Gorban & Karlin
Abstracts & E-prints
E. Chiavazzo, I.V. Karlin, A.N. Gorban, K. Boulouchos,
A new framework of simulation of reactive flows is proposed based on a coupling between accurate reduced reaction mechanism and the lattice Boltzmann representation of the flow phenomena. The model reduction is developed in the setting of slow invariant manifold construction, and the simplest lattice Boltzmann equation is used in order to work out the procedure of coupling of the reduced model with the flow solver. Practical details of constructing slow invariant manifolds of a reaction system under various thermodynamic conditions are reported. The proposed method is validated with the two-dimensional simulation of a premixed counterflow flame in the hydrogen-air mixture.
E. Chiavazzo, I.V. Karlin, and A.N. Gorban,
The Role of Thermodynamics in Model Reduction when Using Invariant Grids, Commun. Comput. Phys., Vol. 8, No. 4 (2010), pp. 701-734.
In the present work, we develop in detail the process leading to reduction of models in chemical kinetics when using the Method of Invariant Grids (MIG). To this end, reduced models (invariant grids) are obtained by refining initial approximations of slow invariant manifolds, and used for integrating smaller and less stiff systems of equations capable to recover the detailed description with high accuracy. Moreover, we clarify the role played by thermodynamics in model reduction, and carry out a comparison between detailed and reduced solutions for a model hydrogen oxidation reaction.
E. Chiavazzo, I. V. Karlin, A. N. Gorban and K Boulouchos,
Combustion simulation via lattice Boltzmann and reduced chemical kinetics, J. Stat. Mech. (2009) P06013.
We present and validate a methodology for coupling reduced models of detailed combustion mechanisms within the lattice Boltzmann framework. A detailed mechanism (9 species, 21 elementary reactions) for modeling reacting mixtures of air and hydrogen is considered and reduced using the method of invariant grids (MIG). In particular, a 2D quasi-equilibrium grid is constructed, further refined via the MIG method, stored in the form of tables and used to simulate a 1D flame propagating freely through a homogeneous premixed mixture. Comparisons between the detailed and reduced models show that the technique presented enables one to achieve a remarkable speedup in the computations with excellent accuracy.
E. Chiavazzo, A.N. Gorban, and I.V. Karlin,
A modern approach to model reduction in chemical kinetics is often based on the notion of slow invariant manifold. The goal of this paper is to give a comparison of various methods of construction of slow invariant manifolds using a simple Michaelis-Menten catalytic reaction. We explore a recently introduced Method of Invariant Grids (MIG) for iteratively solving the invariance equation. Various initial approximations for the grid are considered such as Quasi Equilibrium Manifold, Spectral Quasi Equilibrium Manifold, Intrinsic Low Dimensional Manifold and Symmetric Entropic Intrinsic Low Dimensional Manifold. Slow invariant manifold was also computed using the Computational Singular Perturbation (CSP) method. A comparison between MIG and CSP is also reported.
Complexity in the description of big chemical reaction networks has both structural (number of species and reactions) and temporal (very different reaction rates) aspects. A consistent way to make model reduction is to construct the invariant manifold which describes the asymptotic system behaviour. In this paper we present a discrete analogue of this object: an invariant grid. The invariant grid is introduced independently from the invariant manifold notion and can serve to represent the dynamic system behaviour as well as to approximate the invariant manifold after refinement. The method is designed for pure dissipative systems and widely uses their thermodynamic properties but allows also generalizations for some classes of open systems. The method is illustrated by two examples: the simplest catalytic reaction (Michaelis-Menten mechanism) and the hydrogen oxidation.
A.N. Gorban, I.V. Karlin,
Quasi-Equilibrium Closure Hierarchies for the Boltzmann Equation, Physica A 360 (2006) 325364 GKQEBoltzPhysA2006.pdf LOCAL COPY
In this paper, explicit method of constructing approximations (the Triangle Entropy Method) is developed for nonequilibrium problems. This method enables one to treat any complicated nonlinear functionals that fit best the physics of a problem (such as, for example, rates of processes) as new independent variables.
The work of the method was demonstrated on the Boltzmann's - type kinetics. New macroscopic variables are introduced (moments of the Boltzmann collision integral, or scattering rates). They are treated as independent variables rather than as infinite moment series. This approach gives the complete account of rates of scattering processes. Transport equations for scattering rates are obtained (the second hydrodynamic chain), similar to the usual moment chain (the first hydrodynamic chain). Various examples of the closure of the first, of the second, and of the mixed hydrodynamic chains are considered for the hard spheres model. It is shown, in particular, that the complete account of scattering processes leads to a renormalization of transport coefficients.
The method gives the explicit solution for the closure problem, provides thermodynamic properties of reduced models, and can be applied to any kinetic equation with a thermodynamic Lyapunov function
A.N. Gorban, I.V. Karlin,
Invariance correction to Grad's equations: Where to go beyond approximations? Continuum Mechanics and Thermodynamics, 17(4) (2005), 311335, GorKarCMT_05.pdf, http://arxiv.org/abs/cond-mat/0504221 , LOCAL COPY
We review some recent developments of Grad's approach to solving the Boltzmann equation and creating reduced description. The method of invariant manifold is put forward as a unified principle to establish corrections to Grad's equations. A consistent derivation of regularized Grad's equations in the framework the method of invariant manifold is given. A new class of kinetic models to lift the finite-moment description to a kinetic theory in the whole space is established. Relations of Grad's approach to modern mesoscopic integrators such as the entropic lattice Boltzmann method are also discussed.
A.N. Gorban, I.V. Karlin,
Invariant Manifolds for Physical and Chemical Kinetics, Lect. Notes Phys. 660, Springer,
A collection of methods to derive analytically and to compute numerically the slow invariant manifolds is presented. Among them, iteration methods based on incomplete linearization, relaxation method and the method of invariant grids are developed. The systematic use of thermodynamic structures and of the quasi-chemical representation allows us to construct approximations which are in concordance with physical restrictions.
The following examples of applications are presented: Nonperturbative derivation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for Knudsen numbers Kn~1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of the list of variables) in order to gain more accuracy in description of highly nonequilibrium flows; kinetic theory of phonons; model reduction in chemical kinetics; derivation and numerical implementation of constitutive equations for polymeric fluids; the limits of macroscopic description for polymer molecules, cell division kinetics.
Keywords: Model Reduction; Invariant Manifold; Entropy; Kinetics; Boltzmann Equation; Fokker--Planck Equation; Navier-Stokes Equation; Burnett Equation; Quasi-chemical Approximation; Oldroyd Equation; Polymer Dynamics; Molecular Individualism; Accuracy Estimation; Post-processing.
PACS codes: 05.20.Dd Kinetic theory, 02.30.Mv Approximations and expansions, 02.70.Dh Finite-element and Galerkin methods, 05.70.Ln Nonequilibrium and irreversible thermodynamics.
S. Ansumali, S. Archidiacono,
S. Chikatamarla, A.N. Gorban,
Regularized Kinetic Theory, E-print: http://arxiv.org/abs/cond-mat/0507601
A new approach to model hydrodynamics at the level of one-particle distribution function is presented. The construction is based on the choice of quasi-equilibria pertinent to the physical context of the problem. Kinetic equations for a single component fluid with a given Prandtl number and models of mixtures with a given Schmidt number are derived. A novel realization of these models via an auxiliary kinetic equation is suggested.
Gorban, A.N.;Gorban, P.A.;Karlin, I.V.
Legendre integrators, post-processing and quasiequilibrium J. Non-Newtonian Fluid Mech. 120 (2004) 149-167GoGoKar2004.pdf LOCAL COPY Online: http://arxiv.org/abs/cond-mat/0308488
A toolbox for the development and reduction of the dynamical models of nonequilibrium systems is presented. The main components of this toolbox are: Legendre integrators, dynamical post-processing, and the thermodynamic projector. The thermodynamic projector is the tool to transform almost any anzatz to a thermodynamically consistent model. The post-processing is the cheapestway to improve the solution obtained by the Legendre integrators. Legendre integrators give the opportunity to solve linear equations instead of nonlinear ones for quasiequilibrium (maximum entropy, MaxEnt) approximations. The essentially new element of this toolbox, the method of thermodynamic projector, is demonstrated on application to the FENE-P model of polymer kinetic theory. The multi-peak model of polymer dynamics is developed.
Uniqueness of thermodynamic projector and kinetic basis of molecular individualism Physica A, 336, 2004, 391-432 UniMolIndRepr.pdf LOCAL COPY Online: http://arxiv.org/abs/cond-mat/0309638
Three results are presented: First, we solve the problem of persistence of dissipation for reduction of kinetic models. Kinetic equations with thermodynamic Lyapunov functions are studied. Uniqueness of the thermodynamic projector is proven: There exists only one projector which transforms any vector field equipped with the given Lyapunov function into a vector field with the same Lyapunov function for a given anzatz manifold which is not tangent to the Lyapunov function levels. Second, we use the thermodynamic projector for developing the short memory approximation and coarse-graining for general nonlinear dynamic systems. We prove that in this approximation the entropy production increases. (The theorem about entropy overproduction.) In example, we apply the thermodynamic projector to derive the equations of reduced kinetics for the Fokker-Planck equation. A new class of closures is developed, the kinetic multipeak polyhedra. Distributions of this type are expected in kinetic models with multidimensional instability as universally as the Gaussian distribution appears for stable systems. The number of possible relatively stable states of a nonequilibrium system grows as 2^m, and the number of macroscopic parameters is in order mn, where n is the dimension of configuration space, and m is the number of independent unstable directions in this space. The elaborated class of closures and equations pretends to describe the effects of molecular individualism. This is the third result.
Gorban, A.N.;Karlin, I.V.;Zinovyev, A.Y.
Constructive methods of invariant manifolds for kinetic problems Phys. Rep., 396, 2004, 197-403 PhysRepCorr.pdf LOCAL COPY Online: http://arxiv.org/abs/cond-mat/0311017
The concept of the slow invariant manifold is recognized as the central idea underpinning a transition from micro to macro and model reduction in kinetic theories. We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in the most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the differential equation for a manifold immersed in the phase space (the invariance equation). The equation of motion for immersed manifolds is obtained (the film extension of the dynamics). Invariant manifolds are fixed points for this equation, and slow invariant manifolds are Lyapunov stable fixed points, thus slowness is presented as stability.
A collection of methods to derive analytically and to compute numerically the slow invariant manifolds is presented. Among them, iteration methods based on incomplete linearization, relaxation method and the method of invariant grids are developed. The systematic use of thermodynamics structures and of the quasi-chemical representation allow to construct approximations which are in concordance with physical restrictions.
The following examples of applications are presented: nonperturbative derivation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for Knudsen numbers Kn~1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of list of variables) to gain more accuracy in description of highly nonequilibrium flows; determination of molecules dimension (as diameters of equivalent hard spheres) from experimental viscosity data ; model reduction in chemical kinetics; derivation and numerical implementation of constitutive equations for polymeric fluids; the limits of macroscopic description for polymer molecules, etc.
Gorban, A.N.;Karlin, I.V.;Zinovyev, A.Y.
Invariant grids for reaction kinetics Physica A, 333, 2004 106-154 ChemGrPhA2004.pdf LOCAL COPY Online: http://arxiv.org/abs/cond-mat/0307076
In this paper, we review the construction of low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). MIM is based on a formulation of the condition of invariance as an equation, and its solution by
Method of invariant manifold for chemical kinetics, Chem.
In this paper, we review the construction of low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). The MIM is based on a formulation of the condition of invariance as an equation, and its solution by
Geometry of irreversibility: The film of nonequilibrium states E-print: http://arxiv.org/abs/cond-mat/0308331
A general geometrical framework of nonequilibrium thermodynamics is developed. The notion of macroscopically definable ensembles is developed. The thesis about macroscopically definable ensembles is suggested. This thesis should play the same role in the nonequilibrium thermodynamics, as the Church-Turing thesis in the theory of computability. The primitive macroscopically definable ensembles are described. These are ensembles with macroscopically prepared initial states. The method for computing trajectories of primitive macroscopically definable nonequilibrium ensembles is elaborated. These trajectories are represented as sequences of deformed equilibrium ensembles and simple quadratic models between them. The primitive macroscopically definable ensembles form the manifold in the space of ensembles. We call this manifold the film of nonequilibrium states. The equation for the film and the equation for the ensemble motion on the film are written down. The notion of the invariant film of non-equilibrium states, and the method of its approximate construction transform the the problem of nonequilibrium kinetics into a series of problems of equilibrium statistical physics. The developed methods allow us to solve the problem of macro-kinetics even when there are no autonomous equations of macro-kinetics
Iliya V. Karlin, Larisa
L. Tatarinova, Alexander N. Gorban,
Hans Christian Ottinger
Irreversibility in the short memory approximation Physica A, 327, 2003, 399-424 LOCAL COPY Online: http://arXiv.org/abs/cond-mat/0305419 v1 18 May 2003 KTGOe2003LANL.pdf LOCAL COPY
A recently introduced systematic approach to derivations of the macroscopic dynamics from the underlying microscopic equations of motions in the short-memory approximation [Gorban et al, Phys. Rev. E 63 , 066124 (2001)] is presented in detail. The essence of this method is a consistent implementation of Ehrenfest's idea of coarse-graining, realized via a matched expansion of both the microscopic and the macroscopic motions. Applications of this method to a derivation of the nonlinear Vlasov-Fokker-Planck equation, diffusion equation and hydrodynamic equations of the uid with a long-range mean field interaction are presented in full detail. The advantage of the method is illustrated by the computation of the post-Navier-Stokes approximation of the hydrodynamics which is shown to be stable unlike the Burnett hydrodynamics.
Alexander N. Gorban,
Iliya V. Karlin
Family of additive entropy functions out of thermodynamic limit, Physical Review E 67, 016104, 2003. Online: http://arXiv.org/abs/cond-mat/0205511
We derive a one-parametric family of entropy functions that respect the additivity condition, and which describe effects of finiteness of statistical systems, in particular, distribution functions with long tails. This one-parametric family is different from the Tsallis entropies, and is a convex combination of the Boltzmann- Gibbs-Shannon entropy and the entropy function proposed by Burg. An example of how longer tails are described within the present approach is worked out for the canonical ensemble. We also discuss a possible origin of a hidden statistical dependence, and give explicit recipes on how to construct corresponding generalizations of the master equation.
Gorban A. N., Karlin I. V.
Geometry of irreversibility, in: Recent Developments in Mathematical and Experimental Physics, Volume C: Hydrodynamics and Dynamical Systems, Ed. F. Uribe (Kluwer,
A general geometrical setting of nonequilibrium thermodynamics is developed. The approach is based on the notion of the natural projection which generalizes Ehrenfests' coarse-graining. It is demonstrated how derivations of irreversible macroscopic dynamics from the microscopic theories can be addressed through a study of stability of quasiequilibrium manifolds.
Methods of nonlinear kinetics, Contribution to the "Encyclopedia of Life Support Systems" (EOLSS Publishers,
Nonlinear kinetic equations are reviewed for a wide audience of specialists and postgraduate students in physics, mathematical physics, material science, chemical engineering and interdisciplinary research.
1. The Boltzmann equation
2. Phenomenology of the Boltzmann equation
3. Kinetic models
4. Methods of reduced description
4.1. The Hilbert method
4.2. The Chapman-Enskog method
4.3. The Grad moment method
4.4. Special approximations
4.5. The method of invariant manifold
4.6. Quasi-equilibrium approximations
5. Discrete velocity models
6. Direct simulation
7. Lattice Gas and Lattice Boltzmann models
8. Other kinetic equations
8.1. The Enskog equation for hard spheres
8.2. The Vlasov equation
8.3. The Smoluchowski equation
Gorban A.N., Karlin I.V.
Hydrodynamics from Grad's equations: What can we learn from exact solutions? Annalen der Physics, 2002. Online: http://arXiv.org/abs/cond-mat/0209560 v1 24 Sep 2002. annphys02.pdf LOCAL COPY
A detailed treatment of the classical Chapman-Enskog derivation of hydrodynamics is given in the framework of Grad's moment equations. Grad's systems are considered as the minimal kinetic models where the Chapman-Enskog method can be studied exactly, thereby providing the basis to compare various approximations in extending the hydrodynamic description beyond the Navier-Stokes approximation. Various techniques, such as the method of partial summation, Pad_e approximants, and invariance principle are compared both in linear and nonlinear situations.
Karlin I.V., Grmela M., Gorban A.N.
Duality in nonextensive statistical mechanics. Physical Review E, 2002, Volume 65, 036128. P.1-4. PRE362002.pdf LOCAL COPY
We revisit recent derivations of kinetic equations based on Tsallis entropy concept. The method of kinetic functions is introduced as a standard tool for extensions of classical kinetic equations in the framework of Tsallis statistical mechanics. Our analysis of the Boltzmann equation demonstrates a remarkable relation between thermodynamics and kinetics caused by the deformation of macroscopic observables.
Gorban A.N., Karlin
I.V., Ottinger H.C.
The additive generalization of the Boltzmann entropy, Physical Review E, 2003, Volume 67, 067104, LOCAL COPY. Online: http://arXiv.org/abs/cond-mat/0209319 v1
Gorban A.N., Karlin I.V.
Macroscopic dynamics through coarse-graining: A solvable example, Physical Review E, 2002, Volume 65, 026116, P.1-5. PREEhr02.pdf LOCAL COPY
The recently derived fluctuation-dissipation formula (A. N. Gorban et al., Phys. Rev. E 63, 066124. 2001) is illustrated by the explicit computation for McKeans kinetic model (H. P. McKean, J. Math. Phys. 8, 547. 1967). It is demonstrated that the result is identical, on the one hand, to the sum of the Chapman-Enskog expansion, and, on the other hand, to the exact solution of the invariance equation. The equality between all three results holds up to the crossover from the hydrodynamic to the kinetic domain.
Gorban A.N., Karlin I.V., Ilg P., Ottinger H.C.
Corrections and enhancements of quasi-equilibrium states, J. Non-Newtonian Fluid Mech. 2001, 96, P. 203-219. NonNew01.pdf LOCAL COPY
We give a compact non-technical presentation of two basic principles for reducing the description of nonequilibrium systems based on the quasi-equilibrium approximation. These two principles are: construction of invariant manifolds for the dissipative microscopic dynamics, and coarse-graining for the entropy-conserving microscopic dynamics. Two new results are presented: first, an application of the invariance principle to hybridization of micro-macro integration schemes is introduced, and is illustrated with non-linear dumbbell models; second, Ehrenfests coarse-graining is extended to general quasi-equilibrium approximations, which gives the simplest way to derive dissipative equations from the Liouville equation in the short memory approximation.
Gorban A.N., Karlin I.V., Ottinger H.C., Tatarinova L.L.
Ehrenfest argument extended to a formalism of nonequilibrium thermodynamics, Physical Review E, 2001. Volume 63, 066124, P.1-6. PREEhr01.pdf LOCAL COPY
A general method of constructing dissipative equations is developed, following Ehrenfestsidea of coarse graining. The approach resolves the major issue of discrete time coarse graining versus continuous time macroscopic equations. Proof of the H theorem for macroscopic equations is given, several examples supporting the construction are presented, and generalizations are suggested.
Gorban A.N., Karlin I.V., Zmievskii V.B., Dymova S.V.
Reduced description in the reaction kinetics, Physica A, 2000, 275, P.361-379. GKZD2000.pdf LOCAL COPY
Models of complex reactions in thermodynamically isolated systems often demonstrate evolution towards low-dimensional manifolds in the phase space. For this class of models, we suggest a direct method to construct such manifolds, and thereby to reduce the effective dimension of the problem. The approach realizes the invariance principle of the reduced description, it is based on iterations rather than on a small parameter expansion, it leads to tractable linear problems, and is consistent with thermodynamic requirements. The approach is tested with a model of catalytic reaction.
A. N. Gorban, I. V. Karlin
Schrodinger operator in an overfull set, Europhys. Lett., 42 (2) (1998), 113-117. GK98Shro.pdf LOCAL COPY
Operational simplicity of an expansion of a wave function over a basis in the Hilbert space is an undisputable advantage for many non-relativistic quantum-mechanical computations. However, in certain cases, there are several \natural" bases at one's disposal, and it is not easy to decide which is preferable. Hence, it sounds attractive to use several bases simultaneously, and to represent states as expansions over an overfull set obtained by a junction of their elements. Unfortunately, as is well known, such a representation is not unique, and lacks many convenient properties of full sets (e.g., explicit formulae to compute coeffcients of expansions). Because of this objection, overfull sets are used less frequently than they, perhaps, deserve.
We introduce a variational principle which eliminates this ambiguity, and results in an expansion which provides the best" representation to a given Schrodinger operator.
Karlin I.V., Gorban A.N., Dukek G., Nonnenmacher T. F.
Dynamic correction to moment approximations. Physical Review E, February 1998 Volume 57, Number 2, P.1668-1672. KGDN98.pdf LOCAL COPY
Considering the Grad moment ansatz as a suitable first approximation to a closed finite-moment dynamics, the correction is derived from the Boltzmann equation. The correction consists of two parts, local and nonlocal. Locally corrected thirteen-moment equations are demonstrated to contain exact transport coefficients. Equations resulting from the nonlocal correction give a microscopic justification to some phenomenological theories of extended hydrodynamics.
Karlin I.V., Gorban A.N., Succi S., Boffi V.
Maximum Entropy Principle for Lattice Kinetic Equations. Physical Review Letters Volume 81, Number 1,
The entropy maximum approach to constructing equilibria in lattice kinetic equations is revisited. For a suitable entropy function, we derive explicitly the hydrodynamic local equilibrium, prove the H theorem for lattice Bhatnagar-Gross-Krook models, and develop a systematic method to account for additional constraints.
Short-Wave Limit of Hydrodynamics: A Soluble Example. Physical Review Letters, Volume 77, Number 2,
The Chapman-Enskog series for shear stress is summed up in a closed form for a simple model of Grad moment equations. The resulting linear hydrodynamics is demonstrated to be stable for all wavelengths, and the exact asymptotic of the acoustic spectrum in the short-wave domain is obtained.
N., Karlin I. V.
Scattering rates versus moments: Alternative Grad equations, Physical Review E October 1996 Volume 54, Number 4, P. 3109-3112. pR3109_11996.pdf LOCAL COPY
Scattering rates (moments of collision integral) are treated as independent variables, and as an alternative to moments of the distribution function, to describe the rarefied gas near local equilibrium. A version of the entropy maximum principle is used to derive the Grad-like description in terms of a finite number of scattering rates. The equations are compared to the Grad moment system in the heat nonconductive case. Estimations for hard spheres demonstrate, in particular, some 10% excess of the viscosity coefficient resulting from the scattering rate description, as compared to the Grad moment estimation.
Gorban A. N., Karlin
On “Solid Liquid” limit of Hydrodynamic Equations, Transport theory and Statistical Physics 24 (9) (1995), 1419-1421. GKSolJet95s.pdf LOCAL COPY
An “infinitely viscid threshold” for compressible liquid is described. A rapid compression of a flux amounts to a strong deceleration of particles (particles loose velocity comparable to heat velocity on a distance compatible to the main free path). Such a strong deceleration is described in the frames of hydrodynamic equations by a divergency of viscosity. A fluid becomes solid.
Alexander N. Gorban,
Iliya V. Karlin
Method of invariant manifolds and regularization of acoustic spectra, Transport Theory and Statistical Physics 23 (5) (1994), 559-632. GorbanKarlinTTSP94.pdf LOCAL COPY
A new approach to the problem of reduced description for Boltzmann-type systems is developed. It involves a direct solution of two main problems: thermodynamicity and dynamic invariance of reduced description. A universal construction is introduced, which gives a thermodynamic parameterization of an almost arbitrary approximation. Newton-type procedures of successive approximations are developed which correct dynamic noninvariance. The method is applied to obtain corrections to the local Maxwell manifold using parametrics expansion instead of
Alexander N. Gorban'
, Iliya V. Karlin
General approach to constructing models of the Boltzmann equation, Physica A, 1994, 206, P.401-420. GKPhA94.pdf LOCAL COPY
The problem of thermodynamic parameterization of an arbitrary approximation of reduced description is solved. On the base of this solution a new class of model kinetic equations is constructed that gives a model extension of the chosen approximation to a kinetic model. Model equations describe two processes: rapid relaxation to the chosen approximation along the planes of rapid motions, and the slow motion caused by the chosen approximation. The H-theorem is proved for these models. It is shown, that the rapid process always leads to entropy growth, and also a neighborhood of the approximation is determined inside which the slow process satisfies the H-theorem. Kinetic models for Grad moment approximations and for the Tamm-Mott-Smith approximation are constructed explicitly. In particular, the problem of concordance of the ES-model with the H-theorem is solved.
Alexander N. Gorban'
, Iliya V. Karlin
Thermodynamic parameterization, Physica A, 1992, 190, P.393-404 GKPhA92.pdf LOCAL COPY
A new method of successive construction of a solution is developed for problems of strongly nonequilibrium Boltzmann kinetics beyond normal solutions. Firstly, the method provides dynamic equations for any manifold of distributions where one looks for an approximate solution. Secondly, it gives a successive procedure of obtaining corrections to these approximations. The method requires neither small parameters, nor strong restrictions upon the initial approximation; it involves solutions of linear problems. It is concordant with the H-theorem at every step. In particular, for the Tamm-Mott-Smith approximation, dynamic equations are obtained, an expansion for the strong shock is introduced, and a linear equation for the first correction is found.
N. N. Bugaenko, A. N. Gorban', and I.
Universal expansion of three-particle distribution function, Theoretical and Mathematical Physics, Vol. 88, No. 3, 1991. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 88, No. 3, pp. 430-441, September, 1991.TMF1990.pdf LOCAL COPY
A universal, i.e., not dependent on the Hamiltonian of the two-particle interaction, expansion of the equilibrium three-particle distribution function with respect to the two-particle correlation functions is constructed. A diagram technique that permits systematic calculation of the coefficients of this expansion is proposed. In particular, it is established that allowance for the first four orders in the absence of long-range correlations gives the