John B; Nosaka, Ken K
2012-01-01
Although it is generally accepted that a high load is necessary for muscle hypertrophy, it is possible that a low load with a high velocity results in greater kinematics and kinetics than does a high load with a slow velocity. The purpose of this study was to determine if 2 training loads (35 and 70% 1 repetition maximum [1RM]) equated by volume, differed in terms of their session kinematic and kinetic characteristics. Twelve subjects were recruited in this acute randomized within-subject crossover design study. Two bouts of a half-squat exercise were performed 1 week apart, one with high load-low velocity (HLLV = 3 sets of 12 reps at 70% 1RM) and the other with low-load high-velocity (LLHV = 6 sets of 12 reps at 35% 1RM). Time under tension (TUT), average force, peak force (PF), average power (AP), peak power (PP), work (TW), and total impulse (TI) were calculated and compared between loads for the eccentric and concentric phases. For average eccentric and concentric single repetition values, significantly (p < 0.05) greater (∼15-22%) PP outputs were associated with the LLHV loading, whereas significantly greater (∼7-61%) values were associated with the HLLV condition for most other variables of interest. However, in terms of total session kinematics and kinetics, the LLHV protocol resulted in significantly greater (∼16-61%) eccentric and concentric TUT, PF, AP, PP, and TW. The only variable that was significantly greater for the HLLV protocol than for the LLHV protocol was TI (∼20-24%). From these results, it seems that the LLHV protocol may offer an equal if not better training stimulus for muscular adaptation than the HLLV protocol, because of the greater time under tension, power, force, and work output when the total volume of the exercise is equated.
Acosta, María Lourdes; Sánchez, Asterio; García, Francisco; Molina, Emilio
2007-01-01
Batch cultures were carried out to study the kinetic, stoichiometry, and regulation of glucose and glutamine metabolism of a murine hybridoma line. Asymmetric logistic equations (ALEs) were used to fit total and viable cell density, and nutrient and metabolite/product concentrations. Since these equations were analytically differentiable, specific rates and yield coefficients were readily calculated. Asymmetric logistic equations described satisfactorily uncontrolled batch cultures, including death phase. Specific growth rate showed a Monod-type dependence on initial glucose and glutamine concentrations. Yield coefficients of cell and lactate from glucose, and cell and ammonium from glutamine were all found to change dramatically at low residual glucose and glutamine concentrations. Under stoichiometric glucose limitation, the glucose-to-cell yield increased and glucose-to-lactate yield decreased, indicating a metabolic shift. Under stoichiometric glutamine limitation the glutamine-to-cell and glutamine-to-ammonium yields increased, but also glucose-to-cell yield increased and the glucose-to-lactate yield decreased. Monoclonal antibody production was mainly non-growth associated, independently of glucose and glutamine levels. PMID:19003011
NASA Astrophysics Data System (ADS)
Smolyansky, S. A.; Prozorkevich, A. V.; Maino, G.; Mashnik, S. G.
1999-11-01
A generalized quantum relativistic kinetic equation (RKE) of the Kadanoff-Baym type is obtained on the basis of the Heisenberg equations of motion where the time evolution and space translation are separated from each other by means of the covariant method. The same approach is used also for a covariant modification of the real-time Green's functions method based on the Wigner representation. The suggested approach does not contain arbitrariness' elements and uncertainties which often arise from derivation of RKE on the basis of the motion equations of the Kadanoff-Baym type for the correlation functions in the case of systems with inner degrees of freedom. Possibilities of the proposed method are demonstrated by examples of derivation of RKE of the Vlasov type and collision integrals of the Boltzmann- Uehling-Uhlenbeck (BUU) type in the frame of the σω-version of quantum hadrodynamics, for the simplest case of spin saturated nuclear matter without antinuclear component. Here, the quasiparticle approximation in a covariant performance is used. A generalization of the method for the description of strong non-equilibrium states based on the non-equilibrium statistical operator is then proposed as well.
NASA Astrophysics Data System (ADS)
Lee, W. Wei-Li; Davidson, Ronald C.; Stoltz, Peter
1997-11-01
This paper presents a detailed formulation and analysis of the rate equations for statistically-averaged quantities for an intense nonneutral beam propagating through a periodic solenoidal focusing field. B^sol(x) = B_z(z)hatez - (1/2)B'_z(z)(xhatex + yhate_y), where B_z(z+S) = B_z(z), and S = const. is the axial periodicity length. The anaysis assumes a thin beam with characteristic beam radius rb << S, and is based on the nonlinear Vlasov-Maxwell equations. Particularly important in experimental applications and in numerical simulations schemes, such as the nonlinear δ f- scheme,(Q. Qian, W. Lee, and R. Davidson, Phys. Plasmas 4), 1915 (1997). is an understanding of the self-consistent nonlinear evolution of various quantities averaged over the distribution of beam particles f_b(x,p,t). Self-consistent rate equations are derived for the nonlinear evolution of the mean-square beam radius
Thermostatted kinetic equations as models for complex systems in physics and life sciences.
Bianca, Carlo
2012-12-01
Statistical mechanics is a powerful method for understanding equilibrium thermodynamics. An equivalent theoretical framework for nonequilibrium systems has remained elusive. The thermodynamic forces driving the system away from equilibrium introduce energy that must be dissipated if nonequilibrium steady states are to be obtained. Historically, further terms were introduced, collectively called a thermostat, whose original application was to generate constant-temperature equilibrium ensembles. This review surveys kinetic models coupled with time-reversible deterministic thermostats for the modeling of large systems composed both by inert matter particles and living entities. The introduction of deterministic thermostats allows to model the onset of nonequilibrium stationary states that are typical of most real-world complex systems. The first part of the paper is focused on a general presentation of the main physical and mathematical definitions and tools: nonequilibrium phenomena, Gauss least constraint principle and Gaussian thermostats. The second part provides a review of a variety of thermostatted mathematical models in physics and life sciences, including Kac, Boltzmann, Jager-Segel and the thermostatted (continuous and discrete) kinetic for active particles models. Applications refer to semiconductor devices, nanosciences, biological phenomena, vehicular traffic, social and economics systems, crowds and swarms dynamics.
Yessayan, Lenar; Yee, Jerry; Frinak, Stan; Kwon, David; Szamosfalvi, Balazs
2015-01-01
Concomitant severe metabolic alkalosis, hypernatremia, and kidney failure pose a therapeutic challenge. Hemodialysis to correct azotemia and abnormal electrolytes results in rapid correction of serum sodium, bicarbonate, and urea but presents a risk for dialysis disequilibrium and brain edema. We describe a patient with Zollinger-Ellison syndrome with persistent encephalopathy, severe metabolic alkalosis (highest bicarbonate 81 mEq/L), hypernatremia (sodium 157 mEq/L), and kidney failure despite 30 hours of intravenous crystalloids and proton pump inhibitor. We used continuous renal replacement therapy (RRT) with delivered hourly urea clearance of ~3 L/hour (24 hour sustained low efficiency dialysis with regional citrate anticoagulation protocol at blood flow rate 60 ml/min and dialysate flow rate 400 ml/min). To mitigate a pronounced decrease in plasma osmolality while removing urea from this hypernatremic patient, dialysate sodium was set to start at 155 mEq/L then at 150 mEq/L after 6 hours. Serum bicarbonate, urea, and sodium were slowly corrected over 26 hours. This case demonstrates how to regulate and predict the systemic bicarbonate level using single pool kinetic modeling during convective or diffusive RRT. Kinetic modeling provides a valuable tool for systemic blood pH control in future combined use of extracorporeal CO2 removal and continuous RRT systems.
Thermostatted kinetic equations as models for complex systems in physics and life sciences
NASA Astrophysics Data System (ADS)
Bianca, Carlo
2012-12-01
Statistical mechanics is a powerful method for understanding equilibrium thermodynamics. An equivalent theoretical framework for nonequilibrium systems has remained elusive. The thermodynamic forces driving the system away from equilibrium introduce energy that must be dissipated if nonequilibrium steady states are to be obtained. Historically, further terms were introduced, collectively called a thermostat, whose original application was to generate constant-temperature equilibrium ensembles. This review surveys kinetic models coupled with time-reversible deterministic thermostats for the modeling of large systems composed both by inert matter particles and living entities. The introduction of deterministic thermostats allows to model the onset of nonequilibrium stationary states that are typical of most real-world complex systems. The first part of the paper is focused on a general presentation of the main physical and mathematical definitions and tools: nonequilibrium phenomena, Gauss least constraint principle and Gaussian thermostats. The second part provides a review of a variety of thermostatted mathematical models in physics and life sciences, including Kac, Boltzmann, Jager-Segel and the thermostatted (continuous and discrete) kinetic for active particles models. Applications refer to semiconductor devices, nanosciences, biological phenomena, vehicular traffic, social and economics systems, crowds and swarms dynamics.
Renormalization group equations and matching in a general quantum field theory with kinetic mixing
NASA Astrophysics Data System (ADS)
Fonseca, Renato M.; Malinský, Michal; Staub, Florian
2013-11-01
We work out a set of simple rules for adopting the two-loop renormalization group equations of a generic gauge field theory given in the seminal works of Machacek and Vaughn to the most general case with an arbitrary number of Abelian gauge factors and comment on the extra subtleties possibly encountered upon matching a set of effective gauge theories in such a framework.
Master equation for a kinetic model of a trading market and its analytic solution.
Chatterjee, Arnab; Chakrabarti, Bikas K; Stinchcombe, Robin B
2005-08-01
We analyze an ideal-gas-like model of a trading market with quenched random saving factors for its agents and show that the steady state income (m) distribution P(m) in the model has a power law tail with Pareto index nu exactly equal to unity, confirming the earlier numerical studies on this model. The analysis starts with the development of a master equation for the time development of P(m) . Precise solutions are then obtained in some special cases.
A Kinetic Theory Development of the Equations of Motion of a Diatomic Gas.
1986-09-01
excitation, or nuclear excitation. At moderate temperatures, however, both quantum mechanics and statistical thermodynamics show the rigid rotor to be an...distribution function Yf L,’(P , Nt) in the gas phase * space (PNqN) of systems of N molecules is the fundamental equation of classical statistical mechanics ...and Liquids, John Wiley and Sons., New York, 1954. 3. McQuarrie , D. A., Statistical Ther.modynamics, Harper & Row, New York, 1973. h..-A: 4. Bolz, R. G
Efficient kinetic method for fluid simulation beyond the Navier-Stokes equation.
Zhang, Raoyang; Shan, Xiaowen; Chen, Hudong
2006-10-01
We present a further theoretical extension to the kinetic-theory-based formulation of the lattice Boltzmann method of Shan [J. Fluid Mech. 550, 413 (2006)]. In addition to the higher-order projection of the equilibrium distribution function and a sufficiently accurate Gauss-Hermite quadrature in the original formulation, a regularization procedure is introduced in this paper. This procedure ensures a consistent order of accuracy control over the nonequilibrium contributions in the Galerkin sense. Using this formulation, we construct a specific lattice Boltzmann model that accurately incorporates up to third-order hydrodynamic moments. Numerical evidence demonstrates that the extended model overcomes some major defects existing in conventionally known lattice Boltzmann models, so that fluid flows at finite Knudsen number Kn can be more quantitatively simulated. Results from force-driven Poiseuille flow simulations predict the Knudsen's minimum and the asymptotic behavior of flow flux at large Kn.
NASA Technical Reports Server (NTRS)
Manning, Robert M.
2009-01-01
Based on a theoretical model of the propagation of electromagnetic waves through a hypersonically induced plasma, it has been demonstrated that the classical radiofrequency communications blackout that is experienced during atmospheric reentry can be mitigated through the appropriate control of an external magnetic field of nominal magnitude. The model is based on the kinetic equation treatment of Vlasov and involves an analytical solution for the electric and magnetic fields within the plasma allowing for a description of the attendant transmission, reflection and absorption coefficients. The ability to transmit through the magnetized plasma is due to the magnetic windows that are created within the plasma via the well-known whistler modes of propagation. The case of 2 GHz transmission through a re-entry plasma is considered. The coefficients are found to be highly sensitive to the prevailing electron density and will thus require a dynamic control mechanism to vary the magnetic field as the plasma evolves through the re-entry phase.
NASA Astrophysics Data System (ADS)
Wong, S. K.; Chan, V. S.; Hinton, F. L.
2001-10-01
The classic solution of the linearized drift kinetic equations in neoclassical transport theory for large-aspect-ratio tokamak flux-surfaces relies on the variational principle and the choice of ``localized" distribution functions as trialfunctions.(M.N. Rosenbluth, et al., Phys. Fluids 15) (1972) 116. Somewhat unclear in this approach are the nature and the origin of the ``localization" and whether the results obtained represent the exact leading terms in an asymptotic expansion int he inverse aspect ratio. Using the method of matched asymptotic expansions, we were able to derive the leading approximations to the distribution functions and demonstrated the asymptotic exactness of the existing results. The method is also applied to the calculation of angular momentum transport(M.N. Rosenbluth, et al., Plasma Phys. and Contr. Nucl. Fusion Research, 1970, Vol. 1 (IAEA, Vienna, 1971) p. 495.) and the current driven by electron cyclotron waves.
Regularization of Grad’s 13 -Moment-Equations in Kinetic Gas Theory
2011-01-01
variant of the moment method has been proposed by Eu (1980) and is used, e.g., in Myong (2001). Recently, a maximum- entropy 10-moment system has been used...small amplitude linear waves, the R13 system is linearly stable in time for all modes and wave lengths. The instability of the Burnett system indicates...Boltzmann equation. Related to the problem of global hyperbolicity is the questions of the existence of an entropy law for the R13 system . In the linear
NASA Astrophysics Data System (ADS)
Saylor, Rick D.; Ford, Gregory D.
The integration of systems of ordinary differential equations (ODEs) that arise in atmospheric photochemistry is of significant concern to tropospheric and stratospheric chemistry modelers. As a consequence of the stiff nature of these ODE systems, their solution requires a large fraction of the total computational effort in three-dimensional chemical model simulations. Several integration techniques have been proposed and utilized over the years in an attempt to provide computationally efficient, yet accurate, solutions to chemical kinetics ODES. In this work, we present a comparison of some of these techniques and argue that valid comparisons of ODE solvers must take into account the trade-off between solution accuracy and computational efficiency. Misleading comparison results can be obtained by neglecting the fact that any ODE solution method can be made faster or slower by manipulation of the appropriate error tolerances or time steps. Comparisons among ODE solution techniques should therefore attempt to identify which technique can provide the most accurate solution with the least computational effort over the entire range of behavior of each technique. We present here a procedure by which ODE solver comparisons can achieve this goal. Using this methodology, we compare a variety of integration techniques, including methods proposed by Hesstvedt et al. (1978, Int. J. Chem. Kinet.10, 971-994), Gong and Cho (1993, Atmospheric Environment27A, 2147-2160), Young and Boris (1977, J. phys. Chem.81, 2424-2427) and Hindmarsh (1983, In Scientific Computing (edited by Stepleman R. S. et al.), pp. 55-64. North-Holland, Amsterdam). We find that Gear-type solvers such as the Livermore Solver for ordinary differential equations (LSODE) and the sparse-matrix version of LSODE (LSODES) provide the most accurate solution of our test problems with the least computational effort.
Grima, R
2010-07-21
Chemical master equations provide a mathematical description of stochastic reaction kinetics in well-mixed conditions. They are a valid description over length scales that are larger than the reactive mean free path and thus describe kinetics in compartments of mesoscopic and macroscopic dimensions. The trajectories of the stochastic chemical processes described by the master equation can be ensemble-averaged to obtain the average number density of chemical species, i.e., the true concentration, at any spatial scale of interest. For macroscopic volumes, the true concentration is very well approximated by the solution of the corresponding deterministic and macroscopic rate equations, i.e., the macroscopic concentration. However, this equivalence breaks down for mesoscopic volumes. These deviations are particularly significant for open systems and cannot be calculated via the Fokker-Planck or linear-noise approximations of the master equation. We utilize the system-size expansion including terms of the order of Omega(-1/2) to derive a set of differential equations whose solution approximates the true concentration as given by the master equation. These equations are valid in any open or closed chemical reaction network and at both the mesoscopic and macroscopic scales. In the limit of large volumes, the effective mesoscopic rate equations become precisely equal to the conventional macroscopic rate equations. We compare the three formalisms of effective mesoscopic rate equations, conventional rate equations, and chemical master equations by applying them to several biochemical reaction systems (homodimeric and heterodimeric protein-protein interactions, series of sequential enzyme reactions, and positive feedback loops) in nonequilibrium steady-state conditions. In all cases, we find that the effective mesoscopic rate equations can predict very well the true concentration of a chemical species. This provides a useful method by which one can quickly determine the
NASA Astrophysics Data System (ADS)
Malkov, Ewgenij A.; Poleshkin, Sergey O.; Kudryavtsev, Alexey N.; Shershnev, Anton A.
2016-10-01
The paper presents the software implementation of the Boltzmann equation solver based on the deterministic finite-difference method. The solver allows one to carry out parallel computations of rarefied flows on a hybrid computational cluster with arbitrary number of central processor units (CPU) and graphical processor units (GPU). Employment of GPUs leads to a significant acceleration of the computations, which enables us to simulate two-dimensional flows with high resolution in a reasonable time. The developed numerical code was validated by comparing the obtained solutions with the Direct Simulation Monte Carlo (DSMC) data. For this purpose the supersonic flow past a flat plate at zero angle of attack is used as a test case.
Fu, Mingkun; Perlman, Michael; Lu, Qing; Varga, Csanad
2015-03-25
An accelerated stress approach utilizing the moisture-modified Arrhenius equation and JMP statistical software was utilized to quantitatively assess the solid state stability of an investigational oncology drug MLNA under the influence of temperature (1/T) and humidity (%RH). Physical stability of MLNA under stress conditions was evaluated by using XRPD, DSC, TGA, and DVS, while chemical stability was evaluated by using HPLC. The major chemical degradation product was identified as a hydrolysis product of MLNA drug substance, and was subsequently subjected to an investigation of kinetics based on the isoconversion concept. A mathematical model (ln k=-11,991×(1/T)+0.0298×(%RH)+29.8823) based on the initial linear kinetics observed for the formation of this degradant at all seven stress conditions was built by using the moisture-modified Arrhenius equation and JMP statistical software. Comparison of the predicted versus experimental lnk values gave a mean deviation value of 5.8%, an R(2) value of 0.94, a p-value of 0.0038, and a coefficient of variation of the root mean square error CV(RMSE) of 7.9%. These statistics all indicated a good fit to the model for the stress data of MLNA. Both temperature and humidity were shown to have a statistically significant impact on stability by using effect leverage plots (p-value<0.05 for both 1/T and %RH). Inclusion of a term representing the interaction of relative humidity and temperature (%RH×1/T) was shown not to be justified by using Analysis of Covariance (ANCOVA), which supported the use of the moisture-corrected Arrhenius equation modeling theory. The model was found to be of value to aid setting of specifications and retest period, and storage condition selection. A model was also generated using only four conditions, as an example from a resource saving perspective, which was found to provide a good fit to the entire set of data. Copyright © 2015 Elsevier B.V. All rights reserved.
NASA Astrophysics Data System (ADS)
Qureshi, Mumnuna A.; Zhong, Johnny; Betouras, Joseph J.; Zagoskin, Alexandre M.
2017-03-01
Theoretical description and simulation of large quantum coherent systems out of equilibrium remains a daunting task. Here we are developing an approach to it based on the Pechukas-Yukawa formalism, which is especially convenient in the case of an adiabatically slow external perturbation, though it is not restricted to adiabatic systems. In this formalism the dynamics of energy levels in an externally perturbed quantum system as a function of the perturbation parameter is mapped on that of a fictitious one-dimensional classical gas of particles with cubic repulsion. Equilibrium statistical mechanics of this Pechukas gas allows us to reproduce the random matrix theory of energy levels. In the present work, we develop the nonequilibrium statistical mechanics of the Pechukas gas, starting with the derivation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) chain of equations for the appropriate generalized distribution functions. Sets of approximate kinetic equations can be consistently obtained by breaking this chain at a particular point (i.e., approximating all higher-order distribution functions by the products of the lower-order ones). When complemented by the equations for the level occupation numbers and interlevel transition amplitudes, they allow us to describe the nonequilibrium evolution of the quantum state of the system, which can describe better a large quantum coherent system than the currently used approaches. In particular, we find that corrections to the factorized approximation of the distribution function scale as 1 /N , where N is the number of the "Pechukas gas particles" (i.e., energy levels in the system).
NASA Astrophysics Data System (ADS)
Babintsev, I. A.; Adzhemyan, L. Ts.; Shchekin, A. K.
2017-08-01
Relaxation of micellar systems can be described with the help of the Becker-Döring kinetic difference equations for aggregate concentrations. Passing in these equations to continual description, when the aggregation number is considered as continuous variable and the concentration difference is replaced by the concentration differential, allows one to find analytically the eigenvalues (to whom the inverse times of micellar relaxation are related) and eigenfunctions (or the modes of fast relaxation) of the linearized differential operator of the kinetic equation corresponding to the Fokker-Planck approximation. At this the spectrum of eigenvalues appears to be degenerated at some surfactant concentrations. However, as has been recently found by us, there is no such a degeneracy at numerical determination of the eigenvalues of the matrix of coefficients for the linearized difference Becker-Döring equations. It is shown in this work in the frameworks of the perturbation theory, that taking into account the corrections to the kinetic equation produced by second derivatives at transition from differences to differentials and by deviation of the aggregation work from a parabolic form in the vicinity of the work minimum, lifts the degeneracy of eigenvalues and improves markedly the agreement of concentration-dependent fast relaxation time with the results of the numerical solution of the linearized Becker-Döring difference equations.
Inference for Stochastic Chemical Kinetics Using Moment Equations and System Size Expansion
Thomas, Philipp; Kazeroonian, Atefeh; Theis, Fabian J.; Grima, Ramon; Hasenauer, Jan
2016-01-01
Quantitative mechanistic models are valuable tools for disentangling biochemical pathways and for achieving a comprehensive understanding of biological systems. However, to be quantitative the parameters of these models have to be estimated from experimental data. In the presence of significant stochastic fluctuations this is a challenging task as stochastic simulations are usually too time-consuming and a macroscopic description using reaction rate equations (RREs) is no longer accurate. In this manuscript, we therefore consider moment-closure approximation (MA) and the system size expansion (SSE), which approximate the statistical moments of stochastic processes and tend to be more precise than macroscopic descriptions. We introduce gradient-based parameter optimization methods and uncertainty analysis methods for MA and SSE. Efficiency and reliability of the methods are assessed using simulation examples as well as by an application to data for Epo-induced JAK/STAT signaling. The application revealed that even if merely population-average data are available, MA and SSE improve parameter identifiability in comparison to RRE. Furthermore, the simulation examples revealed that the resulting estimates are more reliable for an intermediate volume regime. In this regime the estimation error is reduced and we propose methods to determine the regime boundaries. These results illustrate that inference using MA and SSE is feasible and possesses a high sensitivity. PMID:27447730
NASA Astrophysics Data System (ADS)
Morawetz, K.
2015-12-01
The coupled kinetic equation for density and spin Wigner functions is derived including spin-orbit coupling, electric and magnetic fields, and self-consistent Hartree mean fields suited for SU(2) transport. The interactions are assumed to be with scalar and magnetic impurities as well as scalar and spin-flip potentials among the particles. The spin-orbit interaction is used in a form suitable for solid state physics with Rashba or Dresselhaus coupling, graphene, extrinsic spin-orbit coupling, and effective nuclear matter coupling. The deficiencies of the two-fluid model are worked out consisting of the appearance of an effective in-medium spin precession. The stationary solution of all these systems shows a band splitting controlled by an effective medium-dependent Zeeman field. The self-consistent precession direction is discussed and a cancellation of linear spin-orbit coupling at zero temperature is reported. The precession of spin around this effective direction caused by spin-orbit coupling leads to anomalous charge and spin currents in an electric field. Anomalous Hall conductivity is shown to consist of the known results obtained from the Kubo formula or Berry phases and a symmetric part interpreted as an inverse Hall effect. Analogously the spin-Hall and inverse spin-Hall effects of spin currents are discussed which are present even without magnetic fields showing a spin accumulation triggered by currents. The analytical dynamical expressions for zero temperature are derived and discussed in dependence on the magnetic field and effective magnetizations. The anomalous Hall and spin-Hall effect changes sign at higher than a critical frequency dependent on the relaxation time.
NASA Astrophysics Data System (ADS)
Shchekin, Alexander K.; Babintsev, Ilya A.; Adzhemyan, Loran Ts.
2016-11-01
Full-time kinetics of self-assembly and disassembly of spherical micelles with their fusion and fission in non-ionic micellar solutions has been considered in detail on the basis of direct numerical solutions of the generalized Smoluchowski equations describing the evolution of the time-dependent concentrations of molecular aggregates for every aggregation number. The cases of instant increase of the monomer concentration up or dilution of a surfactant solution below the critical micelle concentration at large initial deviations from the final equilibrium state have been studied. Different stages in assembly or disassembly of micelles have been described and compared with the results of the stepwise mechanism of monomer attachment-detachment described by the Becker-Döring kinetic equations. A relation of the full-time kinetics to micellar relaxation at small deviations from the equilibrium state has been checked.
Igor D. Kaganovich; Oleg Polomarov
2003-05-19
In low-pressure discharges, when the electron mean free path is larger or comparable with the discharge length, the electron dynamics is essentially non-local. Moreover, the electron energy distribution function (EEDF) deviates considerably from a Maxwellian. Therefore, an accurate kinetic description of the low-pressure discharges requires knowledge of the non-local conductivity operator and calculation of the non-Maxwellian EEDF. The previous treatments made use of simplifying assumptions: a uniform density profile and a Maxwellian EEDF. In the present study a self-consistent system of equations for the kinetic description of nonlocal, non-uniform, nearly collisionless plasmas of low-pressure discharges is derived. It consists of the nonlocal conductivity operator and the averaged kinetic equation for calculation of the non-Maxwellian EEDF. The importance of accounting for the non-uniform plasma density profile on both the current density profile and the EEDF is demonstrated.
NASA Astrophysics Data System (ADS)
Berim, Gersh O.; Ruckenstein, Eli
2003-11-01
A generalized kinetic Ising model is applied to the description of phase transformations in lattice systems. A procedure, based on the conjecture that the probability distribution function of the states of the system is similar to the equilibrium one, is used for closing the infinite chain of kinetic equations. The method is illustrated by treating as an example the one-dimensional Ising model. The predicted rate of phase transformation (RPT) demonstrates various time behaviors dependent upon the details of the interactions between spins and a heat bath. If the parameters W0 and W the reciprocals of which characterize, respectively, the time scales of growth (decay) and splitting (coagulation) of clusters have the same order of magnitude, then the RPT is constant during almost the entire transformation process. For the case W=0, which corresponds to the absence of splitting and coagulation of clusters, the phase transformation follows an exponential law in the final stage and is linear with respect to time during the initial one. It has a similar behavior for W0≫W≠0; however, the RPT in the final stage is much smaller in the last case than for W=0. In the absence of supersaturation, RPT decreases to zero as T→Tc, where Tc(=0 K) is the phase transition temperature for a one-dimensional model. The time-dependent size distribution of clusters is for all times exponential with respect to the cluster size. The average size of the cluster far from both equilibrium and initial state grows linearly in time. Both the above quantities behave in a manner similar to those obtained by Monte Carlo simulations for systems of higher dimension.
NASA Astrophysics Data System (ADS)
Winter, Pierre M.; Rheaume, Michael; Cooksy, Andrew L.
2017-08-01
We have calculated the temperature-dependent rate coefficients of the addition reactions of butadien-2-yl (C4H5) and acroylyl (C3H3O) radicals with ethene (C2H4), carbon monoxide (CO), formaldehyde (H2CO), hydrogen cyanide (HCN), and ketene (H2CCO), in order to explore the balance between kinetic and thermodynamic control in these combustion-related reactions. For the C4H5 radical, the 1,3-diene form of the addition products is more stable than the 1,2-diene, but the 1,2-diene form of the radical intermediate is stabilized by an allylic delocalization, which may influence the relative activation energies. For the reactions combining C3H3O with C2H4, CO, and HCN, the opposite is true: the 1,2-enone form of the addition products is more stable than the 1,3-enone, whereas the 1,3-enone is the slightly more stable radical species. Optimized geometries and vibrational modes were computed with the QCISD/aug-cc-pVDZ level and basis, followed by single-point CCSD(T)-F12a/cc-pVDZ-F12 energy calculations. Our findings indicate that the kinetics in all cases favor reaction along the 1,3 pathway for both the C4H5 and C3H3O systems. The Rice-Ramsperger-Kassel-Marcus (RRKM) microcanonical rate coefficients and subsequent solution of the chemical master equation were used to predict the time-evolution of our system under conditions from 500 K to 2000 K and from 10-5 bar to 10 bars. Despite the 1,3 reaction pathway being more favorable for the C4H5 system, our results predict branching ratios of the 1,2 to 1,3 product as high as 0.48 at 1 bar. Similar results hold for the acroylyl system under these combustion conditions, suggesting that under kinetic control the branching of these reactions may be much more significant than the thermodynamics would suggest. This effect may be partly attributed to the low energy difference between 1,2 and 1,3 forms of the radical intermediate. No substantial pressure-dependence is found for the overall forward reaction rates until pressures
Winter, Pierre M; Rheaume, Michael; Cooksy, Andrew L
2017-08-07
We have calculated the temperature-dependent rate coefficients of the addition reactions of butadien-2-yl (C4H5) and acroylyl (C3H3O) radicals with ethene (C2H4), carbon monoxide (CO), formaldehyde (H2CO), hydrogen cyanide (HCN), and ketene (H2CCO), in order to explore the balance between kinetic and thermodynamic control in these combustion-related reactions. For the C4H5 radical, the 1,3-diene form of the addition products is more stable than the 1,2-diene, but the 1,2-diene form of the radical intermediate is stabilized by an allylic delocalization, which may influence the relative activation energies. For the reactions combining C3H3O with C2H4, CO, and HCN, the opposite is true: the 1,2-enone form of the addition products is more stable than the 1,3-enone, whereas the 1,3-enone is the slightly more stable radical species. Optimized geometries and vibrational modes were computed with the QCISD/aug-cc-pVDZ level and basis, followed by single-point CCSD(T)-F12a/cc-pVDZ-F12 energy calculations. Our findings indicate that the kinetics in all cases favor reaction along the 1,3 pathway for both the C4H5 and C3H3O systems. The Rice-Ramsperger-Kassel-Marcus (RRKM) microcanonical rate coefficients and subsequent solution of the chemical master equation were used to predict the time-evolution of our system under conditions from 500 K to 2000 K and from 10(-5) bar to 10 bars. Despite the 1,3 reaction pathway being more favorable for the C4H5 system, our results predict branching ratios of the 1,2 to 1,3 product as high as 0.48 at 1 bar. Similar results hold for the acroylyl system under these combustion conditions, suggesting that under kinetic control the branching of these reactions may be much more significant than the thermodynamics would suggest. This effect may be partly attributed to the low energy difference between 1,2 and 1,3 forms of the radical intermediate. No substantial pressure-dependence is found for the overall forward reaction rates until pressures
NASA Astrophysics Data System (ADS)
Kernbichler, W.; Kasilov, S. V.; Kapper, G.; Martitsch, A. F.; Nemov, V. V.; Albert, C.; Heyn, M. F.
2016-11-01
NEO-2 is a linearized drift kinetic equation solver for three-dimensional toroidal magnetic fields. It has been designed in order to treat effectively—besides all other regimes—the long mean free path regime, avoiding any simplifications on device geometry or on the Coulomb collision model. The code is based on the field line integration technique combined with a multiple domain decomposition approach, which allows for introduction of an adaptive grid in velocity space. This makes NEO-2 capable of effectively resolving all boundary layers between various classes of trapped particles and passing particles, and also allows for straightforward code parallelization. In stellarators, NEO-2 is used mainly for computations of neoclassical transport coefficients in regimes with slow plasma rotation and for the evaluation of the generalized Spitzer function, which plays the role of a current drive efficiency. In tokamaks with small ideal non-axisymmetric magnetic field perturbations, NEO-2 is used for evaluation of the toroidal torque resulting from these perturbations (neoclassical toroidal viscosity). The limitation to slow plasma rotation pertinent to usage in stellarators has been removed in this case with the help of a quasilinear approach, which is valid due to the smallness of the perturbation field.
NASA Technical Reports Server (NTRS)
Pratt, D. T.
1984-01-01
Conventional algorithms for the numerical integration of ordinary differential equations (ODEs) are based on the use of polynomial functions as interpolants. However, the exact solutions of stiff ODEs behave like decaying exponential functions, which are poorly approximated by polynomials. An obvious choice of interpolant are the exponential functions themselves, or their low-order diagonal Pade (rational function) approximants. A number of explicit, A-stable, integration algorithms were derived from the use of a three-parameter exponential function as interpolant, and their relationship to low-order, polynomial-based and rational-function-based implicit and explicit methods were shown by examining their low-order diagonal Pade approximants. A robust implicit formula was derived by exponential fitting the trapezoidal rule. Application of these algorithms to integration of the ODEs governing homogenous, gas-phase chemical kinetics was demonstrated in a developmental code CREK1D, which compares favorably with the Gear-Hindmarsh code LSODE in spite of the use of a primitive stepsize control strategy.
NASA Technical Reports Server (NTRS)
Pratt, D. T.
1984-01-01
Conventional algorithms for the numerical integration of ordinary differential equations (ODEs) are based on the use of polynomial functions as interpolants. However, the exact solutions of stiff ODEs behave like decaying exponential functions, which are poorly approximated by polynomials. An obvious choice of interpolant are the exponential functions themselves, or their low-order diagonal Pade (rational function) approximants. A number of explicit, A-stable, integration algorithms were derived from the use of a three-parameter exponential function as interpolant, and their relationship to low-order, polynomial-based and rational-function-based implicit and explicit methods were shown by examining their low-order diagonal Pade approximants. A robust implicit formula was derived by exponential fitting the trapezoidal rule. Application of these algorithms to integration of the ODEs governing homogenous, gas-phase chemical kinetics was demonstrated in a developmental code CREK1D, which compares favorably with the Gear-Hindmarsh code LSODE in spite of the use of a primitive stepsize control strategy.
NASA Astrophysics Data System (ADS)
Liu, F.; Schaller, E.; Mott, D. R.
2005-08-01
A major task in many applications of atmospheric chemistry transport problems is the numerical integration of stiff systems of Ordinary Differential Equations (ODEs) describing the chemical transformations. A faster solver that is easier to couple to the other physics in the problem is still needed. The integration method, α-QSS, corresponding to the solver CHEMEQ2 aims at meeting the demands of a process-split, reacting-flow simulation (Mott 2000; Mott and Oran, 2001). However, this integrator has yet to be applied to the numerical integration of kinetic equations in tropospheric chemistry. A zero-dimensional (box) model is developed to test how well CHEMEQ2 works on the tropospheric chemistry equations. This paper presents the testing results. The reference chemical mechanisms herein used are Regional Atmospheric Chemistry Mechanism (RACM) (Stockwell et al., 1997) and its secondary lumped successor Regional Lumped Atmospheric Chemical Scheme (ReLACS) (Crassier et al., 2000). The box model is forced and initialized by the DRY scenarios of Protocol Ver. 2 developed by EUROTRAC (Poppe et al., 2001). The accuracy of CHEMEQ2 is evaluated by comparing the results to solutions obtained with VODE. This comparison is made with parameters of the error tolerance, relative difference with respect to VODE scheme, trade off between accuracy and efficiency, global time step for integration etc. The study based on the comparison concludes that the single-point α-QSS approach is fast and moderately accurate as well as easy to couple to reacting flow simulation models, which makes CHEMEQ2 one of the best candidates for three-dimensional atmospheric Chemistry Transport Modelling (CTM) studies. In addition the RACM mechanism may be replaced by ReLACS mechanism for tropospheric chemistry transport modelling. The testing results also imply that the accuracy for chemistry numerical simulations is highly different from species to species. Therefore ozone is not the good choice for
NASA Astrophysics Data System (ADS)
Ersoy, Mehmet; Lakkis, Omar; Townsend, Philip
2016-04-01
The flow of water in rivers and oceans can, under general assumptions, be efficiently modelled using Saint-Venant's shallow water system of equations (SWE). SWE is a hyperbolic system of conservation laws (HSCL) which can be derived from a starting point of incompressible Navier-Stokes. A common difficulty in the numerical simulation of HSCLs is the conservation of physical entropy. Work by Audusse, Bristeau, Perthame (2000) and Perthame, Simeoni (2001), proposed numerical SWE solvers known as kinetic schemes (KSs), which can be shown to have desirable entropy-consistent properties, and are thus called well-balanced schemes. A KS is derived from kinetic equations that can be integrated into the SWE. In flood risk assessment models the SWE must be coupled with other equations describing interacting meteorological and hydrogeological phenomena such as rain and groundwater flows. The SWE must therefore be appropriately modified to accommodate source and sink terms, so kinetic schemes are no longer valid. While modifications of SWE in this direction have been recently proposed, e.g., Delestre (2010), we depart from the extant literature by proposing a novel model that is "entropy-consistent" and naturally extends the SWE by respecting its kinetic formulation connections. This allows us to derive a system of partial differential equations modelling flow of a one-dimensional river with both a precipitation term and a groundwater flow model to account for potential infiltration and recharge. We exhibit numerical simulations of the corresponding kinetic schemes. These simulations can be applied to both real world flood prediction and the tackling of wider issues on how climate and societal change are affecting flood risk.
NASA Astrophysics Data System (ADS)
Winkler, R. B.; Wilhelm, J.; Winkler, R.
A kinetic description of the dc Hg-Ar mixture plasma of fluorescent lamp discharges is given on the basis of the main microphysical processes and corresponding atomic data. The investigation comprises particle balances for the excited Hg atoms of the triplet 63P0,1,2 and singulet 61P1 state, the balance of the discharge current and an adequate form of the electron Boltzmann equation for such a mixture plasma, the latter including also the various binary collision processes with excited Hg atoms and the Coulomb interaction between the electrons. By simultaneously solving the balances together with the kinetic equation the densities of electrons and excited Hg atoms, the electron energy distribution, the resulting mean energy, mobility and various energy transfer rates of the electrons and the ultraviolet radiation output were determined and in part compared with measured data from the literature. This paper presents and analyses the macroscopic plasma properties mentioned for a wide range of the wall temperature or the corresponding Hg partial pressure variation, where a good agreement of calculated results with experimental data is obtained. In a second paper the change of the macroscopic plasma behaviour due to variation of buffer gas pressure and discharge current will be dealt with.Translated AbstractKinetik des Ar-Hg-Plasmas von Leuchtstofflampenentladungen I. Modell - Grundgleichungen - Hg-PartialdruckvariationDie Arbeit befaßt sich auf der Grundlage der wesentlichen mikrophysikalischen Prozesse und der entsprechenden atomaren Daten mit der kinetischen Beschreibung des de Hg-Ar-Mischplasmas von Leuchtstofflampenentladungen. Die Untersuchung verwendet die Teilchenbilanzen für die angeregten Hg-Atome des Tripletts 63P0,1,2 und des Singuletts 61P1, die Bilanz für den Entladungsstrom und eine adäquate Form der Elektronen-Boltzmann-Gleichung für ein solches Mischplasma, wobei in letzterer auch die verschiedenen Zweierstoßprozesse mit angeregten Hg-Atomen und
Bezerra, Rui M F; Fraga, Irene; Dias, Albino A
2013-01-01
Enzyme kinetic parameters are usually determined from initial rates nevertheless, laboratory instruments only measure substrate or product concentration versus reaction time (progress curves). To overcome this problem we present a methodology which uses integrated models based on Michaelis-Menten equation. The most severe practical limitation of progress curve analysis occurs when the enzyme shows a loss of activity under the chosen assay conditions. To avoid this problem it is possible to work with the same experimental points utilized for initial rates determination. This methodology is illustrated by the use of integrated kinetic equations with the well-known reaction catalyzed by alkaline phosphatase enzyme. In this work nonlinear regression was performed with the Solver supplement (Microsoft Office Excel). It is easy to work with and track graphically the convergence of SSE (sum of square errors). The diagnosis of enzyme inhibition was performed according to Akaike information criterion.
Sharipov, Felix; Kalempa, Denize
2008-10-01
The sound propagation through a rarefied gas is investigated on the basis of the linearized kinetic equation. A plate oscillating in the direction normal to its own plane is considered as a sound wave source. It is assumed a fully established oscillation so that the solution of the kinetic equation depends on time harmonically, while its dependence on the spatial coordinates is obtained numerically. The problem is solved over a wide range of the oscillation speed parameter defined as a ratio of the intermolecular collision frequency to the sound frequency. In order to evaluate the influence of the momentum and energy accommodation coefficients on the solution of the problem, the Cercignani-Lampis scattering kernel is applied as the boundary condition. An analysis of wave characteristics near the source surface shows that they are significantly different from those far from the surface even if the oscillation is slow, i.e., the solution is not harmonic in the space.
NASA Astrophysics Data System (ADS)
Li, David Z.; Anderson, R. H.; Miller, M. D.; Crowell, Ethan
2014-07-01
We examine in detail the method introduced by Sanchez-Castro, Bedell, and Wiegers (SBW) to solve Landau’s linearized kinetic equation, and compare it with the well-known standard method introduced by Abrikosov and Khalatnikov (AK). The SBW approach, hardly known, differs from AK in the way that moments are taken with respect to the angular functions of the Fourier transformed kinetic equation. We compare the SBW and AK solutions for zero-sound and first-sound propagation speeds and attenuation both analytically in the zero and full polarization limits, and numerically at arbitrary polarization using Landau parameters appropriate for thin 3He films. We find that the lesser known method not only yields results in close agreement with the standard method, but in most cases does so with far less analytic and computational effort.
NASA Astrophysics Data System (ADS)
Davidenko, V. D.; Zinchenko, A. S.; Harchenko, I. K.
2016-12-01
Integral equations for the shape functions in the adiabatic, quasi-static, and improved quasi-static approximations are presented. The approach to solving these equations by the Monte Carlo method is described.
Davidenko, V. D. Zinchenko, A. S. Harchenko, I. K.
2016-12-15
Integral equations for the shape functions in the adiabatic, quasi-static, and improved quasi-static approximations are presented. The approach to solving these equations by the Monte Carlo method is described.
NASA Astrophysics Data System (ADS)
Vodolazov, D. Yu.
2017-03-01
Using a kinetic-equation approach, we study the dynamics of electrons and phonons in current-carrying superconducting nanostrips after the absorption of a single photon of the near-infrared or optical range. We find that the larger the Ce/Cph|Tc ratio (where Tc is the critical temperature of a superconductor and Ce and Cph are specific heat capacities of electrons and phonons, respectively), the larger the portion of the photon's energy goes to electrons. The electrons become more strongly heated and hence can thermalize faster during the initial stage of hot-spot formation. The thermalization time τth can be less than 1 ps for superconductors with Ce/Cph|Tc≫1 and a small diffusion coefficient of D ≃0.5 cm2/s when thermalization occurs, mainly due to electron-phonon and phonon-electron scattering in a relatively small volume of approximately ξ2d (ξ is a superconducting coherence length, while d <ξ is a thickness of the strip). For longer time spans, due to diffusion of hot electrons' effective temperature inside the hot spot decreases, the size of the hot spot increases, the superconducting state becomes unstable, and the normal domain spreads in the strip at a current larger than the so-called detection current. We find the dependence of the detection current on the photon's energy, the location of its absorption in the strip, the width of the strip, and the magnetic field, and we compare this dependence with existing experiments. Our results demonstrate that materials with Ce/Cph|Tc≪1 are bad candidates for single-photon detectors due to a small transfer of the photon's energy to electronic system and a large τth . We also predict that even a several-micron-wide dirty superconducting bridge is able to detect a single near-infrared or optical photon if its critical current exceeds 70% of the depairing current and Ce/Cph|Tc≳1 .
NASA Astrophysics Data System (ADS)
Shizgal, Bernie
2016-03-01
The paper by Burini et al. [7] presents an interesting use of the Boltzmann equation of kinetic theory to model real learning processes. The authors provide a comprehensive discussion of the basic concepts involved in their modelling work. The Boltzmann equation as used by physicists and chemists to model a variety of transport processes in many diverse fields is based on the notion of the binary collisions between identifiable particles in the defined system [9]. The particles exchange energy on collision and the distribution function, which depends on the three velocity components and the three spatial coordinates, varies with time. The classical or quantum collision dynamics between particles play a central role in the definition of the kernels in the integral operators that define the Boltzmann equation [8].
Qamar, Shamsul Ahmed, Munshoor
2009-12-20
We present a high order kinetic flux-vector splitting (KFVS) scheme for the numerical solution of a conservative interface-capturing five-equation model of compressible two-fluid flows. This model was initially introduced by Wackers and Koren (2004) . The flow equations are the bulk equations, combined with mass and energy equations for one of the two fluids. The latter equation contains a source term in order to account for the energy exchange. We numerically investigate both one- and two-dimensional flow models. The proposed numerical scheme is based on the direct splitting of macroscopic flux functions of the system of equations. In two space dimensions the scheme is derived in a usual dimensionally split manner. The second order accuracy of the scheme is achieved by using MUSCL-type initial reconstruction and Runge-Kutta time stepping method. For validation, the results of our scheme are compared with those from the high resolution central scheme of Nessyahu and Tadmor . The accuracy, efficiency and simplicity of the KFVS scheme demonstrate its potential for modeling two-phase flows.
NASA Astrophysics Data System (ADS)
Parkhomenko, A. I.; Shalagin, A. M.
2014-11-01
The solution of many problems in light-induced gas kinetics can be simplified significantly using quantum kinetic equations in the context of the so-called one-dimensional approximation, in which the initial equations are averaged over transverse (relative to the direction of radiation) velocities. The errors introduced in such an approach are usually assumed to be small; however, this has been confirmed quantitatively only on the basis of the simplest (two- and three-level) particle models. We analyze the accuracy of the one-dimensional approximation for multilevel particles quantitatively for the light-induced drift (LID) effect in cesium atoms in the atmosphere of inert buffer gases. It is shown that in the case of the so-called "normal" LID, one-dimensional kinetic equations can always be used instead of three-dimensional equations without a risk of losing some important fine details in the dependence of the drift velocity on the radiation frequency. In the case of anomalous LID, the error of the one-dimensional approximation is also insignificant, but it can be disregarded only in the case of light buffer particles. For comparable masses of resonant and buffer particles, the one-dimensional approximation may give a noticeable error in determination of drift velocity amplitudes; however, the positions of drift velocity zeros and extrema depending on radiation-frequency detuning can be described successfully. Results show that the error introduced by using the one-dimensional approximation for multilevel particles turns out to be more significant than for the simplest particle models.
Parkhomenko, A. I. Shalagin, A. M.
2014-11-15
The solution of many problems in light-induced gas kinetics can be simplified significantly using quantum kinetic equations in the context of the so-called one-dimensional approximation, in which the initial equations are averaged over transverse (relative to the direction of radiation) velocities. The errors introduced in such an approach are usually assumed to be small; however, this has been confirmed quantitatively only on the basis of the simplest (two- and three-level) particle models. We analyze the accuracy of the one-dimensional approximation for multilevel particles quantitatively for the light-induced drift (LID) effect in cesium atoms in the atmosphere of inert buffer gases. It is shown that in the case of the so-called “normal” LID, one-dimensional kinetic equations can always be used instead of three-dimensional equations without a risk of losing some important fine details in the dependence of the drift velocity on the radiation frequency. In the case of anomalous LID, the error of the one-dimensional approximation is also insignificant, but it can be disregarded only in the case of light buffer particles. For comparable masses of resonant and buffer particles, the one-dimensional approximation may give a noticeable error in determination of drift velocity amplitudes; however, the positions of drift velocity zeros and extrema depending on radiation-frequency detuning can be described successfully. Results show that the error introduced by using the one-dimensional approximation for multilevel particles turns out to be more significant than for the simplest particle models.
Kurikami, Hiroshi; Malins, Alex; Takeishi, Minoru; Saito, Kimiaki; Iijima, Kazuki
2017-02-17
Radiocesium is an important environmental contaminant in fallout from nuclear reactor accidents and atomic weapons testing. A modified Diffusion-Sorption-Fixation (mDSF) model, based on the advection-dispersion equation, is proposed to describe the vertical migration of radiocesium in soils following fallout. The model introduces kinetics for the reversible binding of radiocesium. We test the model by comparing its results to depth profiles measured in Fukushima Prefecture, Japan, since 2011. The results from the mDSF model are a better fit to the measurement data (as quantified by R(2)) than results from a simple diffusion model and the original DSF model. The introduction of reversible sorption kinetics means that the exponential-shape depth distribution can be reproduced immediately following fallout. The initial relaxation mass depth of the distribution is determined by the diffusion length, which depends on the distribution coefficient, sorption rate and dispersion coefficient. The mDSF model captures the long tails of the radiocesium distribution at large depths, which are caused by different rates for kinetic sorption and desorption. The mDSF model indicates that depth distributions displaying a peak in activity below the surface are possible for soils with high organic matter content at the surface. The mDSF equations thus offers a physical basis for various types of radiocesium depth profiles observed in contaminated environments.
NASA Astrophysics Data System (ADS)
Vosika, Z.; Mitić, V. V.; Vasić, A.; Lazović, G.; Matija, L.; Kocić, Lj. M.
2017-03-01
In this paper, Caputo based Michaelis-Menten kinetic model based on Time Scale Calculus (TSC) is proposed. The main reason for its consideration is a study of tumor cells population growth dynamics. In the particular case discrete-continuous time kinetics, Michaelis-Menten model is numerically treated, using a new algorithm proposed by authors, called multistep generalized difference transformation method (MSGDETM). In addition numerical simulations are performed and is shown that it represents the upgrade of the multi-step variant of generalized differential transformation method (MSGDTM). A possible conditions for its further development are discussed and possible experimental verification is described.
Mulquiney, P J; Kuchel, P W
1999-01-01
Over the last 25 years, several mathematical models of erythrocyte metabolism have been developed. Although these models have identified the key features in the regulation and control of erythrocyte metabolism, many important aspects remain unexplained. In particular, none of these models have satisfactorily accounted for 2,3-bisphosphoglycerate (2,3-BPG) metabolism. 2,3-BPG is an important modulator of haemoglobin oxygen affinity, and hence an understanding of the regulation of 2,3-BPG concentration is important for understanding blood oxygen transport. A detailed, comprehensive, and hence realistic mathematical model of erythrocyte metabolism is presented that can explain the regulation and control of 2,3-BPG concentration and turnover. The model is restricted to the core metabolic pathways, namely glycolysis, the 2,3-BPG shunt and the pentose phosphate pathway (PPP), and includes membrane transport of metabolites, the binding of metabolites to haemoglobin and Mg(2+), as well as pH effects on key enzymic reactions and binding processes. The model is necessarily complex, since it is intended to describe the regulation and control of 2,3-BPG metabolism under a wide variety of physiological and experimental conditions. In addition, since H(+) and blood oxygen tension are important external effectors of 2,3-BPG concentration, it was important that the model take into account the large array of kinetic and binding phenomena that result from changes in these effectors. Through an iterative loop of experimental and simulation analysis many values of enzyme-kinetic parameters of the model were refined to yield close conformity between model simulations and 'real' experimental data. This iterative process enabled a single set of parameters to be found which described well the metabolic behaviour of the erythrocyte under a wide variety of conditions. PMID:10477269
NASA Technical Reports Server (NTRS)
Avissar, Roni; Chen, Fei
1993-01-01
Generated by landscape discontinuities (e.g., sea breezes) mesoscale circulation processes are not represented in large-scale atmospheric models (e.g., general circulation models), which have an inappropiate grid-scale resolution. With the assumption that atmospheric variables can be separated into large scale, mesoscale, and turbulent scale, a set of prognostic equations applicable in large-scale atmospheric models for momentum, temperature, moisture, and any other gaseous or aerosol material, which includes both mesoscale and turbulent fluxes is developed. Prognostic equations are also developed for these mesoscale fluxes, which indicate a closure problem and, therefore, require a parameterization. For this purpose, the mean mesoscale kinetic energy (MKE) per unit of mass is used, defined as E-tilde = 0.5 (the mean value of u'(sub i exp 2), where u'(sub i) represents the three Cartesian components of a mesoscale circulation (the angle bracket symbol is the grid-scale, horizontal averaging operator in the large-scale model, and a tilde indicates a corresponding large-scale mean value). A prognostic equation is developed for E-tilde, and an analysis of the different terms of this equation indicates that the mesoscale vertical heat flux, the mesoscale pressure correlation, and the interaction between turbulence and mesoscale perturbations are the major terms that affect the time tendency of E-tilde. A-state-of-the-art mesoscale atmospheric model is used to investigate the relationship between MKE, landscape discontinuities (as characterized by the spatial distribution of heat fluxes at the earth's surface), and mesoscale sensible and latent heat fluxes in the atmosphere. MKE is compared with turbulence kinetic energy to illustrate the importance of mesoscale processes as compared to turbulent processes. This analysis emphasizes the potential use of MKE to bridge between landscape discontinuities and mesoscale fluxes and, therefore, to parameterize mesoscale fluxes
Liao, Fei; Zhu, Xiao-Yun; Wang, Yong-Mei; Zuo, Yu-Ping
2005-01-31
The estimation of enzyme kinetic parameters by nonlinear fitting reaction curve to the integrated Michaelis-Menten rate equation ln(S(0)/S)+(S(0)-S)/K(m)=(V(m)/K(m))xt was investigated and compared to that by fitting to (S(0)-S)/t=V(m)-K(m)x[ln(S(0)/S)/t] (Atkins GL, Nimmo IA. The reliability of Michaelis-Menten constants and maximum velocities estimated by using the integrated Michaelis-Menten equation. Biochem J 1973;135:779-84) with uricase as the model. Uricase reaction curve was simulated with random absorbance error of 0.001 at 0.075 mmol/l uric acid. Experimental reaction curve was monitored by absorbance at 293 nm. For both CV and deviation <20% by simulation, K(m) from 5 to 100 micromol/l was estimated with Eq. (1) while K(m) from 5 to 50 micromol/l was estimated with Eq. (2). The background absorbance and the error in the lag time of steady-state reaction resulted in negative K(m) with Eq. (2), but did not affect K(m) estimated with Eq. (1). Both equations gave better estimation of V(m). The computation time and the goodness of fit with Eq. (1) were 40-fold greater than those with Eq. (2). By experimentation, Eq. (1) yielded K(m) consistent with the Lineweaver-Burk plot analysis, but Eq. (2) gave many negative parameters. Apparent K(m) by Eq. (1) linearly increased, while V(m) were constant, vs. xanthine concentrations, and the inhibition constant was consistent with the Lineweaver-Burk plot analysis. These results suggested that the integrated rate equation that uses the predictor variable of reaction time was reliable for the estimation of enzyme kinetic parameters and applicable for the characterization of enzyme inhibitors.
NASA Technical Reports Server (NTRS)
Avissar, Roni; Chen, Fei
1993-01-01
Generated by landscape discontinuities (e.g., sea breezes) mesoscale circulation processes are not represented in large-scale atmospheric models (e.g., general circulation models), which have an inappropiate grid-scale resolution. With the assumption that atmospheric variables can be separated into large scale, mesoscale, and turbulent scale, a set of prognostic equations applicable in large-scale atmospheric models for momentum, temperature, moisture, and any other gaseous or aerosol material, which includes both mesoscale and turbulent fluxes is developed. Prognostic equations are also developed for these mesoscale fluxes, which indicate a closure problem and, therefore, require a parameterization. For this purpose, the mean mesoscale kinetic energy (MKE) per unit of mass is used, defined as E-tilde = 0.5 (the mean value of u'(sub i exp 2), where u'(sub i) represents the three Cartesian components of a mesoscale circulation (the angle bracket symbol is the grid-scale, horizontal averaging operator in the large-scale model, and a tilde indicates a corresponding large-scale mean value). A prognostic equation is developed for E-tilde, and an analysis of the different terms of this equation indicates that the mesoscale vertical heat flux, the mesoscale pressure correlation, and the interaction between turbulence and mesoscale perturbations are the major terms that affect the time tendency of E-tilde. A-state-of-the-art mesoscale atmospheric model is used to investigate the relationship between MKE, landscape discontinuities (as characterized by the spatial distribution of heat fluxes at the earth's surface), and mesoscale sensible and latent heat fluxes in the atmosphere. MKE is compared with turbulence kinetic energy to illustrate the importance of mesoscale processes as compared to turbulent processes. This analysis emphasizes the potential use of MKE to bridge between landscape discontinuities and mesoscale fluxes and, therefore, to parameterize mesoscale fluxes
Liu, Ai-Lin; Zhou, Ting; He, Feng-Yun; Xu, Jing-Juan; Lu, Yu; Chen, Hong-Yuan; Xia, Xing-Hua
2006-06-01
We firstly transformed the traditional Michaelis-Menten equation into an off-line form which can be used for evaluating the Michaelis-Menten constant after the enzymatic reaction. For experimental estimation of the kinetics of enzymatic reactions, we have developed a facile and effective method by integrating an enzyme microreactor into direct-printing polymer microchips. Strong nonspecific adsorption of proteins was utilized to effectively immobilize enzymes onto the microchannel wall, forming the integrated on-column enzyme microreactor in a microchip. The properties of the integrated enzyme microreactor were evaluated by using the enzymatic reaction of glucose oxidase (GOx) with its substrate glucose as a model system. The reaction product, hydrogen peroxide, was electrochemically (EC) analyzed using a Pt microelectrode. The data for enzyme kinetics using our off-line form of the Michaelis-Menten equation was obtained (K(m) = 2.64 mM), which is much smaller than that reported in solution (K(m) = 6.0 mM). Due to the hydrophobic property and the native mesoscopic structure of the poly(ethylene terephthalate) film, the immobilized enzyme in the microreactor shows good stability and bioactivity under the flowing conditions.
Blitz, Mark A; Salter, Robert J; Heard, Dwayne E; Seakins, Paul W
2017-03-31
The kinetics of the reaction OH/OD + SO2 have been studied using a laser flash photolysis / laser induced fluorescence technique. Evidence for two-photon photolysis of SO2 at 248 nm is presented and quantified, and which appears to have been evident to some extent in most previous photolysis studies, potentially leading to values for the rate coefficient, k1, that are too large. The kinetics of the reaction OH(v=0) + SO2 (T = 295 K, p = 25 - 300 Torr) were measured under conditions where SO2 photolysis was taken into account. These results, together with literature data, were modelled using a master equation analysis. This analysis highlighted problems with the literature data: the rate coefficients derived from flash photolysis data were generally too high and from the flow tube data too low. Our best estimate of the high-pressure limiting rate coefficient, k1∞ was obtained from selected data and gives a value of (7.8 ± 2.2) × 10(-13) cm(3) molecule(-1) s-1, which is lower than that recommended in the literature. A parameterized form of k1([N2],T) is provided. The OD(v=0) + SO2 (T = 295 K, p = 25 - 300 Torr) data are reported for the first time and master equation analysis reinforces our assignment of k1∞.
NASA Astrophysics Data System (ADS)
Zizin, M. N.; Ivanov, L. D.
2013-12-01
In the present paper, an attempt is made to analyze the accuracy of calculating the effectiveness of the VVER-1000 reactor scram system by means of the inverted solution of the kinetics equation (ISKE). In the numerical studies in the intellectual ShIPR software system, the actuation of the reactor scram system with the possible jamming of one of the two most effective rods is simulated. First, the connection of functionals calculated in the space-time computation in different approximations with the kinetics equation is considered on the theoretical level. The formulas are presented in a manner facilitating their coding. Then, the results of processing of several such functions by the ISKE are presented. For estimating the effectiveness of the VVER-1000 reactor scram system, it is proposed to use the measured currents of ionization chambers (IC) jointly with calculated readings of IC imitators. In addition, the integral of the delayed neutron (DN) generation rate multiplied by the adjoint DN source over the volume of the reactor, calculated for the instant of time when insertion of safety rods ends, is used. This integral is necessary for taking into account the spatial reactivity effects. Reasonable agreement was attained for the considered example between the effectiveness of the scram system evaluated by this method and the values obtained by steady-state calculations as the difference of the reciprocal effective multiplication factors with withdrawn and inserted control rods. This agreement was attained with the use of eight-group DN parameters.
NASA Astrophysics Data System (ADS)
Pan, Liang; Xu, Kun; Li, Qibing; Li, Jiequan
2016-12-01
For computational fluid dynamics (CFD), the generalized Riemann problem (GRP) solver and the second-order gas-kinetic scheme (GKS) provide a time-accurate flux function starting from a discontinuous piecewise linear flow distributions around a cell interface. With the adoption of time derivative of the flux function, a two-stage Lax-Wendroff-type (L-W for short) time stepping method has been recently proposed in the design of a fourth-order time accurate method for inviscid flow [21]. In this paper, based on the same time-stepping method and the second-order GKS flux function [42], a fourth-order gas-kinetic scheme is constructed for the Euler and Navier-Stokes (NS) equations. In comparison with the formal one-stage time-stepping third-order gas-kinetic solver [24], the current fourth-order method not only reduces the complexity of the flux function, but also improves the accuracy of the scheme. In terms of the computational cost, a two-dimensional third-order GKS flux function takes about six times of the computational time of a second-order GKS flux function. However, a fifth-order WENO reconstruction may take more than ten times of the computational cost of a second-order GKS flux function. Therefore, it is fully legitimate to develop a two-stage fourth order time accurate method (two reconstruction) instead of standard four stage fourth-order Runge-Kutta method (four reconstruction). Most importantly, the robustness of the fourth-order GKS is as good as the second-order one. In the current computational fluid dynamics (CFD) research, it is still a difficult problem to extend the higher-order Euler solver to the NS one due to the change of governing equations from hyperbolic to parabolic type and the initial interface discontinuity. This problem remains distinctively for the hypersonic viscous and heat conducting flow. The GKS is based on the kinetic equation with the hyperbolic transport and the relaxation source term. The time-dependent GKS flux function
Yao, Chung-Tay; Lai, Ching-Long; Hsieh, Hsiu-Shan; Chi, Chin-Wen; Yin, Shih-Jiun
2010-09-01
Alcohol dehydrogenase (ADH) catalyzes oxidation of ingested ethanol to acetaldehyde, the first step in hepatic metabolism. The purpose of this study was to establish an ex vivo rat liver perfusion system under defined and verified steady states with respect to the metabolites and the metabolic rates, and to quantitatively correlate the observed rates with simulations based on the kinetic mechanism-based rate equations of rat liver ADH. Class I ADH1 was isolated from male Sprague-Dawley rats and characterized by steady-state kinetics in the Krebs-Ringer perfusion buffer with supplements. Nonrecirculating liver perfusion with constant input of ethanol at near physiological hepatic blood flow rate was performed in situ. Ethanol and the related metabolites acetaldehyde, acetate, lactate, and pyruvate in perfusates were determined. Results of the initial velocity, product, and dead-end inhibition studies showed that rat ADH1 conformed to the Theorell-Chance Ordered Bi Bi mechanism. Steady-state metabolism of ethanol in the perfused liver maintained up to 3h as evidenced by the steady-state levels of ethanol and metabolites in the effluent, and the steady-state ethanol disappearance rates and acetate production rates. The changes of the metabolic rates were qualitatively and in general quantitatively correlated to the results from simulations with the kinetic rate equations of ADH1 under a wide range of ethanol, in the presence of competitive inhibitor 4-methylpyrazole and of uncompetitive inhibitor isobutyramide. Preliminary flux control analysis estimated that apparent C(ADH)(J) in the perfused liver may approximate 0.7 at constant infusion with 1-2 mM ethanol, suggesting that ADH plays a major but not the exclusive role in governing hepatic ethanol metabolism. The reported steady-state rat liver perfusion system may potentially be applicable to other drug or drug-ethanol interaction studies. Copyright © 2010 Elsevier Inc. All rights reserved.
NASA Astrophysics Data System (ADS)
Péraud, Jean-Philippe M.; Hadjiconstantinou, Nicolas G.
2016-01-01
We derive the continuum equations and boundary conditions governing phonon-mediated heat transfer in the limit of a small but finite mean-free path from the asymptotic solution of the linearized Boltzmann equation in the relaxation time approximation. Our approach uses the ratio of the mean-free path to the characteristic system length scale, also known as the Knudsen number, as the expansion parameter to study the effects of boundaries on the breakdown of the Fourier description. We show that, in the bulk, the traditional heat conduction equation using Fourier's law as a constitutive relation is valid at least up to second order in the Knudsen number for steady problems and first order for time-dependent problems. However, this description does not hold within distances on the order of a few mean-free paths from the boundary; this breakdown is a result of kinetic effects that are always present in the boundary vicinity and require solution of a Boltzmann boundary layer problem to be determined. Matching the inner, boundary layer solution to the outer, bulk solution yields boundary conditions for the Fourier description as well as additive corrections in the form of universal kinetic boundary layers; both are found to be proportional to the bulk-solution gradients at the boundary and parametrized by the material model and the phonon-boundary interaction model (Boltzmann boundary condition). Our derivation shows that the traditional no-jump boundary condition for prescribed temperature boundaries and the no-flux boundary condition for diffusely reflecting boundaries are appropriate only to zeroth order in the Knudsen number; at higher order, boundary conditions are of the jump type. We illustrate the utility of the asymptotic solution procedure by demonstrating that it can be used to predict the Kapitza resistance (and temperature jump) associated with an interface between two materials. All results are validated via comparisons with low-variance deviational Monte
Tachibana, Hideki
2015-01-01
Studies of the pressure-dissociation of several amyloid or amyloid-like fibrils have shown that the fibril state is considerably voluminous. Quantitative characterization of the protein fibrillation reaction with respect to volumetric parameters is necessary to elucidate mechanisms of amyloid fibrillation in molecular terms such as protein cavity and hydration. Here we discuss, firstly, basic equations in statics and kinetics of protein polymerization as employed to obtain thermodynamic, volumetric, and kinetic parameters. Equilibrium treatment of the reactions with the scheme such as one-step polymerization, linear-association polymerization, or nucleation-dependent polymerization, and kinetic treatment of seeded linear-polymerization or spontaneous nucleation-elongation polymerization are described. In particular we will detail kinetics of the dissociation of fibrils which have been produced under the linear-association mechanism and therefore the length-distribution of which conforms to a geometric sequence in the degree of polymerization with a common ratio r, which is less than, and usually very close to, unity. In this case, an observed macroscopic rate of dissociation is shown to be a product of the microscopic elementary dissociation rate constant and a factor (1-r), extremely reduced compared with the intrinsic elementary rate. Secondly, we discuss protein conformational states in fibrillogenesis with molecular and volumetric observations reported, such as the unfolded state responsible for the association with seeds and the extension of amyloid fibrils, the transition state in which protein cavity formation and dehydration occur to intermediate levels, and the fibril state in which they occur to final respective levels which, in some cases, depend on the maturity of the fibril.
NASA Astrophysics Data System (ADS)
Piest, Jürgen
1989-06-01
This is the first of a series of three papers which report on an theoretical turbulence investigation. In the present part, the Reynolds equation for the mean velocity field in turbulent shear flow is derived in a systematic way starting from established physical knowledge. A basic problem of contemporary turbulence theory is that, at the hydrodynamic level, there seems to be no way presently to derive systematically the initial probability distribution of the fluctuating momentum density. For this reason, N-particle statistical mechanics is employed in this investigation. The closure problem of continuum turbulence theory is avoided by this method. The technique of deriving transport equations from the Liouville equation by projection operator methods is used for the derivation. Stationary constant density/temperature processes are considered only. The dissipative term of the momemtum transport equation is analyzed in order to obtain the formulas for the laminar and turbulent friction forces. The latter is obtained as a second-order convolution in the mean velocity field. The kernel function is a time integral of an equilibrium triple correlation function; it constitutes a physical “constant” of the fluid which is needed in addition to the viscosity constant. Its calculation has been the object of a separate investigation which will be reported in the second paper. The third paper describes the numerical evaluation and comparison with experiment for the spherical case of the circular jet. In the present state, the theoretical formula does not reproduce the experimental data. This is considered a preliminary result which, in view of the systematic nature of the derivation, offers the possibility to trace it back to the spots where the theoretical structure is still not adequate.
Karpov, S. A.; Potapenko, I. F.
2015-10-15
A stochastic method of simulation of Coulomb interaction is considered. The main idea of the method is to approximate the nonlinear Landau kinetic collision integral by the Boltzmann integral. In its realization, the method can be attributed to a wide class of Monte Carlo-type methods. It is easily combined with the existing particle methods used to simulate collisionless plasmas. This is important for simulation of the dynamics of both laboratory and space plasmas when the mean free path of plasma particles is comparable with the plasma inhomogeneity scale length. Illustrative examples of relaxation of two-temperature plasma being subject to a high-frequency alternating electric field are given, and differences from their classical description are considered. The method satisfies the conservation laws for the number of particles, momentum, and energy and is simple and efficient in implementation.
Koshelev, A. S. Arapov, A. V.; Ovchinnikov, M. A.
2016-12-15
The file-evaluation results of a reactimeter based on the inverse solution to the kinetics equation (ISKE) are presented, which were obtained using an operating hardware-measuring complex with a KNK-4 neutron detector working in the current mode. The processing of power-recording files of the BR-1M, BR-K1, and VIR-2M reactors of the Russian Federal Nuclear Center—All-Russian Research Institute of Experimental Physics, which was performed with the use of Excel simulation of the ISKE formalism, demonstrated the feasibility of implementation of the reactivity monitoring (during the operation of these reactors at stationary power) beginning from the level of ~5 × 10{sup –4}β{sub eff}.
NASA Astrophysics Data System (ADS)
Koshelev, A. S.; Arapov, A. V.; Ovchinnikov, M. A.
2016-12-01
The file-evaluation results of a reactimeter based on the inverse solution to the kinetics equation (ISKE) are presented, which were obtained using an operating hardware-measuring complex with a KNK-4 neutron detector working in the current mode. The processing of power-recording files of the BR-1M, BR-K1, and VIR-2M reactors of the Russian Federal Nuclear Center—All-Russian Research Institute of Experimental Physics, which was performed with the use of Excel simulation of the ISKE formalism, demonstrated the feasibility of implementation of the reactivity monitoring (during the operation of these reactors at stationary power) beginning from the level of 5 × 10-4βeff.
NASA Technical Reports Server (NTRS)
Dyall, Kenneth G.; Faegri, Knut, Jr.
1990-01-01
The paper investigates bounds failure in calculations using Gaussian basis sets for the solution of the one-electron Dirac equation for the 2p1/2 state of Hg(79+). It is shown that bounds failure indicates inadequacies in the basis set, both in terms of the exponent range and the number of functions. It is also shown that overrepresentation of the small component space may lead to unphysical results. It is concluded that it is important to use matched large and small component basis sets with an adequate size and exponent range.
NASA Technical Reports Server (NTRS)
Dyall, Kenneth G.; Faegri, Knut, Jr.
1990-01-01
The paper investigates bounds failure in calculations using Gaussian basis sets for the solution of the one-electron Dirac equation for the 2p1/2 state of Hg(79+). It is shown that bounds failure indicates inadequacies in the basis set, both in terms of the exponent range and the number of functions. It is also shown that overrepresentation of the small component space may lead to unphysical results. It is concluded that it is important to use matched large and small component basis sets with an adequate size and exponent range.
Davidson, R.C.; Chen, C.
1997-08-01
A kinetic description of intense nonneutral beam propagation through a periodic solenoidal focusing field B{sup sol}({rvec x}) is developed. The analysis is carried out for a thin beam with characteristic beam radius r{sub b} {much_lt} S, and directed axial momentum {gamma}{sub b}m{beta}{sub b}c (in the z-direction) large compared with the transverse momentum and axial momentum spread of the beam particles. Making use of the nonlinear Vlasov-Maxwell equations for general distribution function f{sub b}({rvec x},{rvec p},t) and self-consistent electrostatic field consistent with the thin-beam approximation, the kinetic model is used to investigate detailed beam equilibrium properties for a variety of distribution functions. Examples are presented both for the case of a uniform solenoidal focusing field B{sub z}(z) = B{sub 0} = const. and for the case of a periodic solenoidal focusing field B{sub z}(z + S) = B{sub z}(z). The nonlinear Vlasov-Maxwell equations are simplified in the thin-beam approximation, and an alternative Hamiltonian formulation is developed that is particularly well-suited to intense beam propagation in periodic focusing systems. Based on the present analysis, the Vlasov-Maxwell description of intense nonneutral beam propagation through a periodic solenoidal focusing field {rvec B}{sup sol}({rvec x}) is found to be remarkably tractable and rich in physics content. The Vlasov-Maxwell formalism developed here can be extended in a straightforward manner to investigate detailed stability behavior for perturbations about specific choices of beam equilibria.
Jahandar Lashaki, Masoud; Fayaz, Mohammadreza; Niknaddaf, Saeid; Hashisho, Zaher
2012-11-30
This paper investigates the effect of the kinetic diameter (KD) of the reference adsorbate on the accuracy of the Dubinin-Radushkevich (D-R) equation for predicting the adsorption isotherms of organic vapors on microporous activated carbon. Adsorption isotherms for 13 organic compounds on microporous beaded activated carbon were experimentally measured, and predicted using the D-R model and affinity coefficients. The affinity coefficients calculated based on molar volumes, molecular polarizabilities, and molecular parachors were used to predict the isotherms based on four reference compounds (4.3≤KD≤6.8 Å). The results show that the affinity coefficients are independent of the calculation method if the reference and test adsorbates are from the same organic group. Choosing a reference adsorbate with a KD similar to that of the test adsorbate results in better prediction of the adsorption isotherm. The relative error between the predicted and the measured adsorption isotherms increases as the absolute difference in the kinetic diameters of the reference and test adsorbates increases. Finally, the proposed hypothesis was used to explain reports of inconsistent findings among published articles. The results from this study are important because they allow a more accurate prediction of adsorption capacities of adsorbents which allow for better design of adsorption systems.
Chaiken, J; Goodisman, J; Kornilov, Oleg; Peter Toennies, J
2006-08-21
A previously published model of homogeneous nucleation [Villarica et al., J. Chem. Phys. 98, 4610 (1993)] based on the Smoluchowski [Phys. Z. 17, 557 (1916)] equations is used to simulate the experimentally measured size distributions of 4He clusters produced in free jet expansions. The model includes only binary collisions and does not consider evaporative effects, so that binary reactive collisions are rate limiting for formation of all cluster sizes despite the need for stabilization of nascent clusters. The model represents these data very well, accounting in some cases for nearly four orders of magnitude in variation in abundance over cluster sizes ranging up to nearly 100 atoms. The success of the model may be due to particularities of 4He clusters, i.e., their very low coalescence exothermicity, and to the low temperature of 6.7 K at which the data were collected.
Berezhkovskiy, Leonid M
2006-04-01
A common calculation of oral bioavailability is based on the comparison of the areas under the concentration-time curves after intravenous and oral drug administration. It does not take into account that after the oral dosing a drug enters the systemic circulation in different states, that is, as free fraction, protein bound and partitioned into blood cells, and plasma lipids, while after intravenous input it is introduced into the systemic circulation only as a free fraction. Consideration of this difference leads to a novel equation for the oral bioavailability. In general, the traditional calculation overestimates the oral bioavailability. For a widely applied model of a linear pharmacokinetic system with central (plasma) drug elimination it is shown that the traditional calculation of the oral bioavailability could substantially overestimate the true value. If the existence of an immediate equilibrium between different drug fractions in blood is assumed, the obtained equation becomes identical to the traditional one. Thus the deviation of oral bioavailability from the value given by a common calculation appears to be a kinetic phenomenon. The difference could be significant for the drugs with the rate constant of elimination from plasma of the same order of magnitude or greater than the dissociation rate constant of drug-protein complexes, or the off-rate constant of partitioning from the blood cells, if the blood concentration profiles were used to calculate the oral bioavailability.
NASA Astrophysics Data System (ADS)
Volkas, Raymond R.; Wong, Yvonne Y. Y.
2000-11-01
We demonstrate that the relic neutrino asymmetry evolution equation derived from the quantum kinetic equations (QKE's) reduces to the Boltzmann limit that is dependent only on the instantaneous neutrino distribution functions, in the adiabatic limit in conjunction with sufficient damping. An original physical and/or geometrical interpretation of the adiabatic approximation is given, which serves as a convenient visual aid for understanding the sharply contrasting resonance behaviors exhibited by the neutrino ensemble in opposing collision regimes. We also present a classical analogue for the evolution of the difference in the να and νs distribution functions which, in the Boltzmann limit, is akin to the behavior of the generic reaction A⇌B with equal forward and reverse reaction rate constants. A new characteristic quantity, the matter and collision-affected mixing angle of the neutrino ensemble, is identified here for the first time. The role of collisions is revealed to be twofold: (i) to wipe out the inherent oscillations, and (ii) to equilibrate the να and νs distribution functions in the long run. Studies on non-adiabatic evolution and its possible relation to rapid oscillations in lepton number generation are also featured, with the introduction of an adiabaticity parameter for collision-affected oscillations.
Topham, C M; Brocklehurst, K
1992-01-01
1. The suggestion by Segel & Martin [(1988) J. Theor. Biol. 135, 445-453] that the well-known schematic method for the derivation of rate equations for combined quasi-equilibrium-steady-state models proposed by Cha [(1968) J. Biol. Chem. 243, 820-825] may not be generally applicable was shown to be incorrect. By contrast, Cha's method was shown (a) to yield correct initial-rate equations that are exact and (b) not to require any constraints on the relative magnitudes of rate constants for slow steps outside the quasi-equilibrium segments of the kinetic model, including those suggested by Segel & Martin. 2. Examination of residual King-Altman patterns for the general modifier model of Botts & Morales [(1953) Trans. Faraday Soc. 49, 696-707] revealed the reasons for the erroneous conclusions reached by Segel & Martin. The errors arise from the failure to take account of fluxes in parallel pathways that connect the two isolated groups of enzyme species existing in quasi-equilibrium with the modifier. 3. A similar failure to take explicit account of parallel pathways in a thermodynamic box that delayed proper appreciation of the form of pH-dependence of kcat/Km is briefly discussed. PMID:1540141
Wang, Peng; Wang, Lian-Ping; Guo, Zhaoli
2016-10-01
The main objective of this work is to perform a detailed comparison of the lattice Boltzmann equation (LBE) and the recently developed discrete unified gas-kinetic scheme (DUGKS) methods for direct numerical simulation (DNS) of the decaying homogeneous isotropic turbulence and the Kida vortex flow in a periodic box. The flow fields and key statistical quantities computed by both methods are compared with those from the pseudospectral method at both low and moderate Reynolds numbers. The results show that the LBE is more accurate and efficient than the DUGKS, but the latter has a superior numerical stability, particularly for high Reynolds number flows. In addition, we conclude that the DUGKS can adequately resolve the flow when the minimum spatial resolution parameter k_{max}η>3, where k_{max} is the maximum resolved wave number and η is the flow Kolmogorov length. This resolution requirement can be contrasted with the requirements of k_{max}η>1 for the pseudospectral method and k_{max}η>2 for the LBE. It should be emphasized that although more validations should be conducted before the DUGKS can be called a viable tool for DNS of turbulent flows, the present work contributes to the overall assessment of the DUGKS, and it provides a basis for further applications of DUGKS in studying the physics of turbulent flows.
Berglund, Martin; Sunnåker, Mikael; Adiels, Martin; Jirstrand, Mats; Wennberg, Bernt
2012-12-01
Non-linear mixed effects (NLME) models represent a powerful tool to simultaneously analyse data from several individuals. In this study, a compartmental model of leucine kinetics is examined and extended with a stochastic differential equation to model non-steady-state concentrations of free leucine in the plasma. Data obtained from tracer/tracee experiments for a group of healthy control individuals and a group of individuals suffering from diabetes mellitus type 2 are analysed. We find that the interindividual variation of the model parameters is much smaller for the NLME models, compared to traditional estimates obtained from each individual separately. Using the mixed effects approach, the population parameters are estimated well also when only half of the data are used for each individual. For a typical individual, the amount of free leucine is predicted to vary with a standard deviation of 8.9% around a mean value during the experiment. Moreover, leucine degradation and protein uptake of leucine is smaller, proteolysis larger and the amount of free leucine in the body is much larger for the diabetic individuals than the control individuals. In conclusion, NLME models offers improved estimates for model parameters in complex models based on tracer/tracee data and may be a suitable tool to reduce data sampling in clinical studies.
Qian, Hong; Bishop, Lisa M
2010-09-20
We develop the stochastic, chemical master equation as a unifying approach to the dynamics of biochemical reaction systems in a mesoscopic volume under a living environment. A living environment provides a continuous chemical energy input that sustains the reaction system in a nonequilibrium steady state with concentration fluctuations. We discuss the linear, unimolecular single-molecule enzyme kinetics, phosphorylation-dephosphorylation cycle (PdPC) with bistability, and network exhibiting oscillations. Emphasis is paid to the comparison between the stochastic dynamics and the prediction based on the traditional approach based on the Law of Mass Action. We introduce the difference between nonlinear bistability and stochastic bistability, the latter has no deterministic counterpart. For systems with nonlinear bistability, there are three different time scales: (a) individual biochemical reactions, (b) nonlinear network dynamics approaching to attractors, and (c) cellular evolution. For mesoscopic systems with size of a living cell, dynamics in (a) and (c) are stochastic while that with (b) is dominantly deterministic. Both (b) and (c) are emergent properties of a dynamic biochemical network; We suggest that the (c) is most relevant to major cellular biochemical processes such as epi-genetic regulation, apoptosis, and cancer immunoediting. The cellular evolution proceeds with transitions among the attractors of (b) in a "punctuated equilibrium" manner.
Macnamara, Shev; Bersani, Alberto M; Burrage, Kevin; Sidje, Roger B
2008-09-07
Recently the application of the quasi-steady-state approximation (QSSA) to the stochastic simulation algorithm (SSA) was suggested for the purpose of speeding up stochastic simulations of chemical systems that involve both relatively fast and slow chemical reactions [Rao and Arkin, J. Chem. Phys. 118, 4999 (2003)] and further work has led to the nested and slow-scale SSA. Improved numerical efficiency is obtained by respecting the vastly different time scales characterizing the system and then by advancing only the slow reactions exactly, based on a suitable approximation to the fast reactions. We considerably extend these works by applying the QSSA to numerical methods for the direct solution of the chemical master equation (CME) and, in particular, to the finite state projection algorithm [Munsky and Khammash, J. Chem. Phys. 124, 044104 (2006)], in conjunction with Krylov methods. In addition, we point out some important connections to the literature on the (deterministic) total QSSA (tQSSA) and place the stochastic analogue of the QSSA within the more general framework of aggregation of Markov processes. We demonstrate the new methods on four examples: Michaelis-Menten enzyme kinetics, double phosphorylation, the Goldbeter-Koshland switch, and the mitogen activated protein kinase cascade. Overall, we report dramatic improvements by applying the tQSSA to the CME solver.
Qian, Hong; Bishop, Lisa M.
2010-01-01
We develop the stochastic, chemical master equation as a unifying approach to the dynamics of biochemical reaction systems in a mesoscopic volume under a living environment. A living environment provides a continuous chemical energy input that sustains the reaction system in a nonequilibrium steady state with concentration fluctuations. We discuss the linear, unimolecular single-molecule enzyme kinetics, phosphorylation-dephosphorylation cycle (PdPC) with bistability, and network exhibiting oscillations. Emphasis is paid to the comparison between the stochastic dynamics and the prediction based on the traditional approach based on the Law of Mass Action. We introduce the difference between nonlinear bistability and stochastic bistability, the latter has no deterministic counterpart. For systems with nonlinear bistability, there are three different time scales: (a) individual biochemical reactions, (b) nonlinear network dynamics approaching to attractors, and (c) cellular evolution. For mesoscopic systems with size of a living cell, dynamics in (a) and (c) are stochastic while that with (b) is dominantly deterministic. Both (b) and (c) are emergent properties of a dynamic biochemical network; We suggest that the (c) is most relevant to major cellular biochemical processes such as epi-genetic regulation, apoptosis, and cancer immunoediting. The cellular evolution proceeds with transitions among the attractors of (b) in a “punctuated equilibrium” manner. PMID:20957107
Introduction to Kinetic Model Equations
2011-01-01
application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 ∗Dipartimento di Matematica del Politecnico di Milano Piazza Leonardo da Vinci 32...NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Dipartimento di Matematica del Politecnico di Milano Piazza Leonardo da Vinci 32, 20133 Milano
White, Mark D.; McGrail, B. Peter
2005-12-01
flow and transport simulator, STOMP (Subsurface Transport Over Multiple Phases). Prior to these code development activities, the STOMP simulator included sequential and scalable implementations for numerically simulating the injection of supercritical CO2 into deep saline aquifers. Additionally, the sequential implementations included operational modes that considered nonisothermal conditions and kinetic dissolution of CO2 into the saline aqueous phase. This addendum documents the advancement of these numerical simulation capabilities to include reactive transport in the STOMP simulator through the inclusion of the recently PNNL developed batch geochemistry solution module ECKEChem (Equilibrium-Conservation-Kinetic Equation Chemistry). Potential geologic reservoirs for sequestering CO2 include deep saline aquifers, hydrate-bearing formations, depleted or partially depleted natural gas and petroleum reservoirs, and coal beds. The mechanisms for sequestering carbon dioxide in geologic reservoirs include physical trapping, dissolution in the reservoir fluids, hydraulic trapping (hysteretic entrapment of nonwetting fluids), and chemical reaction. This document and the associated code development and verification work are concerned with the chemistry of injecting CO2 into geologic reservoirs. As geologic sequestration of CO2 via chemical reaction, namely precipitation reactions, are most dominate in deep saline aquifers, the principal focus of this document is the numerical simulation of CO2 injection, migration, and geochemical reaction in deep saline aquifers. The ECKEChem batch chemistry module was developed in a fashion that would allow its implementation into all operational modes of the STOMP simulator, making it a more versatile chemistry component. Additionally, this approach allows for verification of the ECKEChem module against more classical reactive transport problems involving aqueous systems.
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)
Plane, J. M. C.; Whalley, C. L.; Frances-Soriano, L.; Goddard, A.; Harvey, J. N.; Glowacki, D. R.; Viggiano, A. A.
2012-07-01
utilized density functional theory along with multireference and explicitly correlated CCSD(T)-F12 electronic structure calculations to examine the lowest lying singlet and triplet surfaces. In addition to mapping stationary points, we used a genetic algorithm to locate minimum energy crossing points between the two surfaces. Simulations of the Ca + O2(a) kinetics were then carried out using a combination of both standard and non-adiabatic Rice-Ramsperger-Kassel-Marcus (RRKM) theory implemented within a weak collision, multiwell master equation model. In terms of atmospheric significance, only in the case of Ca does reaction with O2(a) compete with O3 during the daytime between 85 and 110 km.
Kinetic: A system code for analyzing nuclear thermal propulsion rocket engine transients
NASA Technical Reports Server (NTRS)
Schmidt, Eldon; Lazareth, Otto; Ludewig, Hans
1993-01-01
The topics are presented in viewgraph form and include the following: outline of kinetic code; a kinetic information flow diagram; kinetic neutronic equations; turbopump/nozzle algorithm; kinetic heat transfer equations per node; and test problem diagram.
Stamatakis, Michail; Vlachos, Dionisios G
2011-12-14
Well-mixed and lattice-based descriptions of stochastic chemical kinetics have been extensively used in the literature. Realizations of the corresponding stochastic processes are obtained by the Gillespie stochastic simulation algorithm and lattice kinetic Monte Carlo algorithms, respectively. However, the two frameworks have remained disconnected. We show the equivalence of these frameworks whereby the stochastic lattice kinetics reduces to effective well-mixed kinetics in the limit of fast diffusion. In the latter, the lattice structure appears implicitly, as the lumped rate of bimolecular reactions depends on the number of neighbors of a site on the lattice. Moreover, we propose a mapping between the stochastic propensities and the deterministic rates of the well-mixed vessel and lattice dynamics that illustrates the hierarchy of models and the key parameters that enable model reduction.
NASA Astrophysics Data System (ADS)
Açıkyıldız, Metin; Gürses, Ahmet; Güneş, Kübra; Yalvaç, Duygu
2015-11-01
The present study was designed to compare the linear and non-linear methods used to check the compliance of the experimental data corresponding to the isotherm models (Langmuir, Freundlich, and Redlich-Peterson) and kinetics equations (pseudo-first order and pseudo-second order). In this context, adsorption experiments were carried out to remove an anionic dye, Remazol Brillant Yellow 3GL (RBY), from its aqueous solutions using a commercial activated carbon as a sorbent. The effects of contact time, initial RBY concentration, and temperature onto adsorbed amount were investigated. The amount of dye adsorbed increased with increased adsorption time and the adsorption equilibrium was attained after 240 min. The amount of dye adsorbed enhanced with increased temperature, suggesting that the adsorption process is endothermic. The experimental data was analyzed using the Langmuir, Freundlich, and Redlich-Peterson isotherm equations in order to predict adsorption isotherm. It was determined that the isotherm data were fitted to the Langmuir and Redlich-Peterson isotherms. The adsorption process was also found to follow a pseudo second-order kinetic model. According to the kinetic and isotherm data, it was found that the determination coefficients obtained from linear method were higher than those obtained from non-linear method.
Boyarinov, V. F. Kondrushin, A. E. Fomichenko, P. A.
2014-12-15
Two-dimensional time-dependent finite-difference equations of the surface harmonics method (SHM) for the description of the neutron transport are derived for square-lattice reactors. These equations are implemented in the SUHAM-TD code. Verification of the derived equations and the developed code are performed by the example of known test problems, and the potential and efficiency of the SHM as applied to the solution of the time-dependent neutron transport equation in the diffusion approximation in two-dimensional geometry are demonstrated. These results show the substantial advantage of SHM over direct finite-difference modeling in computational costs.
NASA Astrophysics Data System (ADS)
Rickard, David
1997-01-01
A kinetic study of the reaction FeS+ H2S=FeS2+ H2 where FeS is precipitated FeS, H 2S (aq) is aqueous H 2S, FeS 2 is pyrite, and H 2(g) is hydrogen gas, shows that the rate between 25 and 125°C can be described by the equation dFeS2/dt = k(FeS) >(cH2S) where k the second order rate constant varies between 1.03 × 10 -4L mol -1 s -1 at 25°C and 3.20 × 10 -3 L mol -1 s -1 at 125°C. The rate constant shows a sigmoidal temperature dependence with an average Arrhenius activation energy of 35 kJ mol -1. The reaction is surprisingly fast at ambient temperatures with up to 50% reaction being completed within one day. The direct dependence of the rate on cH 2S (aq) means that the rate is pH dependent for any fixed total sulfide concentration. In typical sulfidic aquatic systems and sediments 9 × 10 -13 to 9 × 10 -8 mol FeS 2 per L sediment will be formed each day by this process. This is equivalent to approximately 3 × 10 -10 to 3 × 10 -5 mol FeS 2 per g sediment per year. At pH = 7, for the same total sulfide and FeS constraints, the rate of pyrite formation 1.5 × 10 -9 to 1.5 × 10 -4 mol FeS 2 per g sediment per year. In hydrothermal systems, such as deep ocean vents, the rate of pyrite formation by oxidation of FeS by H 2S at 125°C assuming a typical H 2S concentration of 1 mM is 3.2 × 10 -6 mol L -1 s -1 per mol FeS. A 1 million tonne pyrite deposit could form from a solution containing 1 mmol FeS and 1 mmol H 2S by this process in 1000 years at a flow rate of 0.3 Ls -1. The fluid would have a H 2 concentration of 3 × 10 -9 M. The process is by far the most rapid of the pyrite-forming reactions hitherto identified. Alternative pyrite-forming processes involving HS -, rather than H 2S, as the reaction requires an additional oxidising agent to maintain electron balance. These pathways may involve reactants such as polysulfides or intermediaries such as greigite, Fe 3S 4. In natural systems, therefore, the H 2S process will tend to be favored in strictly
Solving Simple Kinetics without Integrals
ERIC Educational Resources Information Center
de la Pen~a, Lisandro Herna´ndez
2016-01-01
The solution of simple kinetic equations is analyzed without referencing any topic from differential equations or integral calculus. Guided by the physical meaning of the rate equation, a systematic procedure is used to generate an approximate solution that converges uniformly to the exact solution in the case of zero, first, and second order…
Solving Simple Kinetics without Integrals
ERIC Educational Resources Information Center
de la Pen~a, Lisandro Herna´ndez
2016-01-01
The solution of simple kinetic equations is analyzed without referencing any topic from differential equations or integral calculus. Guided by the physical meaning of the rate equation, a systematic procedure is used to generate an approximate solution that converges uniformly to the exact solution in the case of zero, first, and second order…
NASA Astrophysics Data System (ADS)
Vereshchagin, Gregory V.; Aksenov, Alexey G.
2017-02-01
Preface; Acknowledgements; Acronyms and definitions; Introduction; Part I. Theoretical Foundations: 1. Basic concepts; 2. Kinetic equation; 3. Averaging; 4. Conservation laws and equilibrium; 5. Relativistic BBGKY hierarchy; 6. Basic parameters in gases and plasmas; Part II. Numerical Methods: 7. The basics of computational physics; 8. Direct integration of Boltzmann equations; 9. Multidimensional hydrodynamics; Part III. Applications: 10. Wave dispersion in relativistic plasma; 11. Thermalization in relativistic plasma; 12. Kinetics of particles in strong fields; 13. Compton scattering in astrophysics and cosmology; 14. Self-gravitating systems; 15. Neutrinos, gravitational collapse and supernovae; Appendices; Bibliography; Index.
Tao, W.C.; Tarver, C.M.; Kury, J.W.; Lee, C.G.; Ornellas, D.L.
1993-07-01
Using Fabry-Perot interferometry techniques, we have determined the early time rate of energy release from detonating PETN and TNT explosives filled with 5 to 20 wt % of either 5 {mu}m or 18 {mu}m spherical aluminum with the detonation products, and calculate the extent of reaction at 1--3 {mu}s after the detonation. All of the metal in PETN formulations filled with 5 wt % and 10 wt % of either 5 {mu}m or 18 {mu}m aluminum reacted within 1.5 {mu}s, resulting in an increase of 18--22% in energy compared to pure PETN. For TNT formulations, between 5 to 10 wt % aluminum reacts completely with the same timeframe. A reactive flow hydrodynamic code model based on the Zeldovich-von Neumann-Doring (ZND) description of the reaction zone and subsequent reaction product expansion (Taylor wave) is used to address the reaction rate of the aluminum particles with detonation product gases. The detonation product JWL equation of state is derived from that of pure PETN using a parametric normalization methodology.
Enskog-like kinetic models for vehicular traffic
Klar, A.; Wegener, R.
1997-04-01
In the present paper a general criticism of kinetic equations for vehicular traffic is given. The necessity of introducing an Enskog-type correction into these equations is shown. An Enskog-like kinetic traffic flow equation is presented and fluid dynamic equations are derived. This derivation yields new coefficients for the standard fluid dynamic equations of vehicular traffic. Numerical simulations for inhomogeneous traffic flow situations are shown together with a comparison between kinetic and fluid dynamics models.
Modeling of Reactor Kinetics and Dynamics
Matthew Johnson; Scott Lucas; Pavel Tsvetkov
2010-09-01
In order to model a full fuel cycle in a nuclear reactor, it is necessary to simulate the short time-scale kinetic behavior of the reactor as well as the long time-scale dynamics that occur with fuel burnup. The former is modeled using the point kinetics equations, while the latter is modeled by coupling fuel burnup equations with the kinetics equations. When the equations are solved simultaneously with a nonlinear equation solver, the end result is a code with the unique capability of modeling transients at any time during a fuel cycle.
Understanding Product Optimization: Kinetic versus Thermodynamic Control.
ERIC Educational Resources Information Center
Lin, King-Chuen
1988-01-01
Discusses the concept of kinetic versus thermodynamic control of reactions. Explains on the undergraduate level (1) the role of kinetic and thermodynamic control in kinetic equations, (2) the influence of concentration and temperature upon the reaction, and (3) the application of factors one and two to synthetic chemistry. (MVL)
Understanding Product Optimization: Kinetic versus Thermodynamic Control.
ERIC Educational Resources Information Center
Lin, King-Chuen
1988-01-01
Discusses the concept of kinetic versus thermodynamic control of reactions. Explains on the undergraduate level (1) the role of kinetic and thermodynamic control in kinetic equations, (2) the influence of concentration and temperature upon the reaction, and (3) the application of factors one and two to synthetic chemistry. (MVL)
First Author = C.Z. Cheng; Jay R. Johnson
1998-07-10
A nonlinear kinetic-fluid model for high-beta plasmas with multiple ion species which can be applied to multiscale phenomena is presented. The model embeds important kinetic effects due to finite ion Larmor radius (FLR), wave-particle resonances, magnetic particle trapping, etc. in the framework of simple fluid descriptions. When further restricting to low frequency phenomena with frequencies less than the ion cyclotron frequency the kinetic-fluid model takes a simpler form in which the fluid equations of multiple ion species collapse into single-fluid density and momentum equations and a low frequency generalized Ohm's law. The kinetic effects are introduced via plasma pressure tensors for ions and electrons which are computed from particle distribution functions that are governed by the Vlasov equation or simplified plasma dynamics equations such as the gyrokinetic equation. The ion FLR effects provide a finite parallel electric field, a perpendicular velocity that modifies the ExB drift, and a gyroviscosity tensor, all of which are neglected in the usual one-fluid MHD description. Eigenmode equations are derived which include magnetosphere-ionosphere coupling effects for low frequency waves (e.g., kinetic/inertial Alfven waves and ballooning-mirror instabilities).
ERIC Educational Resources Information Center
Wilson, David B.
1981-01-01
Surveys the research of scientists like Joule, Kelvin, Maxwell, Clausius, and Boltzmann as it comments on the basic conceptual issues involved in the development of a more precise kinetic theory and the idea of a kinetic atom. (Author/SK)
ERIC Educational Resources Information Center
Wilson, David B.
1981-01-01
Surveys the research of scientists like Joule, Kelvin, Maxwell, Clausius, and Boltzmann as it comments on the basic conceptual issues involved in the development of a more precise kinetic theory and the idea of a kinetic atom. (Author/SK)
ERIC Educational Resources Information Center
Moe, Owen; Cornelius, Richard
1988-01-01
Conveys an appreciation of enzyme kinetic analysis by using a practical and intuitive approach. Discusses enzyme assays, kinetic models and rate laws, the kinetic constants (V, velocity, and Km, Michaels constant), evaluation of V and Km from experimental data, and enzyme inhibition. (CW)
ERIC Educational Resources Information Center
Moe, Owen; Cornelius, Richard
1988-01-01
Conveys an appreciation of enzyme kinetic analysis by using a practical and intuitive approach. Discusses enzyme assays, kinetic models and rate laws, the kinetic constants (V, velocity, and Km, Michaels constant), evaluation of V and Km from experimental data, and enzyme inhibition. (CW)
Solving Kinetic Equations on GPU’s
2011-01-01
7 Acknowledgments 23 8 Appendix: CUDA pseudo-codes 27 ∗Dipartimento di Matematica del Politecnico di Milano Piazza Leonardo da Vinci 32, 20133 Milano...PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Dipartimento di Matematica del Politecnico di Milano Piazza Leonardo da Vinci 32, 20133 Milano, Italy 8
Nonlinear gyrokinetic equations
Dubin, D.H.E.; Krommes, J.A.; Oberman, C.; Lee, W.W.
1983-03-01
Nonlinear gyrokinetic equations are derived from a systematic Hamiltonian theory. The derivation employs Lie transforms and a noncanonical perturbation theory first used by Littlejohn for the simpler problem of asymptotically small gyroradius. For definiteness, we emphasize the limit of electrostatic fluctuations in slab geometry; however, there is a straight-forward generalization to arbitrary field geometry and electromagnetic perturbations. An energy invariant for the nonlinear system is derived, and various of its limits are considered. The weak turbulence theory of the equations is examined. In particular, the wave kinetic equation of Galeev and Sagdeev is derived from an asystematic truncation of the equations, implying that this equation fails to consider all gyrokinetic effects. The equations are simplified for the case of small but finite gyroradius and put in a form suitable for efficient computer simulation. Although it is possible to derive the Terry-Horton and Hasegawa-Mima equations as limiting cases of our theory, several new nonlinear terms absent from conventional theories appear and are discussed.
Diffusion Influenced Adsorption Kinetics.
Miura, Toshiaki; Seki, Kazuhiko
2015-08-27
When the kinetics of adsorption is influenced by the diffusive flow of solutes, the solute concentration at the surface is influenced by the surface coverage of solutes, which is given by the Langmuir-Hinshelwood adsorption equation. The diffusion equation with the boundary condition given by the Langmuir-Hinshelwood adsorption equation leads to the nonlinear integro-differential equation for the surface coverage. In this paper, we solved the nonlinear integro-differential equation using the Grünwald-Letnikov formula developed to solve fractional kinetics. Guided by the numerical results, analytical expressions for the upper and lower bounds of the exact numerical results were obtained. The upper and lower bounds were close to the exact numerical results in the diffusion- and reaction-controlled limits, respectively. We examined the validity of the two simple analytical expressions obtained in the diffusion-controlled limit. The results were generalized to include the effect of dispersive diffusion. We also investigated the effect of molecular rearrangement of anisotropic molecules on surface coverage.
A kinetic model of adsorption on solid surfaces
NASA Astrophysics Data System (ADS)
Aoki, Kazuo; Giovangigli, Vincent; Hattori, Masanari
2016-11-01
A kinetic model describing physisorption and chemisorption of gas particles on a crystal surface is introduced. A single kinetic equation is used to model gas and physisorbed particles interacting with an average potential and colliding with phonons. This equation is coupled to a kinetic equation describing localized chemisorbed species. A modified kinetic entropy is introduced for the coupled system and the H theorem is established. Using a fluid scaling and the Chapman-Enskog asymptotic method, fluid boundary conditions for the physisorbed and chemisorbed species are derived from the kinetic model.
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.
THE BERNOULLI EQUATION AND COMPRESSIBLE FLOW THEORIES
The incompressible Bernoulli equation is an analytical relationship between pressure, kinetic energy, and potential energy. As perhaps the simplest and most useful statement for describing laminar flow, it buttresses numerous incompressible flow models that have been developed ...
THE BERNOULLI EQUATION AND COMPRESSIBLE FLOW THEORIES
The incompressible Bernoulli equation is an analytical relationship between pressure, kinetic energy, and potential energy. As perhaps the simplest and most useful statement for describing laminar flow, it buttresses numerous incompressible flow models that have been developed ...
Multiflow approach to plasma kinetics
Ignatov, A. M.
2015-10-15
Instead of the commonly used Vlasov equation, one is able to treat kinetic phenomena in collisionless plasma with the help of the infinite set of hydrodynamic equations. The present paper deals with the linear approximation of multiflow hydrodynamics. It is shown that single-particle and collective excitations analogous to Van Kampen waves are explicitly separated. Expressions for the energy of all eigenmodes are obtained.
Young, C.W.
1997-10-01
In 1967, Sandia National Laboratories published empirical equations to predict penetration into natural earth materials and concrete. Since that time there have been several small changes to the basic equations, and several more additions to the overall technique for predicting penetration into soil, rock, concrete, ice, and frozen soil. The most recent update to the equations was published in 1988, and since that time there have been changes in the equations to better match the expanding data base, especially in concrete penetration. This is a standalone report documenting the latest version of the Young/Sandia penetration equations and related analytical techniques to predict penetration into natural earth materials and concrete. 11 refs., 6 tabs.
NASA Astrophysics Data System (ADS)
Britz, Dieter
In this chapter, we present most of the equations that apply to the systems and processes to be dealt with later. Most of these are expressed as equations of concentration dynamics, that is,concentration of one or more solution species as a function of time, as well as other variables, in the form of differential equations. Fundamentally, these are transport (diffusion-, convection- and migration-) equations but may be complicated by chemical processes occurring heterogeneously (i.e. at the electrode surface - electrochemical reaction) or homogeneously (in the solution bulk - chemical reaction). The transport components are all included in the general Nernst-Planck equation (see also Bard and Faulkner 2001) for the flux J j of species j
NASA Astrophysics Data System (ADS)
Prentis, Jeffrey J.
1996-05-01
One of the most challenging goals of a physics teacher is to help students see that the equations of physics are connected to each other, and that they logically unfold from a small number of basic ideas. Derivations contain the vital information on this connective structure. In a traditional physics course, there are many problem-solving exercises, but few, if any, derivation exercises. Creating an equation poem is an exercise to help students see the unity of the equations of physics, rather than their diversity. An equation poem is a highly refined and eloquent set of symbolic statements that captures the essence of the derivation of an equation. Such a poetic derivation is uncluttered by the extraneous details that tend to distract a student from understanding the essential physics of the long, formal derivation.
Mass Conservation and Chemical Kinetics.
ERIC Educational Resources Information Center
Barbara, Thomas M.; Corio, P. L.
1980-01-01
Presents a method for obtaining all mass conservation conditions implied by a given mechanism in which the conditions are used to simplify integration of the rate equations and to derive stoichiometric relations. Discusses possibilities of faulty inference of kinetic information from a given stoichiometry. (CS)
Mass Conservation and Chemical Kinetics.
ERIC Educational Resources Information Center
Barbara, Thomas M.; Corio, P. L.
1980-01-01
Presents a method for obtaining all mass conservation conditions implied by a given mechanism in which the conditions are used to simplify integration of the rate equations and to derive stoichiometric relations. Discusses possibilities of faulty inference of kinetic information from a given stoichiometry. (CS)
Chemical kinetics of geminal recombination
Levin, P.P.; Khudyakov, I.V.; Brin, E.F.; Kuz'min, V.A.
1988-09-01
The kinetics of geminal recombination of triplet radical pairs formed in photoreduction of benzophenone by p-cresol in glycerin solution was studied by pulsed laser photolysis. The experiments were conducted at several temperatures and in a constant magnetic field of H = 0.34 T. The parameters in six kinetic equations describing geminal recombination were determined with a computer. The values of the sums of the squares of the residual deviations of the approximation were obtained. It was found that the kinetics are best described by the functions proposed by Noyes and Shushin. It was shown that it is necessary to use the mutual diffusion coefficient of the radicals, which is significantly smaller than the sum of the estimations of the experimental values of the radical diffusion coefficients, for describing the kinetics due to the correlations of the molecular motions of the radicals in the cage.
Aerosol kinetic code "AERFORM": Model, validation and simulation results
NASA Astrophysics Data System (ADS)
Gainullin, K. G.; Golubev, A. I.; Petrov, A. M.; Piskunov, V. N.
2016-06-01
The aerosol kinetic code "AERFORM" is modified to simulate droplet and ice particle formation in mixed clouds. The splitting method is used to calculate condensation and coagulation simultaneously. The method is calibrated with analytic solutions of kinetic equations. Condensation kinetic model is based on cloud particle growth equation, mass and heat balance equations. The coagulation kinetic model includes Brownian, turbulent and precipitation effects. The real values are used for condensation and coagulation growth of water droplets and ice particles. The model and the simulation results for two full-scale cloud experiments are presented. The simulation model and code may be used autonomously or as an element of another code.
A Gas-Kinetic Scheme for Reactive Flows
NASA Technical Reports Server (NTRS)
Lian,Youg-Sheng; Xu, Kun
1998-01-01
In this paper, the gas-kinetic BGK scheme for the compressible flow equations is extended to chemical reactive flow. The mass fraction of the unburnt gas is implemented into the gas kinetic equation by assigning a new internal degree of freedom to the particle distribution function. The new variable can be also used to describe fluid trajectory for the nonreactive flows. Due to the gas-kinetic BGK model, the current scheme basically solves the Navier-Stokes chemical reactive flow equations. Numerical tests validate the accuracy and robustness of the current kinetic method.
Fokker Planck equation with fractional coordinate derivatives
NASA Astrophysics Data System (ADS)
Tarasov, Vasily E.; Zaslavsky, George M.
2008-11-01
Using the generalized Kolmogorov-Feller equation with long-range interaction, we obtain kinetic equations with fractional derivatives with respect to coordinates. The method of successive approximations, with averaging with respect to a fast variable, is used. The main assumption is that the correlation function of probability densities of particles to make a step has a power-law dependence. As a result, we obtain a Fokker-Planck equation with fractional coordinate derivative of order 1<α<2.
Mathematics analysis of polymerase chain reaction kinetic curves.
Sochivko, D G; Fedorov, A A; Varlamov, D A; Kurochkin, V E; Petrov, R V
2016-01-01
The paper reviews different approaches to the mathematical analysis of polymerase chain reaction (PCR) kinetic curves. The basic principles of PCR mathematical analysis are presented. Approximation of PCR kinetic curves and PCR efficiency curves by various functions is described. Several PCR models based on chemical kinetics equations are suggested. Decision criteria for an optimal function to describe PCR efficiency are proposed.
The Arrhenius equation revisited.
Peleg, Micha; Normand, Mark D; Corradini, Maria G
2012-01-01
The Arrhenius equation has been widely used as a model of the temperature effect on the rate of chemical reactions and biological processes in foods. Since the model requires that the rate increase monotonically with temperature, its applicability to enzymatic reactions and microbial growth, which have optimal temperature, is obviously limited. This is also true for microbial inactivation and chemical reactions that only start at an elevated temperature, and for complex processes and reactions that do not follow fixed order kinetics, that is, where the isothermal rate constant, however defined, is a function of both temperature and time. The linearity of the Arrhenius plot, that is, Ln[k(T)] vs. 1/T where T is in °K has been traditionally considered evidence of the model's validity. Consequently, the slope of the plot has been used to calculate the reaction or processes' "energy of activation," usually without independent verification. Many experimental and simulated rate constant vs. temperature relationships that yield linear Arrhenius plots can also be described by the simpler exponential model Ln[k(T)/k(T(reference))] = c(T-T(reference)). The use of the exponential model or similar empirical alternative would eliminate the confusing temperature axis inversion, the unnecessary compression of the temperature scale, and the need for kinetic assumptions that are hard to affirm in food systems. It would also eliminate the reference to the Universal gas constant in systems where a "mole" cannot be clearly identified. Unless proven otherwise by independent experiments, one cannot dismiss the notion that the apparent linearity of the Arrhenius plot in many food systems is due to a mathematical property of the model's equation rather than to the existence of a temperature independent "energy of activation." If T+273.16°C in the Arrhenius model's equation is replaced by T+b, where the numerical value of the arbitrary constant b is substantially larger than T and T
Kinetics of excitation in TL and OSL detectors
NASA Astrophysics Data System (ADS)
Mandowski, A.; Orzechowski, J.; Mandowska, E.
2010-10-01
Kinetic equations for the semi-localized transitions (SLT) model are presented describing charge carrier's kinetics for the excitation and fast relaxation stages. The formulation allows for dose dependence studies of thermoluminescence (TL) and optically stimulated luminescence (OSL) detectors based on the SLT model. The sets of equations were solved numerically demonstrating temporal evolution of all variables of the SLT model during excitation and fast relaxation. The influence of the dose rate on excitation kinetics is shown.
Anne, Agnès; Demaille, Christophe
2012-10-16
In the present work, exact kinetic equations describing the action of an enzyme in solution on a substrate attached to a surface have been derived in the framework of the Michaelis-Menten mechanism but without resorting to the often-used steady-state approximation. The here-derived kinetic equations are cast in a workable format, allowing us to introduce a simple and universal procedure for the quantitative analysis of enzyme surface kinetics that is valid for any kinetic situation. The results presented here should allow experimentalists studying the kinetics of enzyme action on immobilized substrates to analyze their data in a perfectly rigorous way.
NASA Astrophysics Data System (ADS)
Viljamaa, Panu; Jacobs, J. Richard; Chris; JamesHyman; Halma, Matthew; EricNolan; Coxon, Paul
2014-07-01
In reply to a Physics World infographic (part of which is given above) about a study showing that Euler's equation was deemed most beautiful by a group of mathematicians who had been hooked up to a functional magnetic-resonance image (fMRI) machine while viewing mathematical expressions (14 May, http://ow.ly/xHUFi).
Homogeneous nucleation kinetics
NASA Technical Reports Server (NTRS)
Rasmussen, D. H.; Appleby, M. R.; Leedom, G. L.; Babu, S. V.; Naumann, R. J.
1983-01-01
Homogeneous nucleation kinetics are rederived in a manner fundamentally similar to the approach of classical nucleation theory with the following modifications and improvements. First, the cluster is a parent phase cluster and does not require energization to the parent state. Second, the thermodynamic potential used to describe phase stability is a continuous function along the pathway of phase decomposition. Third, the kinetics of clustering corresponds directly to the diffusional flux of monomers through the cluster distribution and are formally similar to classical theory with the resulting kinetic equation modified by two terms in the preexponential factor. These terms correct for the influence of a supersaturation dependent clustering within the parent phase and for the influence of an asymmetrical cluster concentration as a function of cluster size at the critical cluster size. Fourth, the supersaturation dependence of the nucleation rate is of the same form as that given by classical nucleation theory. This supersaturation dependence must however be interpreted in terms of a size dependent surface tension. Finally, there are two scaling laws which describe supersaturation to either constant nucleation rate or to the thermodynamically determined physical spinodal.
NASA Astrophysics Data System (ADS)
Heinson, W. R.; Chakrabarti, A.; Sorensen, C. M.
2017-05-01
We demonstrate that kinetic aggregation forms superaggregates that have structures identical to static percolation aggregates, and these superaggregates appear as a separate phase in the size distribution. Diffusion limited cluster-cluster aggregation (DLCA) simulations were performed to yield fractal aggregates with a fractal dimension of 1.8 and superaggregates with a fractal dimension of D = 2.5 composed of these DLCA supermonomers. When properly normalized to account for the DLCA fractal nature of their supermonomers, these superaggregates have the exact same monomer packing fraction, scaling law prefactor, and scaling law exponent (the fractal dimension) as percolation aggregates; these are necessary and sufficient conditions for same structure. The size distribution remains monomodal until these superaggregates form to alter the distribution. Thus the static percolation and the kinetic descriptions of gelation are now unified.
Production of a sterile species: Quantum kinetics
Ho, Chiu Man; Boyanovsky, D.; Ho, C.M.
2007-04-23
Production of a sterile species is studied within an effective model of active-sterile neutrino mixing in a medium in thermal equilibrium. The quantum kinetic equations for the distribution functions and coherences are obtained from two independent methods: the effective action and the quantum master equation. The decoherence time scale for active-sterile oscillations is tau(dec)=2/Gamma(aa), but the evolution of the distribution functions is determined by the two different time scales associated with the damping rates of the quasiparticle modes in the medium: Gamma(1)=Gamma(aa)cos^2theta(m); Gamma(2)=Gamma(aa)sin^2theta(m) where Gamma(aa) is the interaction rate of the active species in the absence of mixing and theta(m) the mixing angle in the medium. These two time scales are widely different away from Mikheyev-Smirnov-Wolfenstein resonances and preclude the kinetic description of active-sterile production in terms of a simple rate equation. We give the complete set of quantum kinetic equations for the active and sterile populations and coherences and discuss in detail the various approximations. A generalization of the active-sterile transition probability in a medium is provided via the quantum master equation. We derive explicitly the usual quantum kinetic equations in terms of the"polarization vector" and show their equivalence to those obtained from the quantum master equation and effective action.
DOE R&D Accomplishments Database
1998-09-21
In the late 1950s to early 1960s Rudolph A. Marcus developed a theory for treating the rates of outer-sphere electron-transfer reactions. Outer-sphere reactions are reactions in which an electron is transferred from a donor to an acceptor without any chemical bonds being made or broken. (Electron-transfer reactions in which bonds are made or broken are referred to as inner-sphere reactions.) Marcus derived several very useful expressions, one of which has come to be known as the Marcus cross-relation or, more simply, as the Marcus equation. It is widely used for correlating and predicting electron-transfer rates. For his contributions to the understanding of electron-transfer reactions, Marcus received the 1992 Nobel Prize in Chemistry. This paper discusses the development and use of the Marcus equation. Topics include self-exchange reactions; net electron-transfer reactions; Marcus cross-relation; and proton, hydride, atom and group transfers.
Group-kinetic theory of turbulence
NASA Technical Reports Server (NTRS)
Tchen, C. M.
1986-01-01
The two phases are governed by two coupled systems of Navier-Stokes equations. The couplings are nonlinear. These equations describe the microdynamical state of turbulence, and are transformed into a master equation. By scaling, a kinetic hierarchy is generated in the form of groups, representing the spectral evolution, the diffusivity and the relaxation. The loss of memory in formulating the relaxation yields the closure. The network of sub-distributions that participates in the relaxation is simulated by a self-consistent porous medium, so that the average effect on the diffusivity is to make it approach equilibrium. The kinetic equation of turbulence is derived. The method of moments reverts it to the continuum. The equation of spectral evolution is obtained and the transport properties are calculated. In inertia turbulence, the Kolmogoroff law for weak coupling and the spectrum for the strong coupling are found. As the fluid analog, the nonlinear Schrodinger equation has a driving force in the form of emission of solitons by velocity fluctuations, and is used to describe the microdynamical state of turbulence. In order for the emission together with the modulation to participate in the transport processes, the non-homogeneous Schrodinger equation is transformed into a homogeneous master equation. By group-scaling, the master equation is decomposed into a system of transport equations, replacing the Bogoliubov system of equations of many-particle distributions. It is in the relaxation that the memory is lost when the ensemble of higher-order distributions is simulated by an effective porous medium. The closure is thus found. The kinetic equation is derived and transformed into the equation of spectral flow.
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.
Non-markovian boltzmann equation
Kremp, D.; Bonitz, M.; Kraeft, W.D.; Schlanges, M.
1997-08-01
A quantum kinetic equation for strongly interacting particles (generalized binary collision approximation, ladder or T-matrix approximation) is derived in the framework of the density operator technique. In contrast to conventional kinetic theory, which is valid on large time scales as compared to the collision (correlation) time only, our approach retains the full time dependencies, especially also on short time scales. This means retardation and memory effects resulting from the dynamics of binary correlations and initial correlations are included. Furthermore, the resulting kinetic equation conserves total energy (the sum of kinetic and potential energy). The second aspect of generalization is the inclusion of many-body effects, such as self-energy, i.e., renormalization of single-particle energies and damping. To this end we introduce an improved closure relation to the Bogolyubov{endash}Born{endash}Green{endash}Kirkwood{endash}Yvon hierarchy. Furthermore, in order to express the collision integrals in terms of familiar scattering quantities (Mo/ller operator, T-matrix), we generalize the methods of quantum scattering theory by the inclusion of medium effects. To illustrate the effects of memory and damping, the results of numerical simulations are presented. {copyright} 1997 Academic Press, Inc.
NASA Astrophysics Data System (ADS)
Marklof, Jens
2010-03-01
One of the central challenges in kinetic theory is the derivation of macroscopic evolution equations--describing, for example, the dynamics of an electron gas--from the underlying fundamental microscopic laws of classical or quantum mechanics. An iconic mathematical model in this research area is the Lorentz gas, which describes an ensemble of non-interacting point particles in an infinite array of spherical scatterers. In the case of a disordered scatterer configuration, the classical results by Gallavotti, Spohn and Boldrighini-Bunimovich-Sinai show that the time evolution of a macroscopic particle cloud is governed, in the limit of small scatterer density (Boltzmann-Grad limit), by the linear Boltzmann equation. In this lecture I will discuss the recent discovery that for a periodic configuration of scatterers the linear Boltzmann equation fails, and the random flight process that emerges in the Boltzmann-Grad limit is substantially more complicated. The key ingredient in the description of the limiting stochastic process is the renormalization dynamics on the space of lattices, a powerful technique that has recently been successfully applied also to other open problems in mathematical physics, including KAM theory and quantum chaos. This lecture is based on joint work with Andreas Strömbergsson, Uppsala.
Kinetics of degenerate atomic gases
NASA Astrophysics Data System (ADS)
Geist, W.; You, L.; Kennedy, T. A. B.
1998-05-01
Using the Uehling-Uhlenbeck, or quantum Boltzmann equation, we discuss the kinetics and evaporative cooling of quantum degenerate gases confined in magnetic traps with cylindrical symmetry. We study the full nonergodic time evolution and compare with results obtained by making the ergodic or continuum energy approximation(C. W. Gardiner, P. Zoller, R. J. Ballagh, M. J. Davis, ``Quantum kinetic theory. Simulation of the quantum Boltzmann master equation'', Phys. Rev. A 56), 575 (1997).. We report evidence of strongly non-ergodic distribution functions, whose relaxation times do not coincide with other characteristic timescales, but depend on trap anisotropy. We also report our study of condensate growth which exhibits the same qualitative behaviour as observed in a recent experiment(H. J. Miesner, D. M. Stamper, M. R. Andrews, D. S. Durfee, S. Inouve, W. Ketterle, ``Bosonic stimulation in the formation of a Bose-Einstein condensate'', (preprint).). Preliminary results for sympathetic cooling of fermions by bosons will also be presented.
2007-01-01
A b. Given an arbitrary relation→, we write →• for the total relation that extends→ by adding a pair a→• a for each a such that there is no b with a→ b...kind of a sort s is denoted by [s]. We write TΣ,k and TΣ,k(~x) to denote respectively the set of ground Σ-terms with kind k and of Σ-terms with kind k...variables. In membership equational logic, subsort relations and operator overloading are just a convenient way of writing corresponding Horn clauses
A kinetic-theory approach to turbulent chemically reacting flows
NASA Technical Reports Server (NTRS)
Chung, P. M.
1976-01-01
The paper examines the mathematical and physical foundations for the kinetic theory of reactive turbulent flows, discussing the differences and relation between the kinetic and averaged equations, and comparing some solutions of the kinetic equations obtained by the Green's function method with those obtained by the approximate bimodal method. The kinetic method described consists essentially in constructing the probability density functions of the chemical species on the basis of solutions of the Langevin stochastic equation for the influence of eddies on the behavior of fluid elements. When the kinetic equations are solved for the structure of the diffusion flame established in a shear layer by the bimodal method, discontinuities in gradients of the mean concentrations at the two flame edges appear. This is a consequence of the bimodal approximation of all distribution functions by two dissimilar half-Maxwellian functions, which is a very crude approximation. These discontinuities do not appear when the solutions are constructed by the Green's function method described here.
Kinetic model of network traffic
NASA Astrophysics Data System (ADS)
Antoniou, I.; Ivanov, V. V.; Kalinovsky, Yu. L.
2002-05-01
We present the first results on the application of the Prigogine-Herman kinetic approach (Kinetic Theory of Vehicular Traffic, American Elsevier Publishing Company, Inc., New York, 1971) to the network traffic. We discuss the solution of the kinetic equation for homogeneous time-independent situations and for the desired speed distribution function, obtained from traffic measurements analysis. For the log-normal desired speed distribution function the solution clearly shows two modes corresponding to individual flow patterns (low-concentration mode) and to collective flow patterns (traffic jam mode). For low-concentration situations we found almost linear dependence of the information flow versus the concentration and that the higher the average speed the lower the concentration at which the optimum flow takes place. When approaching the critical concentration there are no essential differences in the flow for different desired average speeds, whereas for the individual flow regions there are dramatic differences.
Hicks, D.R.; Kraml, M.; Cayen, M.N.; Dubuc, J.; Ryder, S.; Dvornik, D.
1984-10-01
The kinetics of tolrestat, a potent inhibitor of aldose reductase, were examined. Serum concentrations of tolrestat and of total /sup 14/C were measured after dosing normal subjects and subjects with diabetes with /sup 14/C-labeled tolrestat. In normal subjects, tolrestat was rapidly absorbed and disappearance from serum was biphasic. Distribution and elimination t 1/2s were approximately 2 and 10 to 12 hr, respectively, after single and multiple doses. Unchanged tolrestat accounted for the major portion of /sup 14/C in serum. Radioactivity was rapidly and completely excreted in urine and feces in an approximate ratio of 2:1. Findings were much the same in subjects with diabetes. In normal subjects, the kinetics of oral tolrestat were independent of dose in the 10 to 800 mg range. Repetitive dosing did not result in unexpected cumulation. Tolrestat was more than 99% bound to serum protein; it did not compete with warfarin for binding sites but was displaced to some extent by high concentrations of tolbutamide or salicylate.
NASA Astrophysics Data System (ADS)
Avila, Gustavo; Carrington, Tucker
2011-08-01
In this paper we propose and test a method for computing numerically exact vibrational energy levels of a molecule with six atoms. We use a pruned product basis, a non-product quadrature, the Lanczos algorithm, and the exact normal-coordinate kinetic energy operator (KEO) with the πtμπ term. The Lanczos algorithm is applied to a Hamiltonian with a KEO for which μ is evaluated at equilibrium. Eigenvalues and eigenvectors obtained from this calculation are used as a basis to obtain the final energy levels. The quadrature scheme is designed, so that integrals for the most important terms in the potential will be exact. The procedure is tested on C2H4. All 12 coordinates are treated explicitly. We need only ˜1.52 × 108 quadrature points. A product Gauss grid with which one could calculate the same energy levels has at least 5.67 × 1013 points.
Numerical integration of variational equations.
Skokos, Ch; Gerlach, E
2010-09-01
We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom and investigate their efficiency in accurately reproducing well-known properties of chaos indicators such as the Lyapunov characteristic exponents and the generalized alignment indices. We find that the best numerical performance is exhibited by the "tangent map method," a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy. According to this method, a symplectic integrator is used to approximate the solution of the Hamilton equations of motion by the repeated action of a symplectic map S , while the corresponding tangent map TS is used for the integration of the variational equations. A simple and systematic technique to construct TS is also presented.
NASA Astrophysics Data System (ADS)
Tang, J. Y.
2015-09-01
The Michaelis-Menten kinetics and the reverse Michaelis-Menten kinetics are two popular mathematical formulations used in many land biogeochemical models to describe how microbes and plants would respond to changes in substrate abundance. However, the criteria of when to use which of the two are often ambiguous. Here I show that these two kinetics are special approximations to the Equilibrium Chemistry Approximation kinetics, which is the first order approximation to the quadratic kinetics that solves the equation of enzyme-substrate complex exactly for a single enzyme single substrate biogeochemical reaction with the law of mass action and the assumption of quasi-steady-state for the enzyme-substrate complex and that the product genesis from enzyme-substrate complex is much slower than the equilibration between enzyme-substrate complexes, substrates and enzymes. In particular, I showed that the derivation of the Michaelis-Menten kinetics does not consider the mass balance constraint of the substrate, and the reverse Michaelis-Menten kinetics does not consider the mass balance constraint of the enzyme, whereas both of these constraints are taken into account in the Equilibrium Chemistry Approximation kinetics. By benchmarking against predictions from the quadratic kinetics for a wide range of substrate and enzyme concentrations, the Michaelis-Menten kinetics was found to persistently under-predict the normalized sensitivity ∂ ln v / ∂ ln k2+ of the reaction velocity v with respect to the maximum product genesis rate k2+, persistently over-predict the normalized sensitivity ∂ ln v / ∂ ln k1+ of v with respect to the intrinsic substrate affinity k1+, persistently over-predict the normalized sensitivity ∂ ln v / ∂ ln [ E ]T of v with respect the total enzyme concentration [ E ]T and persistently under-predict the normalized sensitivity ∂ ln v / ∂ ln [ S ]T of v with respect to the total substrate concentration [ S ]T. Meanwhile
Tang, J. Y.
2015-09-03
The Michaelis–Menten kinetics and the reverse Michaelis–Menten kinetics are two popular mathematical formulations used in many land biogeochemical models to describe how microbes and plants would respond to changes in substrate abundance. However, the criteria of when to use which of the two are often ambiguous. Here I show that these two kinetics are special approximations to the Equilibrium Chemistry Approximation kinetics, which is the first order approximation to the quadratic kinetics that solves the equation of enzyme-substrate complex exactly for a single enzyme single substrate biogeochemical reaction with the law of mass action and the assumption of quasi-steady-state formore » the enzyme-substrate complex and that the product genesis from enzyme-substrate complex is much slower than the equilibration between enzyme-substrate complexes, substrates and enzymes. In particular, I showed that the derivation of the Michaelis–Menten kinetics does not consider the mass balance constraint of the substrate, and the reverse Michaelis–Menten kinetics does not consider the mass balance constraint of the enzyme, whereas both of these constraints are taken into account in the Equilibrium Chemistry Approximation kinetics. By benchmarking against predictions from the quadratic kinetics for a wide range of substrate and enzyme concentrations, the Michaelis–Menten kinetics was found to persistently under-predict the normalized sensitivity ∂ ln v / ∂ ln k2+ of the reaction velocity v with respect to the maximum product genesis rate k2+, persistently over-predict the normalized sensitivity ∂ ln v / ∂ ln k1+ of v with respect to the intrinsic substrate affinity k1+, persistently over-predict the normalized sensitivity ∂ ln v / ∂ ln [ E ]T of v with respect the total enzyme concentration [ E ]T and persistently under-predict the normalized sensitivity ∂ ln v / ∂ ln [ S ]T of v with respect to the total substrate concentration [ S ]T. Meanwhile, the
Algebraic operator approach to gas kinetic models
NASA Astrophysics Data System (ADS)
Il'ichov, L. V.
1997-02-01
Some general properties of the linear Boltzmann kinetic equation are used to present it in the form ∂ tϕ = - Â†Âϕ with the operators ÂandÂ† possessing some nontrivial algebraic properties. When applied to the Keilson-Storer kinetic model, this method gives an example of quantum ( q-deformed) Lie algebra. This approach provides also a natural generalization of the “kangaroo model”.
Alibrandi, Giuseppe; Fabbrizzi, Luigi; Licchelli, Maurizio; Puglisi, Antonio
2015-01-12
This paper proposes a new type of molecular device that is able to act as an inverse proton sponge to slowly decrease the pH inside a reaction vessel. This makes the automatic monitoring of the concentration of pH-sensitive systems possible. The device is a composite formed of an alkyl chloride, which kinetically produces acidity, and a buffer that thermodynamically modulates the variation in pH value. Profiles of pH versus time (pH-t plots) have been generated under various experimental conditions by computer simulation, and the device has been tested by carrying out automatic spectrophotometric titrations, without using an autoburette. To underline the wide variety of possible applications, this new system has been used to realize and monitor HCl uptake by a di-copper(II) bistren complex in a single run, in a completely automatic experiment. © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
Hybrid fluid/kinetic model for parallel heat conduction
Callen, J.D.; Hegna, C.C.; Held, E.D.
1998-12-31
It is argued that in order to use fluid-like equations to model low frequency ({omega} < {nu}) phenomena such as neoclassical tearing modes in low collisionality ({nu} < {omega}{sub b}) tokamak plasmas, a Chapman-Enskog-like approach is most appropriate for developing an equation for the kinetic distortion (F) of the distribution function whose velocity-space moments lead to the needed fluid moment closure relations. Further, parallel heat conduction in a long collision mean free path regime can be described through a combination of a reduced phase space Chapman-Enskog-like approach for the kinetics and a multiple-time-scale analysis for the fluid and kinetic equations.
Simple Model of Protein Folding Kinetics
NASA Astrophysics Data System (ADS)
Zwanzig, Robert
1995-10-01
A simple model of the kinetics of protein folding is presented. The reaction coordinate is the "correctness" of a configuration compared with the native state. The model has a gap in the energy spectrum, a large configurational entropy, a free energy barrier between folded and partially folded states, and a good thermodynamic folding transition. Folding kinetics is described by a master equation. The folding time is estimated by means of a local thermodynamic equilibrium assumption and then is calculated both numerically and analytically by solving the master equation. The folding time has a maximum near the folding transition temperature and can have a minimum at a lower temperature.
The origins of enzyme kinetics.
Cornish-Bowden, Athel
2013-09-02
The equation commonly called the Michaelis-Menten equation is sometimes attributed to other authors. However, although Victor Henri had derived the equation from the correct mechanism, and Adrian Brown before him had proposed the idea of enzyme saturation, it was Leonor Michaelis and Maud Menten who showed that this mechanism could also be deduced on the basis of an experimental approach that paid proper attention to pH and spontaneous changes in the product after formation in the enzyme-catalysed reaction. By using initial rates of reaction they avoided the complications due to substrate depletion, product accumulation and progressive inactivation of the enzyme that had made attempts to analyse complete time courses very difficult. Their methodology has remained the standard approach to steady-state enzyme kinetics ever since.
Determining enzyme kinetics via isothermal titration calorimetry.
Demarse, Neil A; Killian, Marie C; Hansen, Lee D; Quinn, Colette F
2013-01-01
Isothermal titration calorimetry (ITC) has emerged as a powerful tool for determining the thermodynamic properties of chemical or physical equilibria such as protein-protein, ligand-receptor, and protein-DNA binding interactions. The utility of ITC for determining kinetic information, however, has not been fully recognized. Methods for collecting and analyzing data on enzyme kinetics are discussed here. The step-by-step process of converting the raw heat output rate into the kinetic parameters of the Michaelis-Menten equation is explicitly stated. The hydrolysis of sucrose by invertase is used to demonstrate the capability of the instrument and method.
Kinetic theory of spatially inhomogeneous stellar systems without collective effects
NASA Astrophysics Data System (ADS)
Chavanis, P.-H.
2013-08-01
We review and complete the kinetic theory of spatially inhomogeneous stellar systems when collective effects (dressing of the stars by their polarization cloud) are neglected. We start from the BBGKY hierarchy issued from the Liouville equation and consider an expansion in powers of 1/N in a proper thermodynamic limit. For N → +∞, we obtain the Vlasov equation describing the evolution of collisionless stellar systems like elliptical galaxies. This corresponds to the mean field approximation. At the order 1/N, we obtain a kinetic equation describing the evolution of collisional stellar systems like globular clusters. This corresponds to the weak coupling approximation. This equation coincides with the generalized Landau equation derived from a more abstract projection operator formalism. This equation does not suffer logarithmic divergences at large scales since spatial inhomogeneity is explicitly taken into account. Making a local approximation, and introducing an upper cut-off at the Jeans length, it reduces to the Vlasov-Landau equation which is the standard kinetic equation of stellar systems. Our approach provides a simple and pedagogical derivation of these important equations from the BBGKY hierarchy which is more rigorous for systems with long-range interactions than the two-body encounters theory. Making an adiabatic approximation, we write the generalized Landau equation in angle-action variables and obtain a Landau-type kinetic equation that is valid for fully inhomogeneous stellar systems and is free of divergences at large scales. This equation is less general than the recently derived Lenard-Balescu-type kinetic equation since it neglects collective effects, but it is substantially simpler and could be useful as a first step. We discuss the evolution of the system as a whole and the relaxation of a test star in a bath of field stars. We derive the corresponding Fokker-Planck equation in angle-action variables and provide expressions for the
An introduction to the Dieterici Equation and the van der Waal Equation
NASA Astrophysics Data System (ADS)
Sheldon, John
2003-11-01
The derivation of the ideal gas law by using the kinetic theory of gases is usually presented in an undergraduate physics thermodynamics texts and physical chemistry texts. Following these derivations is the introduction of nonideal effects and the empirical equations of state: the van der Waals equation and the Dieterici equation. These are sometimes are simply given without comment as to the origin of the terms in them. An introduction to a "derivation" of these equations, appropriate for the undergraduate thermodynamics course, is given herein. Empirical equations are not rigorously derived, but rather they are invented, the so-called derivation simply serves to make the empirical terms appear reasonable.The barometric equation is exploited to get an expression for the effective attractive molecular forces. The differential form of the barometric is derived using kinetic theory, then from the barometric equation we get the Dieterici Equation an expansion of the Dieterici Equation, yields the van der Waals Equation of state. The relationship between the empirical constants is also discussed
NASA Astrophysics Data System (ADS)
Xu, Kun; He, Xin; Cai, Chunpei
2008-07-01
It is well known that for increasingly rarefied flowfields, the predictions from continuum formulation, such as the Navier-Stokes equations lose accuracy. For the high speed diatomic molecular flow in the transitional regime, the inaccuracies are partially attributed to the single temperature approximations in the Navier-Stokes equations. Here, we propose a continuum multiple temperature model based on the Bhatnagar-Gross-Krook (BGK) equation for the non-equilibrium flow computation. In the current model, the Landau-Teller-Jeans relaxation model for the rotational energy is used to evaluate the energy exchange between the translational and rotational modes. Due to the multiple temperature approximation, the second viscosity coefficient in the Navier-Stokes equations is replaced by the temperature relaxation term. In order to solve the multiple temperature kinetic model, a multiscale gas-kinetic finite volume scheme is proposed, where the gas-kinetic equation is numerically solved for the fluxes to update the macroscopic flow variables inside each control volume. Since the gas-kinetic scheme uses a continuous gas distribution function at a cell interface for the fluxes evaluation, the moments of a gas distribution function can be explicitly obtained for the multiple temperature model. Therefore, the kinetic scheme is much more efficient than the DSMC method, especially in the near continuum flow regime. For the non-equilibrium flow computations, i.e., the nozzle flow and hypersonic rarefied flow over flat plate, the computational results are validated in comparison with experimental measurements and DSMC solutions.
A new rate equation for desorption at the solid/solution interface
NASA Astrophysics Data System (ADS)
Bashiri, Hadis; Hassani Javanmardi, Alireza
2017-03-01
In this article, a new integrated kinetics Langmuir equation for desorption from the solid surface is derived. This new equation is simple and easy to be used. Several sets of kinetic data points are generated to analyze the accuracy of the new rate equation. By applying theoretical and experimental data, the applicability of the new equation is proved. The analysis of the new equation explains its relation with the pseudo first-order rate equation, and it shows the conditions of its possible application based on Langmuir model. The accuracy of theoretical derivation of pseudo first-order rate equation is proved.
A century of enzyme kinetic analysis, 1913 to 2013.
Johnson, Kenneth A
2013-09-02
This review traces the history and logical progression of methods for quantitative analysis of enzyme kinetics from the 1913 Michaelis and Menten paper to the application of modern computational methods today. Following a brief review of methods for fitting steady state kinetic data, modern methods are highlighted for fitting full progress curve kinetics based upon numerical integration of rate equations, including a re-analysis of the original Michaelis-Menten full time course kinetic data. Finally, several illustrations of modern transient state kinetic methods of analysis are shown which enable the elucidation of reactions occurring at the active sites of enzymes in order to relate structure and function.
A Century of Enzyme Kinetic Analysis, 1913 to 2013
Johnson, Kenneth A.
2013-01-01
This review traces the history and logical progression of methods for quantitative analysis of enzyme kinetics from the 1913 Michaelis and Menten paper to the application of modern computational methods today. Following a brief review of methods for fitting steady state kinetic data, modern methods are highlighted for fitting full progress curve kinetics based upon numerical integration of rate equations, including a re-analysis of the original Michaelis-Menten full time course kinetic data. Finally, several illustrations of modern transient state kinetic methods of analysis are shown which enable the elucidation of reactions occurring at the active sites of enzymes in order to relate structure and function. PMID:23850893
Shore, B.W.
1981-01-30
The equations of motion are discussed which describe time dependent population flows in an N-level system, reviewing the relationship between incoherent (rate) equations, coherent (Schrodinger) equations, and more general partially coherent (Bloch) equations. Approximations are discussed which replace the elaborate Bloch equations by simpler rate equations whose coefficients incorporate long-time consequences of coherence.
Zakharov equations in quantum dusty plasmas
Sayed, F.; Vladimirov, S. V.; Ishihara, O.
2015-08-15
By generalizing the formalism of modulational interactions in quantum dusty plasmas, we derive the kinetic quantum Zakharov equations in dusty plasmas that describe nonlinear coupling of high frequency Langmuir waves to low frequency plasma density variations, for cases of non-degenerate and degenerate plasma electrons.
Kinetic Measurements for Enzyme Immobilization.
Cooney, Michael J
2017-01-01
Enzyme kinetics is the study of the chemical reactions that are catalyzed by enzymes, with a focus on their reaction rates. The study of an enzyme's kinetics considers the various stages of activity, reveals the catalytic mechanism of this enzyme, correlates its value to assay conditions, and describes how a drug or a poison might inhibit the enzyme. Victor Henri initially reported that enzyme reactions were initiated by a bond between the enzyme and the substrate. By 1910, Michaelis and Menten were advancing their work by studying the kinetics of an enzyme saccharase which catalyzes the hydrolysis of sucrose into glucose and fructose. They published their analysis and ever since the Michaelis-Menten equation has been used as the standard to describe the kinetics of many enzymes. Unfortunately, soluble enzymes must generally be immobilized to be reused for long times in industrial reactors. In addition, other critical enzyme properties have to be improved like stability, activity, inhibition by reaction products, and selectivity towards nonnatural substrates. Immobilization is by far the chosen process to achieve these goals.Although the Michaelis-Menten approach has been regularly adapted to the analysis of immobilized enzyme activity, its applicability to the immobilized state is limited by the barriers the immobilization matrix places upon the measurement of compounds that are used to model enzyme kinetics. That being said, the estimated value of the Michaelis-Menten coefficients (e.g., V max, K M) can be used to evaluate effects of immobilization on enzyme activity in the immobilized state when applied in a controlled manner. In this review enzyme activity and kinetics are discussed in the context of the immobilized state, and a few novel protocols are presented that address some of the unique constraints imposed by the immobilization barrier.
Kinetic measurements for enzyme immobilization.
Cooney, Michael J
2011-01-01
Enzyme kinetics is the study of the chemical reactions that are catalyzed by enzymes, with a focus on their reaction rates. The study of an enzyme's kinetics considers the various stages of activity, reveals the catalytic mechanism of the enzyme, correlates its value to assay conditions, and describes how a drug or a poison might inhibit the enzyme. Victor Henri initially reported that enzyme reactions were initiated by a bond between the enzyme and the substrate. By 1910, Michaelis and Menten had advanced this work by studying the kinetics of the enzyme saccharase, which catalyzes the hydrolysis of sucrose into glucose and fructose. They published their analysis, and ever since, the Michaelis-Menten equation has been used as the standard to describe the kinetics of many enzymes. Unfortunately, soluble enzymes must generally be immobilized to be reused for long times in industrial reactors. In addition, other critical enzyme properties have to be improved like stability, activity, inhibition by reaction products, selectivity toward nonnatural substrates. Immobilization is by far the chosen process to achieve these goals.Although the Michaelis-Menten approach has been regularly adopted for the analysis of immobilized enzyme activity, its applicability to the immobilized state is limited by the barriers the immobilization matrix places upon the measurement of compounds that are used to model enzyme kinetics. That being said, the estimated value of the Michaelis-Menten coefficients (e.g., V(max), K(M)) can be used to evaluate effects of immobilization on enzyme activity in the immobilized state when applied in a controlled manner. In this review, enzyme activity and kinetics are discussed in the context of the immobilized state, and a few novel protocols are presented that address some of the unique constraints imposed by the immobilization barrier.
Kinetics of the humid aging of magnetic recording tape
NASA Technical Reports Server (NTRS)
Bertram, H. N.; Cuddihy, E. F.
1982-01-01
The kinetics of the hydrolysis of polyester urethane binders of magnetic recording tape is described. Kinetic data were generated from measurements of acetone-extractable hydrolyzed binder products versus time for various humidity-temperature environments. These data can be described by a linear, single product, reversible rate equation. This equation, coupled with measurements on the effect of hydrolysis on recorded tape performance, is used to predict proper environmental storage conditions for magnetic tape.
Kinetics of the humid aging of magnetic recording tape
NASA Technical Reports Server (NTRS)
Bertram, H. N.; Cuddihy, E. F.
1982-01-01
The kinetics of the hydrolysis of polyester urethane binders of magnetic recording tape is described. Kinetic data were generated from measurements of acetone-extractable hydrolyzed binder products versus time for various humidity-temperature environments. These data can be described by a linear, single product, reversible rate equation. This equation, coupled with measurements on the effect of hydrolysis on recorded tape performance, is used to predict proper environmental storage conditions for magnetic tape.
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.
Simplification of the unified gas kinetic scheme
NASA Astrophysics Data System (ADS)
Chen, Songze; Guo, Zhaoli; Xu, Kun
2016-08-01
The unified gas kinetic scheme (UGKS) is an asymptotic preserving (AP) scheme for kinetic equations. It is superior for transition flow simulation and has been validated in the past years. However, compared to the well-known discrete ordinate method (DOM), which is a classical numerical method solving the kinetic equations, the UGKS needs more computational resources. In this study, we propose a simplification of the unified gas kinetic scheme. It allows almost identical numerical cost as the DOM, but predicts numerical results as accurate as the UGKS. In the simplified scheme, the numerical flux for the velocity distribution function and the numerical flux for the macroscopic conservative quantities are evaluated separately. The equilibrium part of the UGKS flux is calculated by analytical solution instead of the numerical quadrature in velocity space. The simplification is equivalent to a flux hybridization of the gas kinetic scheme for the Navier-Stokes (NS) equations and the conventional discrete ordinate method. Several simplification strategies are tested, through which we can identify the key ingredient of the Navier-Stokes asymptotic preserving property. Numerical tests show that, as long as the collision effect is built into the macroscopic numerical flux, the numerical scheme is Navier-Stokes asymptotic preserving, regardless the accuracy of the microscopic numerical flux for the velocity distribution function.
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.
Scale covariant gravitation. V. Kinetic theory
Hsieh, S.; Canuto, V.M.
1981-09-01
In this paper we construct a scale covariant kinetic theory for particles and photons. The mathematical framework of the theory is given by the tangent bundle of a Weyl manifold. The Liouville equation is then derived. Solutions corresponding to equilibrium distributions are presented and shown to yield thermodynamic results identical to the ones obtained previously.
The correct kinetic Bohm criterion
NASA Astrophysics Data System (ADS)
Czarnetzki, Uwe; Tsankov, Tsanko Vaskov
2016-09-01
Space charge sheaths are characteristic for bounded plasmas and are a key element in plasma-surface interactions. Hence, one of the most fundamental concepts in plasma physics - the Bohm criterion - is related to the definition of a sheath edge. However, its kinetic formulation is stirring controversies for a long time - from questioning its validity at high collisionality to claiming a divergence in its formulation. Here, based on a solution of the Boltzmann equation for ions with charge-exchange collisions and ionization both of these disputes are resolved: 1) The obtained form of the kinetic Bohm criterion removes the divergence in the ionic part. 2) It also introduces a new equally important term that describes collisional and geometric effects. This new term reestablishes the validity of the criterion at high collisionality. 3) It also restores agreement with the fluid counterpart of the criterion. The developed theory is supported by non-invasive spatially resolved measurements and a numerical model.
Kinetic theory of relativistic plasmas
NASA Technical Reports Server (NTRS)
Gould, R. J.
1981-01-01
The thermalization of particle kinetic motion by binary collisions is considered for a plasma with a Boltzmann constant-temperature product approximately equal to 10 to 100 times the product of the electron mass with the square of the speed of light. At this temperature, the principal mechanism for relaxation of electron motion is via radiationless electron-electron collisions (Moller scattering). Ions are nonrelativistic, but are energetic enough so that their Coulomb scattering can be treated in the Born approximation. Relaxation times are computed and Boltzmann-equation Fokker-Planck operators are derived for the various binary-collision processes. The expression for the rate of kinetic energy exchange between electron and ion gases is derived for the case where the gases are at different temperatures.
Kinetic theory of Jeans instability.
Trigger, S A; Ershkovich, A I; van Heijst, G J F; Schram, P P J M
2004-06-01
Kinetic treatment of the Jeans gravitational instability, with collisions taken into account, is presented. The initial-value problem for the distribution function which obeys the kinetic equation, with the collision integral conserving the number of particles, is solved. Dispersion relation is obtained and analyzed. New modes are found. Collisions are shown not to affect the Jeans instability criterion. Although the instability growth rate diminishes, the collisions they cannot quench the instability. However, the oscillation spectrum is modified significantly: even in the neighborhood of the threshold frequency omega=0 (separating stable and unstable modes) the spectrum of oscillations can strongly depend on the collision frequency. Propagating (rather than aperiodic) modes are also found. These modes, however, are strongly damped.
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.
On Coupled Rate Equations with Quadratic Nonlinearities
Montroll, Elliott W.
1972-01-01
Rate equations with quadratic nonlinearities appear in many fields, such as chemical kinetics, population dynamics, transport theory, hydrodynamics, etc. Such equations, which may arise from basic principles or which may be phenomenological, are generally solved by linearization and application of perturbation theory. Here, a somewhat different strategy is emphasized. Alternative nonlinear models that can be solved exactly and whose solutions have the qualitative character expected from the original equations are first searched for. Then, the original equations are treated as perturbations of those of the solvable model. Hence, the function of the perturbation theory is to improve numerical accuracy of solutions, rather than to furnish the basic qualitative behavior of the solutions of the equations. PMID:16592013
Ordinary differential equation for local accumulation time.
Berezhkovskii, Alexander M
2011-08-21
Cell differentiation in a developing tissue is controlled by the concentration fields of signaling molecules called morphogens. Formation of these concentration fields can be described by the reaction-diffusion mechanism in which locally produced molecules diffuse through the patterned tissue and are degraded. The formation kinetics at a given point of the patterned tissue can be characterized by the local accumulation time, defined in terms of the local relaxation function. Here, we show that this time satisfies an ordinary differential equation. Using this equation one can straightforwardly determine the local accumulation time, i.e., without preliminary calculation of the relaxation function by solving the partial differential equation, as was done in previous studies. We derive this ordinary differential equation together with the accompanying boundary conditions and demonstrate that the earlier obtained results for the local accumulation time can be recovered by solving this equation.
Incorporating qualitative knowledge in enzyme kinetic models using fuzzy logic.
Lee, B; Yen, J; Yang, L; Liao, J C
1999-03-20
Modeling of metabolic pathway dynamics requires detailed kinetic equations at the enzyme level. In particular, the kinetic equations must account for metabolite effectors that contribute significantly to the pathway regulation in vivo. Unfortunately, most kinetic rate laws available in the literature do not consider all the effectors simultaneously, and much kinetic information exists in a qualitative or semiquantitative form. In this article, we present a strategy to incorporate such information into the kinetic equation. This strategy uses fuzzy logic-based factors to modify algebraic rate laws that account for partial kinetic characteristics. The parameters introduced by the fuzzy factors are then optimized by use of a hybrid of simplex and genetic algorithms. The resulting model provides a flexible form that can simulate various kinetic behaviors. Such kinetic models are suitable for pathway modeling without complete enzyme mechanisms. Three enzymes in Escherichia coli central metabolism are used as examples: phosphoenolpyruvate carboxylase; phosphoenolpyruvate carboxykinase; and pyruvate kinase I. Results show that, with fuzzy logic-augmented models, the kinetic data can be much better described. In particular, complex behavior, such as allosteric inhibition, can be captured using fuzzy rules. The resulting models, even though they do not provide additional physical meaning in enzyme mechanisms, allow the model to incorporate semiquantitative information in metabolic pathway models.
Kinetic Theory and Fluid Dynamics
NASA Astrophysics Data System (ADS)
Sone, Yoshio
This monograph gives a comprehensive description of the relationship and connections between kinetic theory and fluid dynamics, mainly for a time-independent problem in a general domain. Ambiguities in this relationship are clarified, and the incompleteness of classical fluid dynamics in describing the behavior of a gas in the continuum limit—recently reported as the ghost effect—is also discussed. The approach used in this work engages an audience of theoretical physicists, applied mathematicians, and engineers. By a systematic asymptotic analysis, fluid-dynamic-type equations and their associated boundary conditions that take into account the weak effect of gas rarefaction are derived from the Boltzmann system. Comprehensive information on the Knudsen-layer correction is also obtained. Equations and their boundary conditions are carefully classified depending on the physical context of problems. Applications are presented to various physically interesting phenomena, including flows induced by temperature fields, evaporation and condensation problems, examples of the ghost effect, and bifurcation of flows. Key features: * many applications and physical models of practical interest * experimental works such as the Knudsen compressor are examined to supplement theory * engineers will not be overwhelmed by sophisticated mathematical techniques * mathematicians will benefit from clarity of definitions and precise physical descriptions given in mathematical terms * appendices collect key derivations and formulas, important to the practitioner, but not easily found in the literature Kinetic Theory and Fluid Dynamics serves as a bridge for those working in different communities where kinetic theory or fluid dynamics is important: graduate students, researchers and practitioners in theoretical physics, applied mathematics, and various branches of engineering. The work can be used in graduate-level courses in fluid dynamics, gas dynamics, and kinetic theory; some parts
Coagulation equations with gelation
Hendriks, E.M.; Ernst, M.H.; Ziff, R.M.
1983-06-01
Smoluchowski's equation for rapid coagulation is used to describe the kinetics of gelation, in which the coagulation kernel K/sub i/j models the bonding mechanism. For different classes of kernels we derive criteria for the occurrences of gelation, and obtain critical exponents in the pre- and postgelation stage in terms of the model parameters; we calculate bounds on the time of gelation t/sub c/, and give an exact postgelation solution for the model K/sub i/j = (ij)/sup ..omega../ (..omega..>1/2) and K/sub i/j = ..cap alpha../sup i/+j (..cap alpha..>1). For the model K/sub i/j = i/sup ..omega../+j/sup ..omega../ (..omega..<1, without gelation) initial solutions are given. It is argued that the kernel K/sub i/japprox. (ij)/sup ..omega../ with ..omega..approx. =1-1/d (d is dimensionality) effectively models the sol-gel transformation is polymerizing systems and approximately accounts for the effects of cross-linking and steric hindrance neglected in the classical theory of Flory and Stockmayer (..omega.. = 1). For all ..omega.. the exponents, tau = ..omega..+3/2 and sigma = ..omega..-1/2, ..gamma.. = (3/2-..omega..)/(..omega..-1/2) and ..beta.. = 1, characterize the size distribution, at the slightly below the gel point, under the assumption that scaling is valid.
Equating Error in Observed-Score Equating
ERIC Educational Resources Information Center
van der Linden, Wim J.
2006-01-01
Traditionally, error in equating observed scores on two versions of a test is defined as the difference between the transformations that equate the quantiles of their distributions in the sample and population of test takers. But it is argued that if the goal of equating is to adjust the scores of test takers on one version of the test to make…
Representing Rate Equations for Enzyme-Catalyzed Reactions
ERIC Educational Resources Information Center
Ault, Addison
2011-01-01
Rate equations for enzyme-catalyzed reactions are derived and presented in a way that makes it easier for the nonspecialist to see how the rate of an enzyme-catalyzed reaction depends upon kinetic constants and concentrations. This is done with distribution equations that show how the rate of the reaction depends upon the relative quantities of…
Representing Rate Equations for Enzyme-Catalyzed Reactions
ERIC Educational Resources Information Center
Ault, Addison
2011-01-01
Rate equations for enzyme-catalyzed reactions are derived and presented in a way that makes it easier for the nonspecialist to see how the rate of an enzyme-catalyzed reaction depends upon kinetic constants and concentrations. This is done with distribution equations that show how the rate of the reaction depends upon the relative quantities of…
Kinetic Effects in Dynamic Wetting
NASA Astrophysics Data System (ADS)
Sprittles, James E.
2017-03-01
The maximum speed at which a liquid can wet a solid is limited by the need to displace gas lubrication films in front of the moving contact line. The characteristic height of these films is often comparable to the mean free path in the gas so that hydrodynamic models do not adequately describe the flow physics. This Letter develops a model which incorporates kinetic effects in the gas, via the Boltzmann equation, and can predict experimentally observed increases in the maximum speed of wetting when (a) the liquid's viscosity is varied, (b) the ambient gas pressure is reduced, or (c) the meniscus is confined.
Transport Equations In Tokamak Plasmas
NASA Astrophysics Data System (ADS)
Callen, J. D.
2009-11-01
Tokamak plasma transport equations are usually obtained by flux surface averaging the collisional Braginskii equations. However, tokamak plasmas are not in collisional regimes. Also, ad hoc terms are added for: neoclassical effects on the parallel Ohm's law (trapped particle effects on resistivity, bootstrap current); fluctuation-induced transport; heating, current-drive and flow sources and sinks; small B field non-axisymmetries; magnetic field transients etc. A set of self-consistent second order in gyroradius fluid-moment-based transport equations for nearly axisymmetric tokamak plasmas has been developed recently using a kinetic-based framework. The derivation uses neoclassical-based parallel viscous force closures, and includes all the effects noted above. Plasma processes on successive time scales (and constraints they impose) are considered sequentially: compressional Alfv'en waves (Grad-Shafranov equilibrium, ion radial force balance); sound waves (pressure constant along field lines, incompressible flows within a flux surface); and ion collisions (damping of poloidal flow). Radial particle fluxes are driven by the many second order in gyroradius toroidal angular torques on the plasma fluid: 7 ambipolar collision-based ones (classical, neoclassical, etc.) and 8 non-ambipolar ones (fluctuation-induced, polarization flows from toroidal rotation transients etc.). The plasma toroidal rotation equation [1] results from setting to zero the net radial current induced by the non-ambipolar fluxes. The radial particle flux consists of the collision-based intrinsically ambipolar fluxes plus the non-ambipolar fluxes evaluated at the ambipolarity-enforcing toroidal plasma rotation (radial electric field). The energy transport equations do not involve an ambipolar constraint and hence are more directly obtained. The resultant transport equations will be presented and contrasted with the usual ones. [4pt] [1] J.D. Callen, A.J. Cole, C.C. Hegna, ``Toroidal Rotation In
Transport equations in tokamak plasmas
Callen, J. D.; Hegna, C. C.; Cole, A. J.
2010-05-15
Tokamak plasma transport equations are usually obtained by flux surface averaging the collisional Braginskii equations. However, tokamak plasmas are not in collisional regimes. Also, ad hoc terms are added for neoclassical effects on the parallel Ohm's law, fluctuation-induced transport, heating, current-drive and flow sources and sinks, small magnetic field nonaxisymmetries, magnetic field transients, etc. A set of self-consistent second order in gyroradius fluid-moment-based transport equations for nearly axisymmetric tokamak plasmas has been developed using a kinetic-based approach. The derivation uses neoclassical-based parallel viscous force closures, and includes all the effects noted above. Plasma processes on successive time scales and constraints they impose are considered sequentially: compressional Alfven waves (Grad-Shafranov equilibrium, ion radial force balance), sound waves (pressure constant along field lines, incompressible flows within a flux surface), and collisions (electrons, parallel Ohm's law; ions, damping of poloidal flow). Radial particle fluxes are driven by the many second order in gyroradius toroidal angular torques on a plasma species: seven ambipolar collision-based ones (classical, neoclassical, etc.) and eight nonambipolar ones (fluctuation-induced, polarization flows from toroidal rotation transients, etc.). The plasma toroidal rotation equation results from setting to zero the net radial current induced by the nonambipolar fluxes. The radial particle flux consists of the collision-based intrinsically ambipolar fluxes plus the nonambipolar fluxes evaluated at the ambipolarity-enforcing toroidal plasma rotation (radial electric field). The energy transport equations do not involve an ambipolar constraint and hence are more directly obtained. The 'mean field' effects of microturbulence on the parallel Ohm's law, poloidal ion flow, particle fluxes, and toroidal momentum and energy transport are all included self-consistently. The
Solid State Kinetic Parameters and Chemical Mechanism of the Dehydration of CoCl2.6H2O.
ERIC Educational Resources Information Center
Ribas, Joan; And Others
1988-01-01
Presents an experimental example illustrating the most common methods for the determination of kinetic parameters. Discusses the different theories and equations to be applied and the mechanism derived from the kinetic results. (CW)
Solid State Kinetic Parameters and Chemical Mechanism of the Dehydration of CoCl2.6H2O.
ERIC Educational Resources Information Center
Ribas, Joan; And Others
1988-01-01
Presents an experimental example illustrating the most common methods for the determination of kinetic parameters. Discusses the different theories and equations to be applied and the mechanism derived from the kinetic results. (CW)
Practical steady-state enzyme kinetics.
Lorsch, Jon R
2014-01-01
Enzymes are key components of most biological processes. Characterization of enzymes is therefore frequently required during the study of biological systems. Steady-state kinetics provides a simple and rapid means of assessing the substrate specificity of an enzyme. When combined with site-directed mutagenesis (see Site-Directed Mutagenesis), it can be used to probe the roles of particular amino acids in the enzyme in substrate recognition and catalysis. Effects of interaction partners and posttranslational modifications can also be assessed using steady-state kinetics. This overview explains the general principles of steady-state enzyme kinetics experiments in a practical, rather than theoretical, way. Any biochemistry textbook will have a section on the theory of Michaelis-Menten kinetics, including derivations of the relevant equations. No specific enzymatic assay is described here, although a method for monitoring product formation or substrate consumption over time (an assay) is required to perform the experiments described. © 2014 Elsevier Inc. All rights reserved.
Parabolic approximation method for the mode conversion-tunneling equation
Phillips, C.K.; Colestock, P.L.; Hwang, D.Q.; Swanson, D.G.
1987-07-01
The derivation of the wave equation which governs ICRF wave propagation, absorption, and mode conversion within the kinetic layer in tokamaks has been extended to include diffraction and focussing effects associated with the finite transverse dimensions of the incident wavefronts. The kinetic layer considered consists of a uniform density, uniform temperature slab model in which the equilibrium magnetic field is oriented in the z-direction and varies linearly in the x-direction. An equivalent dielectric tensor as well as a two-dimensional energy conservation equation are derived from the linearized Vlasov-Maxwell system of equations. The generalized form of the mode conversion-tunneling equation is then extracted from the Maxwell equations, using the parabolic approximation method in which transverse variations of the wave fields are assumed to be weak in comparison to the variations in the primary direction of propagation. Methods of solving the generalized wave equation are discussed. 16 refs.
NASA Technical Reports Server (NTRS)
Hamrock, B. J.; Dowson, D.
1981-01-01
Lubricants, usually Newtonian fluids, are assumed to experience laminar flow. The basic equations used to describe the flow are the Navier-Stokes equation of motion. The study of hydrodynamic lubrication is, from a mathematical standpoint, the application of a reduced form of these Navier-Stokes equations in association with the continuity equation. The Reynolds equation can also be derived from first principles, provided of course that the same basic assumptions are adopted in each case. Both methods are used in deriving the Reynolds equation, and the assumptions inherent in reducing the Navier-Stokes equations are specified. Because the Reynolds equation contains viscosity and density terms and these properties depend on temperature and pressure, it is often necessary to couple the Reynolds with energy equation. The lubricant properties and the energy equation are presented. Film thickness, a parameter of the Reynolds equation, is a function of the elastic behavior of the bearing surface. The governing elasticity equation is therefore presented.
Chemical and Biological Kinetics
NASA Astrophysics Data System (ADS)
Emanuel', N. M.
1981-10-01
Examples of the application of the methods and ideas of chemical kinetics in various branches of chemistry and biology are considered and the results of studies on the kinetics and mechanisms of autoxidation and inhibited and catalysed oxidation of organic substances in the liquid phase are surveyed. Problems of the kinetics of the ageing of polymers and the principles of their stabilisation are discussed and certain trends in biological kinetics (kinetics of tumour growth, kinetic criteria of the effectiveness of chemotherapy, problems of gerontology, etc.) are considered. The bibliography includes 281 references.
Variational Derivation of Dissipative Equations
NASA Astrophysics Data System (ADS)
Sogo, Kiyoshi
2017-03-01
A new variational principle is formulated to derive various dissipative equations. Model equations considered are the damping equation, Bloch equation, diffusion equation, Fokker-Planck equation, Kramers equation and Smoluchowski equation. Each equation and its time reversal equation are simultaneously obtained in our variational principle.
NASA Astrophysics Data System (ADS)
Dimakis, Aristophanes; Müller-Hoissen, Folkert
2015-06-01
It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a ''mixed order''. We describe simplex equations (including the Yang-Baxter equation) as realizations of higher Bruhat orders. Correspondingly, a family of ''polygon equations'' realizes higher Tamari orders. They generalize the well-known pentagon equation. The structure of simplex and polygon equations is visualized in terms of deformations of maximal chains in posets forming 1-skeletons of polyhedra. The decomposition of higher Bruhat orders induces a reduction of the N-simplex equation to the (N+1)-gon equation, its dual, and a compatibility equation.
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
Solution of Chemical Master Equations for Nonlinear Stochastic Reaction Networks.
Smadbeck, Patrick; Kaznessis, Yiannis N
2014-08-01
Stochasticity in the dynamics of small reacting systems requires discrete-probabilistic models of reaction kinetics instead of traditional continuous-deterministic ones. The master probability equation is a complete model of randomly evolving molecular populations. Because of its ambitious character, the master equation remained unsolved for all but the simplest of molecular interaction networks. With the first solution of chemical master equations, a wide range of experimental observations of small-system interactions may be mathematically conceptualized.
Solution of Chemical Master Equations for Nonlinear Stochastic Reaction Networks
Smadbeck, Patrick; Kaznessis, Yiannis N.
2014-01-01
Stochasticity in the dynamics of small reacting systems requires discrete-probabilistic models of reaction kinetics instead of traditional continuous-deterministic ones. The master probability equation is a complete model of randomly evolving molecular populations. Because of its ambitious character, the master equation remained unsolved for all but the simplest of molecular interaction networks. With the first solution of chemical master equations, a wide range of experimental observations of small-system interactions may be mathematically conceptualized. PMID:25215268
Power-law spatial dispersion from fractional Liouville equation
Tarasov, Vasily E.
2013-10-15
A microscopic model in the framework of fractional kinetics to describe spatial dispersion of power-law type is suggested. The Liouville equation with the Caputo fractional derivatives is used to obtain the power-law dependence of the absolute permittivity on the wave vector. The fractional differential equations for electrostatic potential in the media with power-law spatial dispersion are derived. The particular solutions of these equations for the electric potential of point charge in this media are considered.
Comparative evaluation of adsorption kinetics of diclofenac and isoproturon by activated carbon.
Torrellas, Silvia A; Rodriguez, Araceli R; Escudero, Gabriel O; Martín, José María G; Rodriguez, Juan G
2015-01-01
Adsorption mechanism of diclofenac and isoproturon onto activated carbon has been proposed using Langmuir and Freundlich isotherms. Adsorption capacity and optimum adsorption isotherms were predicted by nonlinear regression method. Different kinetic equations, pseudo-first-order, pseudo-second-order, intraparticle diffusion model and Bangham kinetic model, were applied to study the adsorption kinetics of emerging contaminants on activated carbon in two aqueous matrices.
Cesium removal and kinetics equilibrium: Precipitation kinetics
Barnes, M.J.
1999-12-17
This task consisted of both non-radioactive and radioactive (tracer) tests examining the influence of potentially significant variables on cesium tetraphenylborate precipitation kinetics. The work investigated the time required to reach cesium decontamination and the conditions that affect the cesium precipitation kinetics.
Traffic flow equations coming from the Grad's method.
NASA Astrophysics Data System (ADS)
Velasco, Rosa M.; Méndez, Alma R.
2006-11-01
The usual Grad's method in kinetic theory of gases is developed to construct a new model in traffic flow problems. This is applied to the kinetic equation called as the Paveri-Fontana equation which tells us how the distribution function evolves in time [1]. We assume a special model for the desired velocity of drivers [2] and the Grad's method provides us with a closure relation in the macroscopic equations. The simulation results for this model allow us to find the behavior of density, mean velocity and the velocity variance in the system. All the results are consistent with the validity region of the kinetic equation and with the qualitative behavior proper to traffic models. We show some comparisons with other models in the literature [3]. [1] S.L Paveri-Fontana; Transp. Res. 9 (1975), 225. [2] R.M. Velasco, W. Marques Jr.; Phys. Rev. E72 (2005), 046102. [3] D. Helbing; Phys. Rev. E51 (1995), 3164.
On the Simultaneous Iron Oxide Reduction and Carburization Kinetics
NASA Astrophysics Data System (ADS)
D'Abreu, Jose Carlos; Kohler, Helio Marques; Falero, Edelink Efrain Tinoco; Otaviano, Mauricio Marcos
Nowadays the most important Direct Reduction — DR processes in shaft furnaces has to deal with carbon fines precipitation and DRI carburization issues. Based in a cooperative research program joining Catholic University (PUC-Rio) and SAMARCO Mining Co, a project dealing with pellets reduction and those two phenomena was established. This work analyzes kinetically the three reactions mentioned before, considering typical values for the operational variables temperature, flowrate, pressure and gas composition, parameters commonly used to control the DRI formation in the Reduction Zone — RZ of the shaft furnaces. From laboratories experimental results, the kinetic equations for those reactions were established and, using the superposition principle, generated a specific global kinetic model for the iron oxide reduction and the soot formation. Finally, using those experimental results and applying a planned statistical factorial analysis for the experiments, the numerical coefficients for each equation were calculated and the correlation factor determined for the proposed global kinetic equation.
Towards adaptive kinetic-fluid simulations of weakly ionized plasmas
NASA Astrophysics Data System (ADS)
Kolobov, V. I.; Arslanbekov, R. R.
2012-02-01
This paper describes an Adaptive Mesh and Algorithm Refinement (AMAR) methodology for multi-scale simulations of gas flows and the challenges associated with extending this methodology for simulations of weakly ionized plasmas. The AMAR method combines Adaptive Mesh Refinement (AMR) with automatic selection of kinetic or continuum solvers in different parts of computational domains. We first review the discrete velocity method for solving Boltzmann and Wang Chang-Uhlenbeck kinetic equations for rarefied gases. Then, peculiarities of AMR implementation with octree Cartesian mesh are discussed. A Unified Flow Solver (UFS) uses AMAR method with adaptive Cartesian mesh to dynamically introduce kinetic patches for multi-scale simulations of gas flows. We describe fluid plasma models with AMR capabilities and illustrate how physical models affect simulation results for gas discharges, especially in the areas where electron kinetics plays an important role. We introduce Eulerian solvers for plasma kinetic equations and illustrate the concept of adaptive mesh in velocity space. Specifics of electron kinetics in collisional plasmas are described focusing on deterministic methods of solving kinetic equations for electrons under different conditions. We illustrate the appearance of distinct groups of electrons in the cathode region of DC discharges and discuss the physical models appropriate for each group. These kinetic models are currently being incorporated into AMAR methodology for multi-scale plasma simulations.
Kinetic energy budgets in areas of intense convection
NASA Technical Reports Server (NTRS)
Fuelberg, H. E.; Berecek, E. M.; Ebel, D. M.; Jedlovec, G. J.
1980-01-01
A kinetic energy budget analysis of the AVE-SESAME 1 period which coincided with the deadly Red River Valley tornado outbreak is presented. Horizontal flux convergence was found to be the major kinetic energy source to the region, while cross contour destruction was the major sink. Kinetic energy transformations were dominated by processes related to strong jet intrusion into the severe storm area. A kinetic energy budget of the AVE 6 period also is presented. The effects of inherent rawinsonde data errors on widely used basic kinematic parameters, including velocity divergence, vorticity advection, and kinematic vertical motion are described. In addition, an error analysis was performed in terms of the kinetic energy budget equation. Results obtained from downward integration of the continuity equation to obtain kinematic values of vertical motion are described. This alternate procedure shows promising results in severe storm situations.
Kinetic properties of fractal stellar media
NASA Astrophysics Data System (ADS)
Chumak, O. V.; Rastorguev, A. S.
2017-01-01
Kinetic processes in fractal stellar media are analysed in terms of the approach developed in our earlier paper involving a generalization of the nearest neighbour and random force distributions to fractal media. Diffusion is investigated in the approximation of scale-dependent conditional density based on an analysis of the solutions of the corresponding Langevin equations. It is shown that kinetic parameters (time-scales, coefficients of dynamic friction, diffusion, etc.) for fractal stellar media can differ significantly both qualitatively and quantitatively from the corresponding parameters for a quasi-uniform random media with limited fluctuations. The most important difference is that in the fractal case, kinetic parameters depend on spatial scalelength and fractal dimension of the medium studied. A generalized kinetic equation for stellar media (fundamental equation of stellar dynamics) is derived in the Fokker-Planck approximation with the allowance for the fractal properties of the spatial stellar density distribution. Also derived are its limit forms that can be used to describe small departures of fractal gravitating medium from equilibrium.
Kinetic and hydrodynamic models of chemotactic aggregation
NASA Astrophysics Data System (ADS)
Chavanis, Pierre-Henri; Sire, Clément
2007-10-01
We derive general kinetic and hydrodynamic models of chemotactic aggregation that describe certain features of the morphogenesis of biological colonies (like bacteria, amoebae, endothelial cells or social insects). Starting from a stochastic model defined in terms of N coupled Langevin equations, we derive a nonlinear mean-field Fokker-Planck equation governing the evolution of the distribution function of the system in phase space. By taking the successive moments of this kinetic equation and using a local thermodynamic equilibrium condition, we derive a set of hydrodynamic equations involving a damping term. In the limit of small frictions, we obtain a hyperbolic model describing the formation of network patterns (filaments) and in the limit of strong frictions we obtain a parabolic model which is a generalization of the standard Keller-Segel model describing the formation of clusters (clumps). Our approach connects and generalizes several models introduced in the chemotactic literature. We discuss the analogy between bacterial colonies and self-gravitating systems and between the chemotactic collapse and the gravitational collapse (Jeans instability). We also show that the basic equations of chemotaxis are similar to nonlinear mean-field Fokker-Planck equations so that a notion of effective generalized thermodynamics can be developed.
Kinetic Models for the Trading of Goods
NASA Astrophysics Data System (ADS)
Toscani, Giuseppe; Brugna, Carlo; Demichelis, Stefano
2013-05-01
In this paper we introduce kinetic equations for the evolution of the probability distribution of two goods among a huge population of agents. The leading idea is to describe the trading of these goods by means of some fundamental rules in price theory, in particular by using Cobb-Douglas utility functions for the binary exchange, and the Edgeworth box for the description of the common exchange area in which utility is increasing for both agents. This leads to a Boltzmann-type equation in which the post-interaction variables depend in a nonlinear way from the pre-interaction ones. Other models will be derived, by suitably linearizing this Boltzmann equation. In presence of uncertainty in the exchanges, it is shown that the solution to some of the linearized kinetic equations develop Pareto tails, where the Pareto index depends on the ratio between the gain and the variance of the uncertainty. In particular, the result holds true for the solution of a drift-diffusion equation of Fokker-Planck type, obtained from the linear Boltzmann equation as the limit of quasi-invariant trades.
Kinetics of Propargyl Radical Dissociation.
Klippenstein, Stephen J; Miller, James A; Jasper, Ahren W
2015-07-16
Due to the prominent role of the propargyl radical for hydrocarbon growth within combustion environments, it is important to understand the kinetics of its formation and loss. The ab initio transition state theory-based master equation method is used to obtain theoretical kinetic predictions for the temperature and pressure dependence of the thermal decomposition of propargyl, which may be its primary loss channel under some conditions. The potential energy surface for the decomposition of propargyl is first mapped at a high level of theory with a combination of coupled cluster and multireference perturbation calculations. Variational transition state theory is then used to predict the microcanonical rate coefficients, which are subsequently implemented within the multiple-well multiple-channel master equation. A variety of energy transfer parameters are considered, and the sensitivity of the thermal rate predictions to these parameters is explored. The predictions for the thermal decomposition rate coefficient are found to be in good agreement with the limited experimental data. Modified Arrhenius representations of the rate constants are reported for utility in combustion modeling.
ERIC Educational Resources Information Center
Blakley, G. R.
1982-01-01
Reviews mathematical techniques for solving systems of homogeneous linear equations and demonstrates that the algebraic method of balancing chemical equations is a matter of solving a system of homogeneous linear equations. FORTRAN programs using this matrix method to chemical equation balancing are available from the author. (JN)
Relativistic Langevin equation for runaway electrons
NASA Astrophysics Data System (ADS)
Mier, J. A.; Martin-Solis, J. R.; Sanchez, R.
2016-10-01
The Langevin approach to the kinetics of a collisional plasma is developed for relativistic electrons such as runaway electrons in tokamak plasmas. In this work, we consider Coulomb collisions between very fast, relativistic electrons and a relatively cool, thermal background plasma. The model is developed using the stochastic equivalence of the Fokker-Planck and Langevin equations. The resulting Langevin model equation for relativistic electrons is an stochastic differential equation, amenable to numerical simulations by means of Monte-Carlo type codes. Results of the simulations will be presented and compared with the non-relativistic Langevin equation for RE electrons used in the past. Supported by MINECO (Spain), Projects ENE2012-31753, ENE2015-66444-R.
Neutral Vlasov kinetic theory of magnetized plasmas
Tronci, Cesare; Camporeale, Enrico
2015-02-15
The low-frequency limit of Maxwell equations is considered in the Maxwell-Vlasov system. This limit produces a neutral Vlasov system that captures essential features of plasma dynamics, while neglecting radiation effects. Euler-Poincaré reduction theory is used to show that the neutral Vlasov kinetic theory possesses a variational formulation in both Lagrangian and Eulerian coordinates. By construction, the new model recovers all collisionless neutral models employed in plasma simulations. Then, comparisons between the neutral Vlasov system and hybrid kinetic-fluid models are presented in the linear regime.
Kinetics and Morphology of Living Polymer Systems
NASA Astrophysics Data System (ADS)
Gopinathan, Ajay; Schwarz, J. M.; Lee, Kun-Chun; Liu, Andrea J.
2004-03-01
The biopolymer F-actin is a key component of the cellular cytoskeleton and is important for cell motility.The actin cytoskeleton is not static but evolves via kinetic processes such as actin polymerization, depolymerization and branching. Various proteins are used to regulate these processses in order for the cell to crawl. Here we model the kinetics using a rate equation approach and focus on the resulting steady-state morphology. We determine such characteristics as the length distribution of filaments, density of branches and crawling rate as functions of the concentration of actin and associated proteins.
Kinetic Approach for Laser-Induced Plasmas
Omar, Banaz; Rethfeld, Baerbel
2008-10-22
Non-equilibrium distribution functions of electron gas and phonon gas excited with ultrashort intense laser pulses are calculated for laser-induced plasmas occurring in solids. The excitation during femtosecond irradiation and the subsequent thermalization of the free electrons, as well as the dynamics of phonons are described by kinetic equations. The microscopic collision processes, such as absorption by inverse bremsstrahlung, electron-electron collisions, and electron-phonon interactions are considered by complete Boltzmann collision integrals. We apply our kinetic approach for gold by taking s-band electron into account and compare it with the case of excitation of d-band electrons.
Kinetic Approach for Laser-Induced Plasmas
NASA Astrophysics Data System (ADS)
Omar, Banaz; Rethfeld, Bärbel
2008-10-01
Non-equilibrium distribution functions of electron gas and phonon gas excited with ultrashort intense laser pulses are calculated for laser-induced plasmas occurring in solids. The excitation during femtosecond irradiation and the subsequent thermalization of the free electrons, as well as the dynamics of phonons are described by kinetic equations. The microscopic collision processes, such as absorption by inverse bremsstrahlung, electron-electron collisions, and electron-phonon interactions are considered by complete Boltzmann collision integrals. We apply our kinetic approach for gold by taking s-band electron into account and compare it with the case of excitation of d-band electrons.
Kinetics equation replacement function for a particular continuous intake scenario.
Potter, Charles A
2013-01-01
An important paper by Skrable et al. included a retention function for compartment contents during a continuous intake, including the same time variable in both the numerator and denominator of the replacement function. In fact, the time in the denominator should have been represented as a constant describing the ultimate period length for the continuous intake, whether greater than, less than, or equal to the time variable for the associated measurement.
Solutions of the chemical kinetic equations for initially inhomogeneous mixtures.
NASA Technical Reports Server (NTRS)
Hilst, G. R.
1973-01-01
Following the recent discussions by O'Brien (1971) and Donaldson and Hilst (1972) of the effects of inhomogeneous mixing and turbulent diffusion on simple chemical reaction rates, the present report provides a more extensive analysis of when inhomogeneous mixing has a significant effect on chemical reaction rates. The analysis is then extended to the development of an approximate chemical sub-model which provides much improved predictions of chemical reaction rates over a wide range of inhomogeneities and pathological distributions of the concentrations of the reacting chemical species. In particular, the development of an approximate representation of the third-order correlations of the joint concentration fluctuations permits closure of the chemical sub-model at the level of the second-order moments of these fluctuations and the mean concentrations.
Kinetic-equation approach to diffusive superconducting hybrid devices
Stoof, T.H.; Nazarov, Y.V.
1996-06-01
We present calculations of the temperature-dependent electrostatic and chemical potential distributions in disordered normal metal-superconductor structures. We show that they differ appreciably in the presence of a superconducting terminal and propose an experiment to measure these two different potential distributions. We also compute the resistance change in these structures due to a recently proposed mechanism which causes a finite effect at zero temperature. The relative resistance change due to this effect is of the order of the interaction parameter in the normal metal. Finally a detailed calculation of the resistance change due to the temperature dependence of Andreev reflection in diffusive systems is presented. We find that the maximal magnitude due to this thermal effect is in general much larger than the magnitude of the interaction effect. {copyright} {ital 1996 The American Physical Society.}
Ф-order kinetics of photoreversible-drug reactions.
Maafi, Mounir; Maafi, Wassila
2014-08-25
Drug photodegradation data are usually treated by zero-, first- or second-order kinetic equations. Such treatments would lack reliability since the aforementioned equations have been originally developed for pure thermal reactions. In this respect, it has recently been shown that unimolecular photodegradations obey Ф-order kinetics (Maafi and Maafi, 2013). However, no similar information is, thus far, available for other reactions including photoreversible AB(2Ф) systems. This paper aims at filling this gap for AB(2Ф) kinetics. Runge-Kutta numerical integration data for photoreversible reactions traces were combined with a template equation in order to derive an optimized (semi-empirical) integrated rate-law equation for AB(2Ф) reactions. The proposed model equation was test by examining its ability to fit synthetic Runge-Kutta data that have not been used for the optimization. The obtained fitting parameters are then compared to their theoretical counterparts. Both an integrated rate-law and an analytical equation for the overall reaction rate-constant were set for photoreversible drug reactions. The values of overall reaction rate-constant and initial velocity obtained theoretically correlated well with those obtained by fitting the kinetic traces of reactions with the derived integrated rate-law. AB(2Ф) photodegradation reactions have been proven to obey Ф-order kinetics. The equation proposed describes faithfully their kinetic behaviour in solution. The formula of the overall rate-constant involves both reagents' characteristics and experimental parameters. These equations facilitated the rationalisation and prediction of the individual effects of each reaction parameter. Specially, our results proved a self-photostabilisation with increasing initial drug-concentration and demonstrated the potential for actinometry of drugs obeying AB(2Ф) mechanism. Copyright © 2014 Elsevier B.V. All rights reserved.
Resonance Van Hove singularities in wave kinetics
NASA Astrophysics Data System (ADS)
Shi, Yi-Kang; Eyink, Gregory L.
2016-10-01
Wave kinetic theory has been developed to describe the statistical dynamics of weakly nonlinear, dispersive waves. However, we show that systems which are generally dispersive can have resonant sets of wave modes with identical group velocities, leading to a local breakdown of dispersivity. This shows up as a geometric singularity of the resonant manifold and possibly as an infinite phase measure in the collision integral. Such singularities occur widely for classical wave systems, including acoustical waves, Rossby waves, helical waves in rotating fluids, light waves in nonlinear optics and also in quantum transport, e.g. kinetics of electron-hole excitations (matter waves) in graphene. These singularities are the exact analogue of the critical points found by Van Hove in 1953 for phonon dispersion relations in crystals. The importance of these singularities in wave kinetics depends on the dimension of phase space D =(N - 2) d (d physical space dimension, N the number of waves in resonance) and the degree of degeneracy δ of the critical points. Following Van Hove, we show that non-degenerate singularities lead to finite phase measures for D > 2 but produce divergences when D ≤ 2 and possible breakdown of wave kinetics if the collision integral itself becomes too large (or even infinite). Similar divergences and possible breakdown can occur for degenerate singularities, when D - δ ≤ 2, as we find for several physical examples, including electron-hole kinetics in graphene. When the standard kinetic equation breaks down, then one must develop a new singular wave kinetics. We discuss approaches from pioneering 1971 work of Newell & Aucoin on multi-scale perturbation theory for acoustic waves and field-theoretic methods based on exact Schwinger-Dyson integral equations for the wave dynamics.
Gao, Jeff Y
2012-12-18
A dissolution model that integrates the solid-liquid interface kinetics and the mass transport kinetics is introduced. Such a model reduces to the Noyes-Whitney equation under special conditions, but offers expanded range of applicability and flexibility fitting dissolution profiles when interfacial kinetics and interfacial concentration deviate from the assumptions implied in the Noyes-Whitney equation. General solutions to the integrated dissolution model derived for noninteractive solutes as well as for solutes participating in ionization equilibrium are discussed. Parameters defining the integrated dissolution model are explained conceptually along with practical ways for their determinations. Conditions under which the model exhibits supersaturation features are elaborated. Simulated dissolution profiles using the integrated dissolution model for published experimental data exhibiting supersaturation features are illustrated.
Reflected kinetics model for nuclear space reactor kinetics and control scoping calculations
Washington, K.E.
1986-01-01
Renewed interest in space nuclear applications has motivated the study of a specialized reactor kinetics model. Consideration of a kinetics model favorable for study of the feasibility of automatic control of these devices is warranted. The need to bridge this gap between reactor kinetics and automatic control in conjunction with the control drum design characteristic of next generation paper space reactors inspired the development of a new Reflected Kinetics (RK) model. An extension of the conventional point-kinetics (PK) model was done in order to explicitly correlate reactivity and the reflector/absorber control drums characteristic of space nuclear reactor designs. Open-loop computations and numerical comparison to analytic PK equations indicated that the RK model is a functional alternative to equivalent bare point kinetics in the analysis of moderate transients. Variations in the RK reflector-to-core transfer probabilities and coolant flow rate do indeed drive the transient differently than the lumped insertion of equivalent reactivity amounts in the core. These computations illustrated the potential importance of the utilization of variable coolant flow rate to aid control in space reactor systems limited by minimal drum reactivity worth. Additionally the Doppler reactivity shutdown mechanism was concluded to be the primarily reliable means of safety shutdown in such systems. The structure of the RK equations proved to be advantageous for integration of automatic control.
Eulerian action principles for linearized reduced dynamical equations
Brizard, A. )
1994-08-01
New Eulerian action principles for the linearized gyrokinetic Maxwell--Vlasov equations and the linearized kinetic-magnetohydrodynamic (kinetic-MHD) equations are presented. The variational fields for the linearized gyrokinetic Vlasov--Maxwell equations are the perturbed electromagnetic potentials ([phi][sub 1],[bold A][sub 1]) and the gyroangle-independent gyrocenter (gy) function [ital S][sub gy], while the variational fields for the linearized kinetic-MHD equations are the ideal MHD fluid displacement [xi] and the gyroangle-independent drift-kinetic (dk) function [ital S][sub dk] (defined as the drift-kinetic limit of [ital S][sub gy]). According to the Lie-transform approach to Vlasov perturbation theory, [ital S][sub gy] generates first-order perturbations in the gyrocenter distribution [ital F][sub 1][equivalent to][l brace][ital S][sub gy], [ital F][sub 0][r brace][sub gc], where [ital F][sub 1] satisfies the linearized gyrokinetic Vlasov equation and [l brace] , [r brace][sub gc] denotes the unperturbed guiding-center (gc) Poisson bracket. Previous quadratic variational forms were constructed [ital ad] [ital hoc] from the linearized equations, and required the linearized gyrokinetic (or drift-kinetic) Vlasov equation to be solved [ital a] [ital priori] (e.g., by integration along an unperturbed guiding-center orbit) through the use of the normal-mode and ballooning-mode representations. The presented action principles ignore these requirements and, thus, apply to more general perturbations.
Enzyme kinetics of oxidative metabolism: cytochromes P450.
Korzekwa, Ken
2014-01-01
The cytochrome P450 enzymes (CYPs) are the most important enzymes in the oxidative metabolism of hydrophobic drugs and other foreign compounds (xenobiotics). The versatility of these enzymes results in some unusual kinetic properties, stemming from the simultaneous interaction of multiple substrates with the CYP active site. Often, the CYPs display kinetics that deviate from standard hyperbolic saturation or inhibition kinetics. Non-Michaelis-Menten or "atypical" saturation kinetics include sigmoidal, biphasic, and substrate inhibition kinetics (see Chapter 3 ). Interactions between substrates include competitive inhibition, noncompetitive inhibition, mixed inhibition, partial inhibition, activation, and activation followed by inhibition (see Chapter 4 ). Models and equations that can result in these kinetic profiles will be presented and discussed.
The quasicontinuum Fokker-Plank equation
Alexander, Francis J
2008-01-01
We present a regularized Fokker-Planck equation with more accurate short-time and high-frequency behavior for continuous-time, discrete-state systems. The regularization preserves crucial aspects of state-space discreteness lost in the standard Kramers-Moyal expansion. We apply the method to a simple example of biochemical reaction kinetics and to a two-dimensional symmetric random walk, and suggest its application to more complex systerns.
Unified fluid/kinetic description of magnetized plasmas
Chang, Zuoyang; Callen, J.D.
1991-06-01
Unified fluid/kinetic equations for the plasma perturbed density ({tilde n}), parallel flow velocity ({tilde u}{sub {parallel}}) and temperature ({tilde T}) are developed in a sheared slab geometry by calculating the fluid moment closure relations kinetically. At first, a set of (unclosed) nonlinear perturbed fluid equations for {tilde n}, {tilde u}{sub {parallel}} and {tilde T} is developed using a drift ordering analysis and a new gyroviscous force ({del} {center dot} {product}{sub g}). Thereafter, to develop linear closure relations for b {center dot} {del} {center dot} {tilde product}{sub {parallel}} and {tilde q}{sub {parallel}}, a drift-kinetic version of a Chapman-Enskog-like (CEL) equation is developed and solved by using a moment approach and a physically realistic collision operator (Lorentz scattering operator plus the momentum restoring terms.) The resultant closure relations for b {center dot} {del} {center dot} {tilde product}{sub {parallel}} and {tilde q}{sub {parallel}} unify both the fluid and kinetic approaches. In the collisional fluid limit the equations reduce to the well-known Braginskii equations. In the adiabatic limit they reproduce the usual kinetic results, including Landau damping. It is shown that the CEL approach is more compatible with a fluid-like description of plasmas than the usual drift/gyro kinetic approach. A remarkable simplification of these complicated closure relations is achieved by using single power of plasma dispersion functions with modified arguments. The results are compared with other recently developed Landau damping models and shown to be more accurate, complete and physically meaningful. The resultant set of nonlinear fluid/kinetic equations (with linear closure relations) will be applied to various microinstabilities in tokamak plasmas and drift type microturbulence in a separate paper. 19 refs., 7 refs., 1 tab.
Kinetic theory of runaway air-breakdown
Roussel-Dupre, R.A.; Gurevich, A.V.; Tunnell, T.; Milikh, G.M.
1993-09-01
The kinetic theory for a new air breakdown mechanism advanced in a previous paper is developed. The relevant form of the Boltzmann equation is derived and the particle orbits in both velocity space and configuration space are computed. A numerical solution of the Boltzmann equation, assuming a spatially uniform electric field, is obtained and the temporal evolution of the electron velocity distribution function is described. The results of our analysis are used to estimate the magnitude of potential x-ray emissions from discharges in thunderstorms.
Kinetic theory of runaway air breakdown
Roussel-Dupre, R.A. ); Gurevich, A.V. ); Tunnell, T. ); Milikh, G.M. )
1994-03-01
The kinetic theory for an air breakdown mechanism advanced in a previous paper [Phys. Lett. A 165, 463 (1992)] is developed. The relevant form of the Boltzmann equation is derived and the particle orbits in both velocity space and configuration space are computed. A numerical solution of the Boltzmann equation, assuming a spatially uniform electric field, is obtained and the temporal evolution of the electron velocity distribution function is described. The results of our analysis are used to estimate the magnitude of potential x-ray emissions from discharges in thunderstorms.
Zakynthinaki, Maria S.
2015-01-01
The objective of the present study was to formulate a simple and at the same time effective mathematical model of heart rate kinetics in response to movement (exercise). Based on an existing model, a system of two coupled differential equations which give the rate of change of heart rate and the rate of change of exercise intensity is used. The modifications introduced to the existing model are justified and discussed in detail, while models of blood lactate accumulation in respect to time and exercise intensity are also presented. The main modification is that the proposed model has now only one parameter which reflects the overall cardiovascular condition of the individual. The time elapsed after the beginning of the exercise, the intensity of the exercise, as well as blood lactate are also taken into account. Application of the model provides information regarding the individual’s cardiovascular condition and is able to detect possible changes in it, across the data recording periods. To demonstrate examples of successful numerical fit of the model, constant intensity experimental heart rate data sets of two individuals have been selected and numerical optimization was implemented. In addition, numerical simulations provided predictions for various exercise intensities and various cardiovascular condition levels. The proposed model can serve as a powerful tool for a complete means of heart rate analysis, not only in exercise physiology (for efficiently designing training sessions for healthy subjects) but also in the areas of cardiovascular health and rehabilitation (including application in population groups for which direct heart rate recordings at intense exercises are not possible or not allowed, such as elderly or pregnant women). PMID:25876164
Modelling heart rate kinetics.
Zakynthinaki, Maria S
2015-01-01
The objective of the present study was to formulate a simple and at the same time effective mathematical model of heart rate kinetics in response to movement (exercise). Based on an existing model, a system of two coupled differential equations which give the rate of change of heart rate and the rate of change of exercise intensity is used. The modifications introduced to the existing model are justified and discussed in detail, while models of blood lactate accumulation in respect to time and exercise intensity are also presented. The main modification is that the proposed model has now only one parameter which reflects the overall cardiovascular condition of the individual. The time elapsed after the beginning of the exercise, the intensity of the exercise, as well as blood lactate are also taken into account. Application of the model provides information regarding the individual's cardiovascular condition and is able to detect possible changes in it, across the data recording periods. To demonstrate examples of successful numerical fit of the model, constant intensity experimental heart rate data sets of two individuals have been selected and numerical optimization was implemented. In addition, numerical simulations provided predictions for various exercise intensities and various cardiovascular condition levels. The proposed model can serve as a powerful tool for a complete means of heart rate analysis, not only in exercise physiology (for efficiently designing training sessions for healthy subjects) but also in the areas of cardiovascular health and rehabilitation (including application in population groups for which direct heart rate recordings at intense exercises are not possible or not allowed, such as elderly or pregnant women).
Drift kinetic theory of neoclassical tearing mode physics
NASA Astrophysics Data System (ADS)
Wilson, Howard; Connor, Jack; Hill, Peter; Imada, Koki
2016-10-01
Orbit averaged equations for the particle responses to a small magnetic island are derived, expanding the drift kinetic equation in the ratio of island width to tokamak plasma minor radius, assumed small. Analytic solutions demonstrate that the particles follow drift orbits which have the same geometry as the magnetic island flux surfaces, but are shifted radially by an amount that is proportional to the poloidal Larmor radius (in opposite directions for opposite signs of parallel velocity). The distribution function is flattened across these drift island structures, rather than across the magnetic island. Numerical solutions of our equations confirm the existence of the drift orbits. We employ a model momentum-conserving collision operator to evaluate the consequences for neoclassical tearing mode threshold physics, implementing numerical solutions to our orbit-averaged drift kinetic equations in a ``Modified Rutherford Equation''. Supported by the EPSRC, Grant Number EP/N009363/1.
Kinetics of helium bubble formation in nuclear materials
Bonilla, L L; Carpio, A; Neu, J C; Wolfer, W G
2005-10-13
The formation and growth of helium bubbles due to self-irradiation in plutonium has been modeled by a discrete kinetic equations for the number densities of bubbles having k atoms. Analysis of these equations shows that the bubble size distribution function can be approximated by a composite of: (1) the solution of partial differential equations describing the continuum limit of the theory but corrected to take into account the effects of discreteness, and (2) a local expansion about the advancing leading edge of the distribution function in size space. Both approximations contribute to the memory term in a close integrodifferential equation for the monomer concentration of single helium atoms. The present theory is compared to the numerical solution of the full kinetic model and to previous approximation of Schaldach and Wolfer involving a truncated system of moment equations.
Single wall penetration equations
NASA Technical Reports Server (NTRS)
Hayashida, K. B.; Robinson, J. H.
1991-01-01
Five single plate penetration equations are compared for accuracy and effectiveness. These five equations are two well-known equations (Fish-Summers and Schmidt-Holsapple), two equations developed by the Apollo project (Rockwell and Johnson Space Center (JSC), and one recently revised from JSC (Cour-Palais). They were derived from test results, with velocities ranging up to 8 km/s. Microsoft Excel software was used to construct a spreadsheet to calculate the diameters and masses of projectiles for various velocities, varying the material properties of both projectile and target for the five single plate penetration equations. The results were plotted on diameter versus velocity graphs for ballistic and spallation limits using Cricket Graph software, for velocities ranging from 2 to 15 km/s defined for the orbital debris. First, these equations were compared to each other, then each equation was compared with various aluminum projectile densities. Finally, these equations were compared with test results performed at JSC for the Marshall Space Flight Center. These equations predict a wide variety of projectile diameters at a given velocity. Thus, it is very difficult to choose the 'right' prediction equation. The thickness of a single plate could have a large variation by choosing a different penetration equation. Even though all five equations are empirically developed with various materials, especially for aluminum alloys, one cannot be confident in the shield design with the predictions obtained by the penetration equations without verifying by tests.
Modeling the Kinetics of Root Gravireaction
NASA Astrophysics Data System (ADS)
Kondrachuk, Alexander V.; Starkov, Vyacheslav N.
2011-02-01
The known "sun-flower equation" (SFE), which was originally proposed to model root circumnutating, was used to describe the simplest tip root graviresponse. Two forms of the SFE (integro-differential and differential-delayed) were solved, analyzed and compared with each other. The numerical solutions of these equations were found to be matching with arbitrary accuracy. The analysis of the solutions focused on time-lag effects on the kinetics of tip root bending. The results of the modeling are in good correlation with an experiment at the initial stages of root tips graviresponse. Further development of the model calls for its systematic comparison with some specially designed experiments, which would include measuring the kinetics of root tip bending before gravistimulation over the period of time longer than the time lag.
Transport equations for partially ionized reactive plasma in magnetic field
NASA Astrophysics Data System (ADS)
Zhdanov, V. M.; Stepanenko, A. A.
2016-06-01
Transport equations for partially ionized reactive plasma in magnetic field taking into account the internal degrees of freedom and electronic excitation of plasma particles are derived. As a starting point of analysis the kinetic equation with a binary collision operator written in the Wang-Chang and Uhlenbeck form and with a reactive collision integral allowing for arbitrary chemical reactions is used. The linearized variant of Grad's moment method is applied to deduce the systems of moment equations for plasma and also full and reduced transport equations for plasma species nonequilibrium parameters.
Transport equations for partially ionized reactive plasma in magnetic field
Zhdanov, V. M.; Stepanenko, A. A.
2016-06-08
Transport equations for partially ionized reactive plasma in magnetic field taking into account the internal degrees of freedom and electronic excitation of plasma particles are derived. As a starting point of analysis the kinetic equation with a binary collision operator written in the Wang-Chang and Uhlenbeck form and with a reactive collision integral allowing for arbitrary chemical reactions is used. The linearized variant of Grad’s moment method is applied to deduce the systems of moment equations for plasma and also full and reduced transport equations for plasma species nonequilibrium parameters.
Transport equations for partially ionized reactive plasma in magnetic field
NASA Astrophysics Data System (ADS)
Zhdanov, V. M.; Stepanenko, A. A.
2016-06-01
Transport equations for partially ionized reactive plasma in magnetic field taking into account the internal degrees of freedom and electronic excitation of plasma particles are derived. As a starting point of analysis the kinetic equation with a binary collision operator written in the Wang-Chang and Uhlenbeck form and with a reactive collision integral allowing for arbitrary chemical reactions is used. The linearized variant of Grad's moment method is applied to deduce the systems of moment equations for plasma and also full and reduced transport equations for plasma species nonequilibrium parameters.
Kinetic-energy-momentum tensor in electrodynamics
NASA Astrophysics Data System (ADS)
Sheppard, Cheyenne J.; Kemp, Brandon A.
2016-01-01
We show that the Einstein-Laub formulation of electrodynamics is invalid since it yields a stress-energy-momentum (SEM) tensor that is not frame invariant. Two leading hypotheses for the kinetic formulation of electrodynamics (Chu and Einstein-Laub) are studied by use of the relativistic principle of virtual power, mathematical modeling, Lagrangian methods, and SEM transformations. The relativistic principle of virtual power is used to demonstrate the field dynamics associated with energy relations within a relativistic framework. Lorentz transformations of the respective SEM tensors demonstrate the relativistic frameworks for each studied formulation. Mathematical modeling of stationary and moving media is used to illustrate the differences and discrepancies of specific proposed kinetic formulations, where energy relations and conservation theorems are employed. Lagrangian methods are utilized to derive the field kinetic Maxwell's equations, which are studied with respect to SEM tensor transforms. Within each analysis, the Einstein-Laub formulation violates special relativity, which invalidates the Einstein-Laub SEM tensor.
Kinetics of microbial growth on pentachlorophenol.
Klecka, G M; Maier, W J
1985-01-01
Batch and fed-batch experiments were conducted to examine the kinetics of pentachlorophenol utilization by an enrichment culture of pentachlorophenol-degrading bacteria. The Haldane modification of the Monod equation was found to describe the relationship between the specific growth rate and substrate concentration. Analysis of the kinetic parameters indicated that the maximum specific growth rate and yield coefficients are low, with values of 0.074 h-1 and 0.136 g/g, respectively. The Monod constant (Ks) was estimated to be 60 micrograms/liter, indicating a high affinity of the microorganisms for the substrate. However, high concentrations (KI = 1,375 micrograms/liter) were shown to be inhibitory for metabolism and growth. These kinetic parameters can be used to define the optimal conditions for the removal of pentachlorophenol in biological treatment systems. PMID:3977315
Interpretation of Bernoulli's Equation.
ERIC Educational Resources Information Center
Bauman, Robert P.; Schwaneberg, Rolf
1994-01-01
Discusses Bernoulli's equation with regards to: horizontal flow of incompressible fluids, change of height of incompressible fluids, gases, liquids and gases, and viscous fluids. Provides an interpretation, properties, terminology, and applications of Bernoulli's equation. (MVL)
Reflections on Chemical Equations.
ERIC Educational Resources Information Center
Gorman, Mel
1981-01-01
The issue of how much emphasis balancing chemical equations should have in an introductory chemistry course is discussed. The current heavy emphasis on finishing such equations is viewed as misplaced. (MP)
Study of kinetic effects arising in simulations of Farley-Buneman instability
Kovalev, D. V.; Smirnov, A. P.; Dimant, Ya. S.
2009-05-15
The Farley-Buneman instability, which has been observed in the E region of the Earth's ionosphere, is studied using fluid equations for electrons, a four-dimensional (in coordinate-velocity space) kinetic equation for ions, and Poisson's equation. Numerical simulations with allowance for Landau damping show that the Farley-Buneman instability results in anisotropy of the ion velocity distribution function.
Propyl propionate methanolysis kinetics: experiment and modeling.
Povarov, Vladimir G; Keresten, Andrey A
2013-09-26
In concentrated solutions, the reaction rate constant calculated from the law of mass action depends on a mixture composition. In this study, we suggest a new approach based on the dynamic equilibrium principle and modified equations of van Rysselberghe and de Donder. Experimental results obtained for propyl propionate methanolysis are in good agreement with calculated ones in the entire area of studied compositions. Such a method allows one to describe kinetics of reversible chemical reactions on both sides of the equilibrium.
Random equations in aerodynamics
NASA Technical Reports Server (NTRS)
Bharucha-Reid, A. T.
1984-01-01
Literature was reviewed to identify aerodynamic models which might be treated by probablistic methods. The numerical solution of some integral equations that arise in aerodynamical problems were investigated. On the basis of the numerical studies a qualitative theory of random integral equations was developed to provide information on the behavior of the solutions of these equations (in particular, boundary and asymptotic behavior, and stability) and their statistical properties without actually obtaining explicit solutions of the equations.
Parametrically defined differential equations
NASA Astrophysics Data System (ADS)
Polyanin, A. D.; Zhurov, A. I.
2017-01-01
The paper deals with nonlinear ordinary differential equations defined parametrically by two relations. It proposes techniques to reduce such equations, of the first or second order, to standard systems of ordinary differential equations. It obtains the general solution to some classes of nonlinear parametrically defined ODEs dependent on arbitrary functions. It outlines procedures for the numerical solution of the Cauchy problem for parametrically defined differential equations.
Integrable nonlinear relativistic equations
NASA Astrophysics Data System (ADS)
Hadad, Yaron
This work focuses on three nonlinear relativistic equations: the symmetric Chiral field equation, Einstein's field equation for metrics with two commuting Killing vectors and Einstein's field equation for diagonal metrics that depend on three variables. The symmetric Chiral field equation is studied using the Zakharov-Mikhailov transform, with which its infinitely many local conservation laws are derived and its solitons on diagonal backgrounds are studied. It is also proven that it is equivalent to a novel equation that poses a fascinating similarity to the Sinh-Gordon equation. For the 1+1 Einstein equation the Belinski-Zakharov transformation is explored. It is used to derive explicit formula for N gravitational solitons on arbitrary diagonal background. In particular, the method is used to derive gravitational solitons on the Einstein-Rosen background. The similarities and differences between the attributes of the solitons of the symmetric Chiral field equation and those of the 1+1 Einstein equation are emphasized, and their origin is pointed out. For the 1+2 Einstein equation, new equations describing diagonal metrics are derived and their compatibility is proven. Different gravitational waves are studied that naturally extend the class of Bondi-Pirani-Robinson waves. It is further shown that the Bondi-Pirani-Robinson waves are stable with respect to perturbations of the spacetime. Their stability is closely related to the stability of the Schwarzschild black hole and the relation between the two allows to conjecture about the stability of a wide range of gravitational phenomena. Lastly, a new set of equations that describe weak gravitational waves is derived. This new system of equations is closely and fundamentally connected with the nonlinear Schrodinger equation and can be properly called the nonlinear Schrodinger-Einstein equations. A few preliminary solutions are constructed.
Kinetic approach for describing biological systems
NASA Astrophysics Data System (ADS)
Aristov, V. V.; Ilyin, O. V.
2016-11-01
We attempt to consider a biological structure as an open nonequilibrium system the properties of which can be described on the basis of kinetic approach with the help of appropriate kinetic equations. This approach allows us to evaluate in principle scales of sizes and to connect these values to the inner characteristics of the processes of kinetic interaction and advection. One can compare the results with some empirical data concerning these characteristics for bio-systems, in particular mammals, and also for some parts of the systems, say sizes of green leaves. A sense of the nonequilibrium entropy as a measure of complexity of bio-organisms is discussed. Besides the estimations of bio-systems on a global scale, possible methods to describe restricted regions (associated e.g. with living cells) as nonequilibrium open structure with specific boundaries are also discussed. A new boundary 1D problem is formulated and solved for kinetic equations with the membrane-like boundaries conditions. Non-classical transport properties in the system are found.
ERIC Educational Resources Information Center
Fay, Temple H.
2002-01-01
We investigate the pendulum equation [theta] + [lambda][squared] sin [theta] = 0 and two approximations for it. On the one hand, we suggest that the third and fifth-order Taylor series approximations for sin [theta] do not yield very good differential equations to approximate the solution of the pendulum equation unless the initial conditions are…
Kinetic theory for interacting Luttinger liquids
NASA Astrophysics Data System (ADS)
Buchhold, Michael; Diehl, Sebastian
2015-10-01
We derive a closed set of equations for the kinetics and non-equilibrium dynamics of interacting Luttinger Liquids with cubic resonant interactions. In the presence of these interactions, the Luttinger phonons become dressed but still well defined quasi-particles, characterized by a life-time much larger then the inverse energy. This enables the separation of forward time dynamics and relative time dynamics into slow and fast dynamics and justifies the so-called Wigner approximation, which can be seen as a "local-time approximation" for the relative dynamics. Applying field theoretical methods in the Keldysh framework, i.e. kinetic and Dyson-Schwinger equations, we derive a closed set of dynamic equations, describing the kinetics of normal and anomalous phonon densities, the phonon self-energy and vertex corrections for a Gaussian non-equilibrium initial state. In the limit of low phonon densities, the results from self-consistent Born approximation are recaptured, including Andreev's scaling solution for the quasi-particle life-time in a thermal state. As an application, we compute the relaxation of an excited state to its thermal equilibrium. While the intermediate time dynamics displays exponentially fast relaxation, the last stages of thermalization are governed by algebraic laws. This can be traced back to the importance of energy and momentum conservation at the longest times, which gives rise to dynamical slow modes.
A note on the reverse Michaelis-Menten kinetics
Wang, Gangsheng; Post, Wilfred M
2013-01-01
We theoretically derive a general equation describing the enzyme kinetics that can be further simplified to the typical Michaelis-Menten (M-M) kinetics and the reverse M-M equation (RM-M) proposed by Schimel and Weintraub (2003). We discuss the conditions under which the RM-M is valid with this theoretical derivation. These conditions are contrary to the assumptions of Schimel and Weintraub (2003) and limit the applicability of the model in field soil environments. Nonetheless, Schimel and Weintraub s RM-M model is useful and has the ability to produce a non-linear response of SOM decomposition to enzyme concentration consistent with observations. Regardless of the theoretical basis, if we assume that the M-M and the RM-M could be equivalent, our sensitivity analysis indicates that enzyme plays a more sensitive role in the M-M kinetics compared with in the RM-M kinetics.
Numerical methods for high-dimensional probability density function equations
Cho, H.; Venturi, D.; Karniadakis, G.E.
2016-01-15
In this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables. These types of equations arise naturally in many different areas of mathematical physics, e.g., in particle systems (Liouville and Boltzmann equations), stochastic dynamical systems (Fokker–Planck and Dostupov–Pugachev equations), random wave theory (Malakhov–Saichev equations) and coarse-grained stochastic systems (Mori–Zwanzig equations). We propose three different classes of new algorithms addressing high-dimensionality: The first one is based on separated series expansions resulting in a sequence of low-dimensional problems that can be solved recursively and in parallel by using alternating direction methods. The second class of algorithms relies on truncation of interaction in low-orders that resembles the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations. The third class of algorithms is based on high-dimensional model representations, e.g., the ANOVA method and probabilistic collocation methods. A common feature of all these approaches is that they are reducible to the problem of computing the solution to high-dimensional equations via a sequence of low-dimensional problems. The effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations, with up to 120 variables.
Numerical methods for high-dimensional probability density function equations
NASA Astrophysics Data System (ADS)
Cho, H.; Venturi, D.; Karniadakis, G. E.
2016-01-01
In this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables. These types of equations arise naturally in many different areas of mathematical physics, e.g., in particle systems (Liouville and Boltzmann equations), stochastic dynamical systems (Fokker-Planck and Dostupov-Pugachev equations), random wave theory (Malakhov-Saichev equations) and coarse-grained stochastic systems (Mori-Zwanzig equations). We propose three different classes of new algorithms addressing high-dimensionality: The first one is based on separated series expansions resulting in a sequence of low-dimensional problems that can be solved recursively and in parallel by using alternating direction methods. The second class of algorithms relies on truncation of interaction in low-orders that resembles the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations. The third class of algorithms is based on high-dimensional model representations, e.g., the ANOVA method and probabilistic collocation methods. A common feature of all these approaches is that they are reducible to the problem of computing the solution to high-dimensional equations via a sequence of low-dimensional problems. The effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations, with up to 120 variables.
NASA Astrophysics Data System (ADS)
Karkheck, John; Stell, George
1981-08-01
A kinetic mean-field theory for the evolution of the one-particle distribution function is derived from maximizing the entropy. For a potential with a hard-sphere core plus tail, the resulting theory treats the hard-core part as in the revised Enskog theory. The tail, weighted by the hard-sphere pair distribution function, appears linearly in a mean-field term. The kinetic equation is accompanied by an entropy functional for which an H theorem was proven earlier. The revised Enskog theory is obtained by setting the potential tail to zero, the Vlasov equation is obtained by setting the hard-sphere diameter to zero, and an equation of the Enskog-Vlasov type is obtained by effecting the Kac limit on the potential tail. At equilibrium, the theory yields a radial distribution function that is given by the hard-sphere reference system and thus furnishes through the internal energy a thermodynamic description which is exact to first order in inverse temperature. A second natural route to thermodynamics (from the momentum flux which yields an approximate equation of state) gives somewhat different results; both routes coincide and become exact in the Kac limit. Our theory furnishes a conceptual basis for the association in the heuristically based modified Enskog theory (MET) of the contact value of the radial distribution function with the ''thermal pressure'' since this association follows from our theory (using either route to thermodynamics) and moreover becomes exact in the Kac limit. Our transport theory is readily extended to the general case of a soft repulsive core, e.g., as exhibited by the Lennard-Jones potential, via by-now-standard statistical-mechanical methods involving an effective hard-core potential, thus providing a self-contained statistical-mechanical basis for application to such potentials that is lacking in the standard versions of the MET. We obtain very good agreement with experiment for the thermal conductivity and shear viscosity of several
Kinetic behaviour of zymogen activation processes in the presence of an inhibitor.
Varón, R; Manjabacas, M C; García-Moreno, M; Valero, E; Garcia-Canovas, F
1993-01-01
A global kinetic analysis of a general zymogen activation model, where not only the activating but also the activated enzyme suffer an irreversible inhibition is presented. A reaction in which the enzyme acts upon a substrate is coupled to monitor the process. In addition, we determined the corresponding kinetic equations for a number of particular cases of the general model studied. Finally, a kinetic data analysis and a procedure to discriminate among the different mechanisms considered, which are based on the kinetic equations obtained, are suggested. PMID:8452535
Tang, J. Y.
2015-12-01
The Michaelis–Menten kinetics and the reverse Michaelis–Menten kinetics are two popular mathematical formulations used in many land biogeochemical models to describe how microbes and plants would respond to changes in substrate abundance. However, the criteria of when to use either of the two are often ambiguous. Here I show that these two kinetics are special approximations to the equilibrium chemistry approximation (ECA) kinetics, which is the first-order approximation to the quadratic kinetics that solves the equation of an enzyme–substrate complex exactly for a single-enzyme and single-substrate biogeochemical reaction with the law of mass action and the assumption of a quasi-steadymore » state for the enzyme–substrate complex and that the product genesis from enzyme–substrate complex is much slower than the equilibration between enzyme–substrate complexes, substrates, and enzymes. In particular, I show that the derivation of the Michaelis–Menten kinetics does not consider the mass balance constraint of the substrate, and the reverse Michaelis–Menten kinetics does not consider the mass balance constraint of the enzyme, whereas both of these constraints are taken into account in deriving the equilibrium chemistry approximation kinetics. By benchmarking against predictions from the quadratic kinetics for a wide range of substrate and enzyme concentrations, the Michaelis–Menten kinetics was found to persistently underpredict the normalized sensitivity ∂ ln v / ∂ ln k2+ of the reaction velocity v with respect to the maximum product genesis rate k2+, persistently overpredict the normalized sensitivity ∂ ln v / ∂ ln k1+ of v with respect to the intrinsic substrate affinity k1+, persistently overpredict the normalized sensitivity ∂ ln v / ∂ ln [E]T of v with respect the total enzyme concentration [E]T, and persistently underpredict the normalized sensitivity ∂ ln v / ∂ ln [S]T of v with respect to the total substrate concentration
NASA Astrophysics Data System (ADS)
Tang, J. Y.
2015-12-01
The Michaelis-Menten kinetics and the reverse Michaelis-Menten kinetics are two popular mathematical formulations used in many land biogeochemical models to describe how microbes and plants would respond to changes in substrate abundance. However, the criteria of when to use either of the two are often ambiguous. Here I show that these two kinetics are special approximations to the equilibrium chemistry approximation (ECA) kinetics, which is the first-order approximation to the quadratic kinetics that solves the equation of an enzyme-substrate complex exactly for a single-enzyme and single-substrate biogeochemical reaction with the law of mass action and the assumption of a quasi-steady state for the enzyme-substrate complex and that the product genesis from enzyme-substrate complex is much slower than the equilibration between enzyme-substrate complexes, substrates, and enzymes. In particular, I show that the derivation of the Michaelis-Menten kinetics does not consider the mass balance constraint of the substrate, and the reverse Michaelis-Menten kinetics does not consider the mass balance constraint of the enzyme, whereas both of these constraints are taken into account in deriving the equilibrium chemistry approximation kinetics. By benchmarking against predictions from the quadratic kinetics for a wide range of substrate and enzyme concentrations, the Michaelis-Menten kinetics was found to persistently underpredict the normalized sensitivity ∂ ln v / ∂ ln k2+ of the reaction velocity v with respect to the maximum product genesis rate k2+, persistently overpredict the normalized sensitivity ∂ ln v / ∂ ln k1+ of v with respect to the intrinsic substrate affinity k1+, persistently overpredict the normalized sensitivity ∂ ln v / ∂ ln [E]T of v with respect the total enzyme concentration [E]T, and persistently underpredict the normalized sensitivity ∂ ln v / ∂ ln [S]T of v with respect to the total substrate concentration [S
Master equation analysis of deterministic chemical chaos
NASA Astrophysics Data System (ADS)
Wang, Hongli; Li, Qianshu
1998-05-01
The underlying microscopic dynamics of deterministic chemical chaos was investigated in this paper. We analyzed the master equation for the Williamowski-Rössler model by direct stochastic simulation as well as in the generating function representation. Simulation within an ensemble revealed that in the chaotic regime the deterministic mass action kinetics is related neither to the ensemble mean nor to the most probable value within the ensemble. Cumulant expansion analysis of the master equation also showed that the molecular fluctuations do not admit bounded values but increase linearly in time infinitely, indicating the meaninglessness of the chaotic trajectories predicted by the phenomenological equations. These results proposed that the macroscopic description is no longer useful in the chaotic regime and a more microscopic description is necessary in this circumstance.
Exact Pressure Evolution Equation for Incompressible Fluids
NASA Astrophysics Data System (ADS)
Tessarotto, M.; Ellero, M.; Aslan, N.; Mond, M.; Nicolini, P.
2008-12-01
An important aspect of computational fluid dynamics is related to the determination of the fluid pressure in isothermal incompressible fluids. In particular this concerns the construction of an exact evolution equation for the fluid pressure which replaces the Poisson equation and yields an algorithm which is a Poisson solver, i.e., it permits to time-advance exactly the same fluid pressure without solving the Poisson equation. In fact, the incompressible Navier-Stokes equations represent a mixture of hyperbolic and elliptic pde's, which are extremely hard to study both analytically and numerically. This amounts to transform an elliptic type fluid equation into a suitable hyperbolic equation, a result which usually is reached only by means of an asymptotic formulation. Besides being a still unsolved mathematical problem, the issue is relevant for at least two reasons: a) the proliferation of numerical algorithms in computational fluid dynamics which reproduce the behavior of incompressible fluids only in an asymptotic sense (see below); b) the possible verification of conjectures involving the validity of appropriate equations of state for the fluid pressure. Another possible motivation is, of course, the ongoing quest for efficient numerical solution methods to be applied for the construction of the fluid fields {ρ,V,p}, solutions of the initial and boundary-value problem associated to the incompressible N-S equations (INSE). In this paper we intend to show that an exact solution to this problem can be achieved adopting the approach based on inverse kinetic theory (IKT) recently developed for incompressible fluids by Tessarotto et al. [7, 6, 7, 8, 9]. In particular we intend to prove that the evolution of the fluid fields can be achieved by means of a suitable dynamical system, to be identified with the so-called Navier-Stokes (N-S) dynamical system. As a consequence it is found that the fluid pressure obeys a well-defined evolution equation. The result appears
The Nonlinear Magnetosphere: Expressions in MHD and in Kinetic Models
NASA Technical Reports Server (NTRS)
Hesse, Michael; Birn, Joachim
2011-01-01
Like most plasma systems, the magnetosphere of the Earth is governed by nonlinear dynamic evolution equations. The impact of nonlinearities ranges from large scales, where overall dynamics features are exhibiting nonlinear behavior, to small scale, kinetic, processes, where nonlinear behavior governs, among others, energy conversion and dissipation. In this talk we present a select set of examples of such behavior, with a specific emphasis on how nonlinear effects manifest themselves in MHD and in kinetic models of magnetospheric plasma dynamics.
Consistent description of kinetics and hydrodynamics of dusty plasma
Markiv, B.; Tokarchuk, M.
2014-02-15
A consistent statistical description of kinetics and hydrodynamics of dusty plasma is proposed based on the Zubarev nonequilibrium statistical operator method. For the case of partial dynamics, the nonequilibrium statistical operator and the generalized transport equations for a consistent description of kinetics of dust particles and hydrodynamics of electrons, ions, and neutral atoms are obtained. In the approximation of weakly nonequilibrium process, a spectrum of collective excitations of dusty plasma is investigated in the hydrodynamic limit.
Effects of Fluid Environment on Microbial Uptake Kinetics
1990-09-26
filtration of a KINETIC MODEL solution consisting of 200-jLL of prefiltered cells with an Assuming Monod kinetics, the change in nutrient mass i absorbance...advective flow past the microorganisms. This hypothesis was based on a biohydrodynamical uptake model derived from mass transfer and coagulation theories...hypothesis was based on a biohydrodynamical uptake model derived from mass transfer and coagulation theories (Logan and Hunt 1987,1988). From equations
ERIC Educational Resources Information Center
Golicnik, Marko
2011-01-01
The Michaelis-Menten rate equation can be found in most general biochemistry textbooks, where the time derivative of the substrate is a hyperbolic function of two kinetic parameters (the limiting rate "V", and the Michaelis constant "K"[subscript M]) and the amount of substrate. However, fundamental concepts of enzyme kinetics can be difficult to…
ERIC Educational Resources Information Center
Golicnik, Marko
2011-01-01
The Michaelis-Menten rate equation can be found in most general biochemistry textbooks, where the time derivative of the substrate is a hyperbolic function of two kinetic parameters (the limiting rate "V", and the Michaelis constant "K"[subscript M]) and the amount of substrate. However, fundamental concepts of enzyme kinetics can be difficult to…
ENZO: A Web Tool for Derivation and Evaluation of Kinetic Models of Enzyme Catalyzed Reactions
Bevc, Staš; Konc, Janez; Stojan, Jure; Hodošček, Milan; Penca, Matej; Praprotnik, Matej; Janežič, Dušanka
2011-01-01
We describe a web tool ENZO (Enzyme Kinetics), a graphical interface for building kinetic models of enzyme catalyzed reactions. ENZO automatically generates the corresponding differential equations from a stipulated enzyme reaction scheme. These differential equations are processed by a numerical solver and a regression algorithm which fits the coefficients of differential equations to experimentally observed time course curves. ENZO allows rapid evaluation of rival reaction schemes and can be used for routine tests in enzyme kinetics. It is freely available as a web tool, at http://enzo.cmm.ki.si. PMID:21818304
A method for BPS equations of vortices
NASA Astrophysics Data System (ADS)
Nata Atmaja, A.
2017-05-01
We develop a new method for obtaining the BPS equations of static vortices motivated by the results of the On-Shell method on the standard Maxwell-Higgs model and its Born-Infeld-Higgs model [1]. Our method relies on the existence of what we shall call a BPS energy function Q as such the total energy of BPS vortices EBPS are simply given by an integral of total differential of the BPS energy function, EBPS = ∫ dQ. Imposing a condition that the effective fields are independent of each other, we may define a BPS Lagrangian LBPS by EBPS ≡ - ∫d2 x LBPS. Equating this BPS Lagrangian with the corresponding effective Lagrangian, the equation is expected to be a sum of positive-semidefinite functions Leff -LBPS = ∑iN Ai2 = 0, where N is the number of effective fields. Solving this equation by parts would yields the desired BPS equations. With our method, the various known BPS equations of vortices are derived in a relatively simple procedure. We show that in all models considered here, the BPS energy function is given by a general formula Q = 2 πaF (f), where a and f are the effective fields for the gauge field and scalar field, and F‧ (f) = ± 2 f w (f), with w is an overall coupling of the scalar field's kinetic term.
A closure scheme for chemical master equations.
Smadbeck, Patrick; Kaznessis, Yiannis N
2013-08-27
Probability reigns in biology, with random molecular events dictating the fate of individual organisms, and propelling populations of species through evolution. In principle, the master probability equation provides the most complete model of probabilistic behavior in biomolecular networks. In practice, master equations describing complex reaction networks have remained unsolved for over 70 years. This practical challenge is a reason why master equations, for all their potential, have not inspired biological discovery. Herein, we present a closure scheme that solves the master probability equation of networks of chemical or biochemical reactions. We cast the master equation in terms of ordinary differential equations that describe the time evolution of probability distribution moments. We postulate that a finite number of moments capture all of the necessary information, and compute the probability distribution and higher-order moments by maximizing the information entropy of the system. An accurate order closure is selected, and the dynamic evolution of molecular populations is simulated. Comparison with kinetic Monte Carlo simulations, which merely sample the probability distribution, demonstrates this closure scheme is accurate for several small reaction networks. The importance of this result notwithstanding, a most striking finding is that the steady state of stochastic reaction networks can now be readily computed in a single-step calculation, without the need to simulate the evolution of the probability distribution in time.
A closure scheme for chemical master equations
Smadbeck, Patrick; Kaznessis, Yiannis N.
2013-01-01
Probability reigns in biology, with random molecular events dictating the fate of individual organisms, and propelling populations of species through evolution. In principle, the master probability equation provides the most complete model of probabilistic behavior in biomolecular networks. In practice, master equations describing complex reaction networks have remained unsolved for over 70 years. This practical challenge is a reason why master equations, for all their potential, have not inspired biological discovery. Herein, we present a closure scheme that solves the master probability equation of networks of chemical or biochemical reactions. We cast the master equation in terms of ordinary differential equations that describe the time evolution of probability distribution moments. We postulate that a finite number of moments capture all of the necessary information, and compute the probability distribution and higher-order moments by maximizing the information entropy of the system. An accurate order closure is selected, and the dynamic evolution of molecular populations is simulated. Comparison with kinetic Monte Carlo simulations, which merely sample the probability distribution, demonstrates this closure scheme is accurate for several small reaction networks. The importance of this result notwithstanding, a most striking finding is that the steady state of stochastic reaction networks can now be readily computed in a single-step calculation, without the need to simulate the evolution of the probability distribution in time. PMID:23940327
Deng, De-Ming; Chang, Cheng-Hung
2015-05-14
Conventional studies of biomolecular behaviors rely largely on the construction of kinetic schemes. Since the selection of these networks is not unique, a concern is raised whether and under which conditions hierarchical schemes can reveal the same experimentally measured fluctuating behaviors and unique fluctuation related physical properties. To clarify these questions, we introduce stochasticity into the traditional lumping analysis, generalize it from rate equations to chemical master equations and stochastic differential equations, and extract the fluctuation relations between kinetically and thermodynamically equivalent networks under intrinsic and extrinsic noises. The results provide a theoretical basis for the legitimate use of low-dimensional models in the studies of macromolecular fluctuations and, more generally, for exploring stochastic features in different levels of contracted networks in chemical and biological kinetic systems.
Metal reduction kinetics in Shewanella.
Lall, Raman; Mitchell, Julie
2007-10-15
Metal reduction kinetics have been studied in cultures of dissimilatory metal reducing bacteria which include the Shewanella oneidensis strain MR-1. Estimation of system parameters from time-series data faces obstructions in the implementation depending on the choice of the mathematical model that captures the observed dynamics. The modeling of metal reduction is often based on Michaelis-Menten equations. These models are often developed using initial in vitro reaction rates and seldom match with in vivo reduction profiles. For metal reduction studies, we propose a model that is based on the power law representation that is effectively applied to the kinetics of metal reduction. The method yields reasonable parameter estimates and is illustrated with the analysis of time-series data that describes the dynamics of metal reduction in S.oneidensis strain MR-1. In addition, mixed metal studies involving the reduction of Uranyl (U(VI)) to the relatively insoluble tetravalent form (U(IV)) by S. alga strain (BR-Y) were studied in the presence of environmentally relevant iron hydrous oxides. For mixed metals, parameter estimation and curve fitting are accomplished with a generalized least squares formulation that handles systems of ordinary differential equations and is implemented in Matlab. It consists of an optimization algorithm (Levenberg-Marquardt, LSQCURVEFIT) and a numerical ODE solver. Simulation with the estimated parameters indicates that the model captures the experimental data quite well. The model uses the estimated parameters to predict the reduction rates of metals and mixed metals at varying concentrations. Supplementary data are available at Bioinformatics online.
Kinetic modelling of mitochondrial translation.
Korla, Kalyani; Mitra, Chanchal K
2014-01-01
Mitochondrial genome contains 13 protein coding genes, all being part of the oxidative phosphorylation complexes. The process of translation of these protein coding mRNAs in mitochondrial matrix is a good miniature model of translation in cytoplasm. In this work, we have simulated three phases of mitochondrial translation viz. initiation, elongation and termination (including ribosome recycling). The kinetic equations for these phases have been deduced based on the information available in literature. Various factors involved in the process have been included explicitly. Kinetic simulation was done using Octave, open source software. Scripts were written individually for each phase. Initiation begins with mitoribosome, mRNA, fMet-tRNA and initiation factors. The final product of the initiation script, the initiation complex, was introduced as the start point in the successive step, i.e. elongation. Elongation is a particular extensive process where the various aminoacyl-tRNAs already present in the matrix check for matching with the triplet codon in A-site of mitoribosome. This script consists of two parts: one with the time behaviour of the factors involved in the molecular process (using ordinary differential equation solver) and the other including the reading of triplet codon on the mRNA and incorporating the corresponding aminoacyl-tRNA, and then at each step elongating the peptide chain (using loops and conditions). The peptide chain thus formed in the elongation step (in the loops and conditions segment) was released in the termination step. This was followed by mitoribosome recycling where the mitoribosome reached the native state and was ready for the next cycle of translation.
Fractional kinetics in drug absorption and disposition processes.
Dokoumetzidis, Aristides; Macheras, Panos
2009-04-01
We explore the use of fractional order differential equations for the analysis of datasets of various drug processes that present anomalous kinetics, i.e. kinetics that are non-exponential and are typically described by power-laws. A fractional differential equation corresponds to a differential equation with a derivative of fractional order. The fractional equivalents of the "zero-" and "first-order" processes are derived. The fractional zero-order process is a power-law while the fractional first-order process is a Mittag-Leffler function. The latter behaves as a stretched exponential for early times and as a power-law for later times. Applications of these two basic results for drug dissolution/release and drug disposition are presented. The fractional model of dissolution is fitted successfully to datasets taken from literature of in vivo dissolution curves. Also, the proposed pharmacokinetic model is fitted to a dataset which exhibits power-law terminal phase. The Mittag-Leffler function describes well the data for small and large time scales and presents an advantage over empirical power-laws which go to infinity as time approaches zero. The proposed approach is compared conceptually with fractal kinetics, an alternative approach to describe datasets with non exponential kinetics. Fractional kinetics offers an elegant description of anomalous kinetics, with a valid scientific basis, since it has already been applied in problems of diffusion in other fields, and describes well the data.
Hybrid Parallelization of Adaptive MHD-Kinetic Module in Multi-Scale Fluid-Kinetic Simulation Suite
Borovikov, Sergey; Heerikhuisen, Jacob; Pogorelov, Nikolai
2013-04-01
The Multi-Scale Fluid-Kinetic Simulation Suite has a computational tool set for solving partially ionized flows. In this paper we focus on recent developments of the kinetic module which solves the Boltzmann equation using the Monte-Carlo method. The module has been recently redesigned to utilize intra-node hybrid parallelization. We describe in detail the redesign process, implementation issues, and modifications made to the code. Finally, we conduct a performance analysis.
Extended monod kinetics for substrate, product, and cell inhibition.
Han, K; Levenspiel, O
1988-08-05
A generalized form of Monod kinetics is proposed to account for all kinds of product, cell, and substrate inhibition. This model assumes that there exists a critical inhibitor concentration above which cells cannot grow, and that the constants of the Monod equation are functions of this limiting inhibitor concentration. Methods for evaluating the constants of this rate form are presented. Finally the proposed kinetic form is compared with the available data in the literature, which unfortunately is very sparse. In all cases, this equation form fitted the data very well.
The Continuous Coagulation-FragmentationEquations with Diffusion
NASA Astrophysics Data System (ADS)
Laurençot, Philippe; Mischler, Stéphane
Existence of global weak solutions to the continuous coagulation-fragmentation equations with diffusion is investigated when the kinetic coefficients satisfy a detailed balance condition or the coagulation coefficient enjoys a monotonicity condition. Our approach relies on weak and strong compactness methods in L1 in the spirit of the DiPerna-Lions theory for the Boltzmann equation. Under the detailed balance condition the large-time behaviour is also studied.
The SQG Equation as a Geodesic Equation
NASA Astrophysics Data System (ADS)
Washabaugh, Pearce
2016-12-01
We demonstrate that the surface quasi-geostrophic (SQG) equation given by θ_t + < u, nabla θrangle = 0,quad θ = nabla × (-Δ)^{-1/2} u, is the geodesic equation on the group of volume-preserving diffeomorphisms of a Riemannian manifold M in the right-invariant {dot{H}^{-1/2}} metric. We show by example, that the Riemannian exponential map is smooth and non-Fredholm, and that the sectional curvature at the identity is unbounded of both signs.
Ion kinetic transport in TJ-II
Velasco, J. L.; Tarancon, A.; Castejon, F.; Fernandez, L. A.; Martin-Mayor, V.
2008-11-02
The ion Drift Kinetic Equation (DKE) which describes the ion collisional transport is solved for the TJ-II device plasmas. This non-linear equation is computed by performing a mean field iterative calculation. In each step of the calculation, a Fokker-Planck equation is solved by means of the Langevin approach: one million particles are followed in a realistic TJ-II magnetic configuration, taking into account collisions and electric field. This allows to avoid the assumptions made in the usual neoclassical approach, namely considering radially narrow particle trajectories, diffusive transport, energy conservation and infinite parallel transport. As a consequence, global features of transport, not present in the customary neoclassical models, appear: non-diffusive transport and asymmetries on the magnetic surfaces.
Kinetic modeling of non-ideal explosives
Fried, L E; Howard, W M; Souers, P C
1999-03-01
We have implemented a Wood-Kirkwood kinetic detonation model based on multi-species equations of state and multiple reaction rate laws. Finite rate laws are used for the slowest chemical reactions, while other reactions are given infinite rates and are kept in constant thermodynamic equilibrium. We model a wide range of ideal and non-ideal composite energetic materials. In addition, we develop an exp-6 equation of state for the product fluids that reproduces a wide range experimental shock Hugoniot and static compression data. For unreacted solids, including solid and liquid Al and Al{sub 2}O{sub 3}, we use a Murnaghan form for the equation of state. We find that we can replicate experimental detonation velocities to within a few per cent for a wide range of explosives, while obtaining good agreement with estimated reaction zone lengths. The detonation velocity as a function of charge radius is also correctly reproduced.
Alpha-effect dynamos with zero kinetic helicity.
Rädler, Karl-Heinz; Brandenburg, Axel
2008-02-01
A simple explicit example of a Roberts-type dynamo is given in which the alpha effect of mean-field electrodynamics exists in spite of pointwise vanishing kinetic helicity of the fluid flow. In this way, it is shown that alpha-effect dynamos do not necessarily require nonzero kinetic helicity. A mean-field theory of Roberts-type dynamos is established within the framework of the second-order correlation approximation. In addition, numerical solutions of the original dynamo equations are given that are independent of any approximation of that kind. Both theory and numerical results demonstrate the possibility of dynamo action in the absence of kinetic helicity.
{alpha}-effect dynamos with zero kinetic helicity
Raedler, Karl-Heinz; Brandenburg, Axel
2008-02-15
A simple explicit example of a Roberts-type dynamo is given in which the {alpha} effect of mean-field electrodynamics exists in spite of pointwise vanishing kinetic helicity of the fluid flow. In this way, it is shown that {alpha}-effect dynamos do not necessarily require nonzero kinetic helicity. A mean-field theory of Roberts-type dynamos is established within the framework of the second-order correlation approximation. In addition, numerical solutions of the original dynamo equations are given that are independent of any approximation of that kind. Both theory and numerical results demonstrate the possibility of dynamo action in the absence of kinetic helicity.
The Einstein-Vlasov System/Kinetic Theory.
Andréasson, Håkan
2011-01-01
The main purpose of this article is to provide a guide to theorems on global properties of solutions to the Einstein-Vlasov system. This system couples Einstein's equations to a kinetic matter model. Kinetic theory has been an important field of research during several decades in which the main focus has been on non-relativistic and special relativistic physics, i.e., to model the dynamics of neutral gases, plasmas, and Newtonian self-gravitating systems. In 1990, Rendall and Rein initiated a mathematical study of the Einstein-Vlasov system. Since then many theorems on global properties of solutions to this system have been established. This paper gives introductions to kinetic theory in non-curved spacetimes and then the Einstein-Vlasov system is introduced. We believe that a good understanding of kinetic theory in non-curved spacetimes is fundamental to a good comprehension of kinetic theory in general relativity.
Fractional chemotaxis diffusion equations.
Langlands, T A M; Henry, B I
2010-05-01
We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modeling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macromolecular crowding. The mesoscopic models are formulated using continuous time random walk equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macromolecular crowding or other obstacles.
NASA Astrophysics Data System (ADS)
Kostov, Ivan; Serban, Didina; Volin, Dmytro
2008-08-01
We give a realization of the Beisert, Eden and Staudacher equation for the planar Script N = 4 supersymetric gauge theory which seems to be particularly useful to study the strong coupling limit. We are using a linearized version of the BES equation as two coupled equations involving an auxiliary density function. We write these equations in terms of the resolvents and we transform them into a system of functional, instead of integral, equations. We solve the functional equations perturbatively in the strong coupling limit and reproduce the recursive solution obtained by Basso, Korchemsky and Kotański. The coefficients of the strong coupling expansion are fixed by the analyticity properties obeyed by the resolvents.
Kinetic theory for electrostatic waves due to transverse velocity shears
NASA Technical Reports Server (NTRS)
Ganguli, G.; Lee, Y. C.; Palmadesso, P. J.
1988-01-01
A kinetic theory in the form of an integral equation is provided to study the electrostatic oscillations in a collisionless plasma immersed in a uniform magnetic field and a nonuniform transverse electric field. In the low temperature limit the dispersion differential equation is recovered for the transverse Kelvin-Helmholtz modes for arbitrary values of K parallel, where K parallel is the component of the wave vector in the direction of the external magnetic field assumed in the z direction. For higher temperatures the ion-cyclotron-like modes described earlier in the literature by Ganguli, Lee and Plamadesso are recovered. In this article, the integral equation is reduced to a second-order differential equation and a study is made of the kinetic Kelvin-Helmholtz and ion-cyclotron-like modes that constitute the two branches of oscillation in a magnetized plasma including a transverse inhomogeneous dc electric field.
A study of the Sabatier-methanation reaction kinetics
NASA Technical Reports Server (NTRS)
Verostko, C. E.; Forsythe, R. K.
1974-01-01
The kinetics of the Sabatier methanation reaction, the reduction of carbon dioxide with hydrogen to methane and water, was investigated for 58 percent nickel on kieselguhr catalyst and 20 percent ruthenium on alumina catalyst. Differential rate data from an experimental program were correlated with a power function rate equation both for forward and reverse reactions. The kinetic parameters of activation energy, frequency rate constant and reaction order were determined for the rate equation. The values of these parameters were obtained from an Arrhenius plot of the experimental differential rate data. Also the carbon monoxide side reaction effect was measured and included in the correlation of parameters. The reaction was found to fit the rate equation experimentally within the temperature range 421 K, where the reaction effectively begins, the 800 K where the reaction rate drops and departs from the rate equation form.
A study of the Sabatier-methanation reaction kinetics
NASA Technical Reports Server (NTRS)
Verostko, C. E.; Forsythe, R. K.
1974-01-01
The kinetics of the Sabatier methanation reaction, the reduction of carbon dioxide with hydrogen to methane and water, was investigated for 58 percent nickel on kieselguhr catalyst and 20 percent ruthenium on alumina catalyst. Differential rate data from an experimental program were correlated with a power function rate equation both for forward and reverse reactions. The kinetic parameters of activation energy, frequency rate constant and reaction order were determined for the rate equation. The values of these parameters were obtained from an Arrhenius plot of the experimental differential rate data. Also the carbon monoxide side reaction effect was measured and included in the correlation of parameters. The reaction was found to fit the rate equation experimentally within the temperature range 421 K, where the reaction effectively begins, the 800 K where the reaction rate drops and departs from the rate equation form.
Solving Ordinary Differential Equations
NASA Technical Reports Server (NTRS)
Krogh, F. T.
1987-01-01
Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.
Television Tracker Range Equation
NASA Astrophysics Data System (ADS)
Huan-Wen, Zhu
1987-05-01
The paper gives an approximate television tracker range equation based on the concept of the radiology and signal-to-noise of television system, and describes the physical process and mathematical method of reckoning range equation. The range equation is useful to the desing and development of a system. This paper also discusses the demand and selection standard of the television tracker system to the imaging device and gives some possible approaches to increase the range.
Rubel's universal differential equation
Duffin, R. J.
1981-01-01
Fourth-order differential equations such as 16y′my′2 - 32ymyny′ + 17y03 = 0 are developed. It is shown that the equation is “universal” in the sense that any continuous function can be approximated with arbitrary accuracy over the whole x axis by a solution y(x) of the equation. This solution is a piecewise polynomial of degree 9 and of class C4. PMID:16593068
Hyperbolic type transport equations
NASA Astrophysics Data System (ADS)
García-Colín, L. S.; Olivares-Robles, M. A.
1995-02-01
In recent years hyperbolic type transport equations have acquired a great deal of importance in problems ranging from theoretical physics to biology. In spite of their greater mathematical difficulty as compared with their parabolic type analogs arising from the framework of Linear Irreversible Thermodynamics, they have, in many ways, superseded the latter ones. Although the use of this type of equations is well known since the last century through the telegraphist equation of electromagnetic theory, their use in studying several problems in transport theory is hardly fifty years old. In fact the first appearance of a hyperbolic type transport equation for the problem of heat conduction dates back to Cattaneos' work in 1948. Three years later, in 1951 S. Goldstein showed how in the theory of stochastic processes this type of an equation is obtained in the continuous limit of a one-dimensional persistent random walk problem. After that, other phenomenological derivations have been offered for such equations. The main purpose of this paper is to critically discuss a derivation of a hyperbolic type Fokker-Planck equation recently presented using the same ideas as M.S. Green did in 1952 to provide the stochastic foundations of irreversible statistical mechanics. Arguments are given to show that such an equation as well as transport equations derived from it by taking appropriate averages are at most approximate and that a much more detailed analysis is required before asserting their validity.
Grima, Ramon
2011-11-01
The mesoscopic description of chemical kinetics, the chemical master equation, can be exactly solved in only a few simple cases. The analytical intractability stems from the discrete character of the equation, and hence considerable effort has been invested in the development of Fokker-Planck equations, second-order partial differential equation approximations to the master equation. We here consider two different types of higher-order partial differential approximations, one derived from the system-size expansion and the other from the Kramers-Moyal expansion, and derive the accuracy of their predictions for chemical reactive networks composed of arbitrary numbers of unimolecular and bimolecular reactions. In particular, we show that the partial differential equation approximation of order Q from the Kramers-Moyal expansion leads to estimates of the mean number of molecules accurate to order Ω(-(2Q-3)/2), of the variance of the fluctuations in the number of molecules accurate to order Ω(-(2Q-5)/2), and of skewness accurate to order Ω(-(Q-2)). We also show that for large Q, the accuracy in the estimates can be matched only by a partial differential equation approximation from the system-size expansion of approximate order 2Q. Hence, we conclude that partial differential approximations based on the Kramers-Moyal expansion generally lead to considerably more accurate estimates in the mean, variance, and skewness than approximations of the same order derived from the system-size expansion.
A steady-state kinetic analysis of the prolyl-4-hydroxylase mechanism.
Soskel, N T; Kuby, S A
1981-01-01
Published kinetic data by Kivirikko, et al. on the prolyl-4-hydroxylase reaction have been re-evaluated using the overall steady-state velocity equation in the forward and reverse directions for an ordered ter ter kinetic mechanism. Qualitatively, the published data for prolyl-4-hydroxylase appear to fit the predicted patterns for this kinetic mechanism. More kinetic data are needed to confirm these results and to quantitate the kinetic parameters but, tentatively, the order of substrate addition would appear to be alpha-ketoglutarate, oxygen, and peptide; and the order of product release would be hydroxylated peptide (or collagen), carbon dioxide, and succinate.
Chemical gas-dynamics beyond Wang Chang-Uhlenbeck's kinetics
Kolesnichenko, Evgeniy G.; Gorbachev, Yuriy E.
2014-12-09
Wang Chang-Uhlenbeck equation does not give possibility to take into account intermolecular processes such as redistribution of the energy among different degrees of freedom. The modification of the generalized Wang Chang-Uhlenbeck equation including such processes is proposed. It allows to study for instance the kinetics of non-radiative transitions. Limitations of this approach are connected with the requirements of absence of polarization of rotational momentum and phases of intermolecular vibrations.
Chemical gas-dynamics beyond Wang Chang-Uhlenbeck's kinetics
NASA Astrophysics Data System (ADS)
Kolesnichenko, Evgeniy G.; Gorbachev, Yuriy E.
2014-12-01
Wang Chang-Uhlenbeck equation does not give possibility to take into account intermolecular processes such as redistribution of the energy among different degrees of freedom. The modification of the generalized Wang Chang-Uhlenbeck equation including such processes is proposed. It allows to study for instance the kinetics of non-radiative transitions. Limitations of this approach are connected with the requirements of absence of polarization of rotational momentum and phases of intermolecular vibrations.
Quantifying uncertainty in the chemical master equation
NASA Astrophysics Data System (ADS)
Bayati, Basil S.
2017-06-01
We describe a novel approach to quantifying the uncertainty inherent in the chemical kinetic master equation with stochastic coefficients. A stochastic collocation method is coupled to an analytical expansion of the master equation to analyze the effects of both extrinsic and intrinsic noise. The method consists of an analytical moment-closure method resulting in a large set of differential equations with stochastic coefficients that are in turn solved via a Smolyak sparse grid collocation method. We discuss the error of the method relative to the dimension of the model and clarify which methods are most suitable for the problem. We apply the method to two typical problems arising in chemical kinetics with time-independent extrinsic noise. Additionally, we show agreement with classical Monte Carlo simulations and calculate the variance over time as the sum of two expectations. The method presented here has better convergence properties for low to moderate dimensions than standard Monte Carlo methods and is therefore a superior alternative in this regime.
Macroscopic equations for the adiabatic piston.
Cencini, Massimo; Palatella, Luigi; Pigolotti, Simone; Vulpiani, Angelo
2007-11-01
A simplified version of a classical problem in thermodynamics--the adiabatic piston--is discussed in the framework of kinetic theory. We consider the limit of gases whose relaxation time is extremely fast so that the gases contained in the left and right chambers of the piston are always in equilibrium (that is, the molecules are uniformly distributed and their velocities obey the Maxwell-Boltzmann distribution) after any collision with the piston. Then by using kinetic theory we derive the collision statistics, from which we obtain a set of ordinary differential equations for the evolution of the macroscopic observables (namely, the piston average velocity and position, the velocity variance, and the temperatures of the two compartments). The dynamics of these equations is compared with simulations of an ideal gas and a microscopic model of a gas devised to verify the assumptions used in the derivation. We show that the equations predict an evolution for the macroscopic variables that catches the basic features of the problem. The results here presented recover those derived, using a different approach, by Gruber, Pache, and Lesne [J. Stat. Phys. 108, 669 (2002); Gruber, Pache, and Lesne,J. Stat. Phys.112, 1177 (2003)].