A.N. Gorban and O. Radulescu
Dynamic and Static Limitation in Multiscale Reaction Networks
Revisited, Advances in Chemical Engineering 34, 103-173. 2008

  The concept of the limiting step gives the limit simplification: the whole network behaves as a single step. This is the most popular approach for model simplification in chemical kinetics. However, in its elementary form this idea is applicable only to the simplest linear cycles in steady states. For simple cycles the nonstationary behavior is also limited by a single step, but not the same step that limits the stationary rate. In this chapter, we develop a general theory of static and dynamic limitation for all linear multiscale networks. Our main mathematical tools are auxiliary discrete dynamical systems on finite sets and specially developed algorithms of ‘‘cycles surgery’’ for reaction graphs. New estimates of eigenvectors for diagonally dominant matrices are used.
  Multiscale ensembles of reaction networks with well-separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors (‘‘modes’’) is presented. In particular, we prove that for systems with well-separated constants eigenvalues are real (damped oscillations are improbable). For systems with modular structure, we propose the selection of such modules that it is possible to solve the kinetic equation for every module in the explicit form. All such ‘‘solvable’’ networks are described. The obtained multiscale approximations, that we call ‘‘dominant systems’’ are computationally cheap and robust. These dominant systems can be used for direct computation of steady states and relaxation dynamics, especially when kinetic information is incomplete, for design of experiments and mining of experimental data, and could serve as a robust first approximation in perturbation theory or for preconditioning.

A. N. Gorban
Selection Theorem for Systems with Inheritance
Math. Model. Nat. Phenom., Vol. 2, No. 4, 2007, pp. 1-45.
The original publication is available at www.edpsciences.org

  The problem of finite-dimensional asymptotics of infinite-dimensional dynamic systems is studied. A non-linear kinetic system with conservation of supports for distributions has generically finite-dimensional asymptotics. Such systems are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and economics. This conservation of support has a biological interpretation: inheritance. The finite-dimensional asymptotics demonstrates effects of natural selection. Estimations of the asymptotic dimension are presented. After some initial time, solution of a kinetic equation with conservation of support becomes a finite set of narrow peaks that become increasingly narrow over time and move increasingly slowly. It is possible that these peaks do not tend to fixed positions, and the path covered tends to infinity as t→∞. The drift equations for peak motion are obtained. Various types of distribution stability are studied: internal stability (stability with respect to perturbations that do not extend the support), external stability or uninvadability (stability with respect to strongly small perturbations that extend the support), and stable realizability (stability with respect to small shifts and extensions of the density peaks). Models of self-synchronization of cell division are studied, as an example of selection in systems with additional symmetry. Appropriate construction of the notion of typicalness in infinite-dimensional space is discussed, and the notion of “completely thin” sets is introduced.

R. A. Brownlee, A. N. Gorban, and J. Levesley
Nonequilibrium entropy limiters in lattice Boltzmann methods
Physica A: Statistical Mechanics and its Applications
Volume 387, Issues 2-3, 15 January 2008, Pages 385-406

  We construct a system of nonequilibrium entropy limiters for the lattice Boltzmann methods (LBM). These limiters erase spurious oscillations without blurring of shocks, and do not affect smooth solutions. In general, they do the same work for LBM as flux limiters do for finite differences, finite volumes and finite elements methods, but for LBM the main idea behind the construction of nonequilibrium entropy limiter schemes is to transform a field of a scalar quantity — nonequilibrium entropy. There are two families of limiters: (i) based on restriction of nonequilibrium entropy (entropy “trimming”) and (ii) based on filtering of nonequilibrium entropy (entropy filtering). The physical properties of LBM provide some additional benefits: the control of entropy production and accurate estimation of introduced artificial dissipation are possible. The constructed limiters are tested on classical numerical examples: 1D athermal shock tubes with an initial density ratio 1:2 and the 2D lid-driven cavity for Reynolds numbers between 2000 and 7500 on a coarse 100?100 grid. All limiter constructions are applicable both for entropic and for non-entropic equilibria.

A.N. Gorban, O. Radulescu
Dynamic and static limitation in reaction networks, revisited
http://arxiv.org/abs/physics/0703278  [physics.chem-ph]

   The concept of limiting step gives the limit simplification: the whole network behaves as a single step. This is the most popular approach for model simplification in chemical kinetics. However, in its simplest form this idea is applicable only to the simplest linear cycles in steady states. For such the simplest cycles the nonstationary behaviour is also limited by a single step, but not the same step that limits the stationary rate. In this paper, we develop a general theory of static and dynamic limitation for all linear multiscale networks, not only for simple cycles. Our main mathematical tools are auxiliary discrete dynamical systems on finite sets and specially developed algorithms of ``cycles surgery" for reaction graphs. New estimates of eigenvectors for diagonally dominant matrices are used.
   Multiscale ensembles of reaction networks with well separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors (``modes") is presented. In particular, we proved that for systems with well separated constants eigenvalues are real (damped oscillations are improbable). For systems with modular structure, we propose to select such modules that it is possible to solve the kinetic equation for every module in the explicit form. All such ``solvable" networks are described. The obtained multiscale approximations that we call ``dominant systems" are computationally cheap and robust. These dominant systems can be used for direct computation of steady states and relaxation dynamics, especially when kinetic information is incomplete, for design of experiments and mining of experimental data, and could serve as a robust first approximation in perturbation theory or for preconditioning.

R.A. Brownlee, A.N. Gorban, J. Levesley
Nonequilibrium entropy limiters in lattice Boltzmann methods
arXiv:0704.0043v1 [cond-mat.stat-mech]

  We construct a system of nonequilibrium entropy limiters for the lattice Boltzmann methods (LBM). These limiters erase spurious oscillations without blurring of shocks, and do not affect smooth solutions. In general, they do the same work for LBM as flux limiters do for finite differences, finite volumes and finite elements methods, but for LBM the main idea behind the construction of nonequilibrium entropy limiter schemes is to transform a field of a scalar quantity - nonequilibrium entropy. There are two families of limiters: (i) based on restriction of nonequilibrium entropy (entropy "trimming") and (ii) based on filtering of nonequilibrium entropy (entropy filtering). The physical properties of LBM provide some additional benefits: the control of entropy production and accurate estimate of introduced artificial dissipation are possible. The constructed limiters are tested on classical numerical examples: 1D athermal shock tubes with an initial density ratio 1:2 and the 2D lid-driven cavity for Reynolds numbers Re between 2000 and 7500 on a coarse 100*100 grid. All limiter constructions are applicable for both entropic and non-entropic quasiequilibria.

R. A. Brownlee, A. N. Gorban, and J. Levesley
Stability and stabilization of the lattice Boltzmann method
Phys. Rev. E 75, 036711 (2007)(17 pages)

  We revisit the classical stability versus accuracy dilemma for the lattice Boltzmann methods (LBM). Our goal is a stable method of second-order accuracy for fluid dynamics based on the lattice Bhatnager-Gross-Krook method (LBGK). The LBGK scheme can be recognized as a discrete dynamical system generated by free flight and entropic involution. In this framework the stability and accuracy analysis are more natural. We find the necessary and sufficient conditions for second-order accurate fluid dynamics modeling. In particular, it is proven that in order to guarantee second-order accuracy the distribution should belong to a distinguished surface—the invariant film (up to second order in the time step). This surface is the trajectory of the (quasi)equilibrium distribution surface under free flight. The main instability mechanisms are identified. The simplest recipes for stabilization add no artificial dissipation (up to second order) and provide second-order accuracy of the method. Two other prescriptions add some artificial dissipation locally and prevent the system from loss of positivity and local blowup. Demonstration of the proposed stable LBGK schemes are provided by the numerical simulation of a one-dimensional (1D) shock tube and the unsteady 2D flow around a square cylinder up to Reynolds number Re~20,000.

E. Chiavazzo, A.N. Gorban, and I.V. Karlin
Comparison of Invariant Manifolds for Model Reduction in Chemical Kinetics
Commun. Comput. Phys. Vol. 2, No. 5 (2007), pp. 964-992

  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.

A.N. Gorban
Order–disorder separation: Geometric revision
Physica A, Volume 374, Issue 1, 15 January 2007, Pages 85-102

  After Boltzmann and Gibbs, the notion of disorder in statistical physics relates to ensembles, not to individual states. This disorder is measured by the logarithm of ensemble volume, the entropy. But recent results about measure concentration effects in analysis and geometry allow us to return from the ensemble-based point of view to a state-based one, at least, partially. In this paper, the order–disorder problem is represented as a problem of relation between distance and measure. The effect of strong order–disorder separation for multiparticle systems is described: the phase space could be divided into two subsets, one of them (set of disordered states) has almost zero diameter, the second one has almost zero measure. The symmetry with respect to permutations of particles is responsible for this type of concentration. Dynamics of systems with strong order–disorder separation has high average acceleration squared, which can be interpreted as evolution through a series of collisions (acceleration-dominated dynamics). The time arrow direction from order to disorder follows from the strong order–disorder separation. But, inverse, for systems in space of symmetric configurations with “sticky boundaries” the way back from disorder to order is typical (Natural selection). Recommendations for mining of molecular dynamics results are also presented.

R. A. Brownlee, A. N. Gorban, and J. Levesley
Stabilization of the lattice Boltzmann method using the Ehrenfests' coarse-graining idea
Phys. Rev. E 74, 037703 (2006)

  The lattice Boltzmann method (LBM) and its variants have emerged as promising, computationally efficient and increasingly popular numerical methods for modeling complex fluid flow. However, it is acknowledged that the method can demonstrate numerical instabilities, e.g., in the vicinity of shocks. We propose a simple technique to stabilize the LBM by monitoring the difference between microscopic and macroscopic entropy. Populations are returned to their equilibrium states if a threshold value is exceeded. We coin the name Ehrenfests' steps for this procedure in homage to the vehicle that we use to introduce the procedure, namely, the Ehrenfests' coarse-graining idea.

A.N. Gorban, B.M. Kaganovich, S.P. Filippov, A.V. Keiko, V.A. Shamansky, I.A. Shirkalin
Thermodynamic Equilibria and Extrema: Analysis of Attainability Regions and Partial Equilibria
Springer, Berlin - Heidelberg - New York, 2006.

Ed. by Alexander N. Gorban, Nikolaos  Kazantzis, Ioannis G. Kevrekidis, Hans Christian Ottinger, Constantinos Theodoropoulos
Model Reduction and Coarse--Graining Approaches for Multiscale Phenomena
Springer, Berlin - Heidelberg - New York, 2006

A. Gorban, I. Karlin, A. Zinovyev
Invariant Grids: Method of Complexity Reduction in Reaction Networks
Complexus, V. 2, 110–127, 2006

  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
Basic Types of Coarse-Graining
e-print http://arxiv.org/abs/cond-mat/0602024

  42 pgs, 11 figs. A talk given at the research workshop: "Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena,"
We consider two basic types of coarse-graining: the Ehrenfest's coarse-graining and its extension to a general principle of non-equilibrium thermodynamics, and the coarse-graining based on uncertainty of dynamical models and $\epsilon$-motions (orbits). Non-technical discussion of basic notions and main coarse-graining theorems are presented: the theorem about entropy overproduction for the Ehrenfest's coarse-graining and its generalizations, both for conservative and for dissipative systems, and the theorems about stable properties and the Smale order for $\epsilon$-motions of general dynamical systems including structurally unstable systems. A brief discussion of two other types, coarse-graining by rounding and by small noise, is also presented. Computational kinetic models of macroscopic dynamics are considered. We construct a theoretical basis for these kinetic models using generalizations of the Ehrenfest's coarse-graining.

A.N. Gorban, I.V. Karlin
Quasi-Equilibrium Closure Hierarchies for the Boltzmann Equation
Physica A 360 (2006) 325–364

   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), 311–335, http://arxiv.org/abs/cond-mat/0504221

  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, Berlin, Heidelberg, 2005 (498 pages).
[Preface-Contents-Introduction(pdf)]
[Review in Bull. London Math. Soc. 38 (2006) (pdf)]
[Review in Zentralblatt Math. (2006) (pdf)]
[Editorial Reviews(htm)]
Russian web-site with this book

  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 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.

A.N. Gorban
Order--disorder separation: Geometric revision
E-print: http://arxiv.org/abs/cond-mat/0507644

  After Boltzmann and Gibbs, the notion of disorder in statistical physics relates to ensembles, not to individual states. This disorder is measured by the logarithm of ensemble volume, the entropy. But recent results about measure concentration effects in analysis and geometry allow us to return from the ensemble--based point of view to a state--based one, at least, partially. In this paper, the order--disorder problem is represented as a problem of relation between distance and measure. The effect of strong order--disorder separation for multiparticle systems is described: the phase space could be divided into two subsets, one of them (set of disordered states) has almost zero diameter, the second one has almost zero measure. The symmetry with respect to permutations of particles is responsible for this type of concentration. Dynamics of systems with strong order--disorder separation has high average acceleration squared, which can be interpreted as evolution through a series of collisions (acceleration--dominated dynamics). The time arrow direction from order to disorder follows from the strong order--disorder separation. But, inverse, for systems in space of symmetric configurations with ``sticky boundaries" the way back from disorder to order is typical (Natural selection). Recommendations for mining of molecular dynamics results are presented also.

S. Ansumali, S. Archidiacono, S. Chikatamarla, A.N. Gorban, I.V. Karlin
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.

A.N. Gorban, G.S. Yablonsky
Thermodynamic theory of kinetic overshoots
IMACS2005 extended abstract, E-print: http://arxiv.org/abs/physics/0505135

  Everything that is not prohibited is permissible. So, what is prohibited in the course of chemical reactions, heat transfer and other dissipative processes? Is it possible to "overshoot" the equilibrium, and if yes, then how far? Thermodynamically allowed and prohibited trajectories of processes are discussed by the example of effects of equilibrium encircling. The complete theory of thermodynamically accessible states is presented. The space of all thermodynamically admissible paths is presented by projection on the "thermodynamic tree", that is the tree of the related thermodynamic potential (entropy, free energy, free enthalpy) in the balance polyhedron. The stationary states and limit points for open systems are localized too.

Gorban, A.N.
Singularities of transition processes in dynamical systems: Qualitative theory of critical delays Electron
J. Diff. Eqns. Monograph 5, 2004, 55 p.
Online: http://ejde.math.txstate.edu/Monographs/05/abstr.html

  This monograph presents a systematic analysis of the singularities in the transition processes for dynamical systems. We study general dynamical systems, with dependence on a parameter, and construct relaxation times that depend on three variables: Initial conditions x, parameters k of the system, and accuracy e of the relaxation. We study the singularities of relaxation times as functions of (x,k) under fixed e, and then classify the bifurcations (explosions) of limit sets. We study the relationship between singularities of relaxation times and bifurcations of limit sets. An analogue of the Smale order for general dynamical systems under perturbations is constructed. It is shown that the perturbations simplify the situation: the interrelations between the singularities of relaxation times and other peculiarities of dynamics for general dynamical system under small perturbations are the same as for the Morse-Smale systems.

Gorban, A.N.;Gorban, P.A.;Karlin, I.V.
Legendre integrators, post-processing and quasiequilibrium
J. Non-Newtonian Fluid Mech. 120 (2004) 149-167
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.

Gorban, A.N.;Karlin, I.V.
Uniqueness of thermodynamic projector and kinetic basis of molecular individualism
Physica A, 336, 2004,  391-432
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
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
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 Newton iterations. A grid-based version of MIM is developed. Generalizations to open systems are suggested. The set of methods covered makes it possible to effectively reduce description in chemical kinetics. The most essential new element of this paper is the systematic consideration of a discrete analogue of the slow (stable) positively invariant manifolds for dissipative systems, {invariant grids}. We describe the Newton method and the relaxation method for the invariant grids construction. The problem of the grid correction is fully decomposed into the problems of the grid's nodes correction. The edges between the nodes appears only in the calculation of the tangent spaces. This fact determines high computational efficiency of the invariant grids method.

Gorban, A.N.;Karlin, I.V.
Method of invariant manifold for chemical kinetics
Chem. Eng. Sci.. 58, 2003, 4751-4768
NEW: Elsevier Most Cited Paper Award for this paper DIPLOMA

  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 Newton iterations. A review of existing alternative methods is extended by a thermodynamically consistent version of the method of intrinsic low-dimensional manifolds. A grid-based version of the MIM is developed, and model extensions of low-dimensional dynamics are described. Generalizations to open systems are suggested. The set of methods covered makes it possible to effectively reduce description in chemical kinetics

Gorban A. N., Karlin I. V.
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

Gorban A. N. , Karlin I. V., Zinovyev A. Yu.
Invariant grids for reaction kinetics
Русское название:Инвариантные сетки для химической кинетики
Preprint, June 2003

 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 Newton iterations. A grid-based version of MIM is developed. Generalizations to open systems are suggested. The set of methods covered makes it possible to effectively reduce description in chemical kinetics.
  The most essential new element of this paper is the systematic consideration of a discrete analogue of the slow (stable) positively invariant manifolds for dissipative systems, invariant grids. We describe the Newton method and the relaxation method for the invariant grids construction. The problem of the grid correction is fully decomposed into the problems of the grid's nodes correction. The edges between the nodes appears only in the calculation of the tangent spaces. This fact determines high computational efficiency of the invariant grids method.

Gorban A. N. , Karlin I. V.  [EN]
Quasi-Equilibrium Closure Hierarchies for The Boltzmann Equation
Русское название:Иерархии квазиравновесных аппроксимаций для уравнения Больцмана
E-print,http://arXiv.org/abs/cond-mat/0305599 v1 26 May 2003

 Explicit method of constructing of approximations (Triangle Entropy Method) is developed for strongly nonequilibrium problems of Boltzmann's--type kinetics, i.e. when standard moment variables are insufficient. This method enables one to treat any complicated nonlinear functionals that fit the physics of a problem (such as, for example, rates of processes) as new independent variables.
 The method is applied to the problem of derivation of hydrodynamics from the Boltzmann equation. New macroscopic variables are introduced (moments of the Boltzmann collision integral, or collision moments). 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). Using the triangle entropy method, three different types of the macroscopic description are considered. The first type involves only moments of distribution functions, and results coincide with those of the Grad method in the Maximum Entropy version. The second type of description involves only collision moments. Finally, the third type involves both the moments and the collision moments (the mixed description). The second and the mixed hydrodynamics are sensitive to the choice of the collision model. The second hydrodynamics is equivalent to the first hydrodynamics only for Maxwell molecules, and the mixed hydrodynamics exists for all types of collision models excluding Maxwell molecules. 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 paper gives English translation of the first part of the paper: Gorban, A. N., Karlin, I. V., Quasi-equilibrium approximation and non-standard expansions in the theory of the Boltzmann kinetic equation, in: "Mathematical Modelling in Biology and Chemistry. New Approaches", ed. R. G. Khlebopros, Nauka, Novosibirsk, P.69-117 (1992) [in Russian].

Iliya V. Karlin, Larisa L. Tatarinova, Alexander N. Gorban, Hans Christian Ottinger
Irreversibility in the short memory approximation
Русское название:Необратимость в приближении короткой памяти
Physica A, 327, 2003, 399-424  Online: http://arXiv.org/abs/cond-mat/0305419  v1 18 May 2003

 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. Received 27 March 2002; published 16 January 2003.
DOI: 10.1103/PhysRevE.67.016104
PACS number(s): 05.70.Ln, 05.20.Dd

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.

Горбань А.Н., Карлин И.В.
Геометрия необратимости: натуральный проектор и пленка неравновесных состояний.
V Всероссийский семинар "Моделирование неравновесных систем". Лекция. Красноярск: ИПЦ КГТУ, 2002. 64 с.

Конспект лекции, прочитанной на V Всероссийском семинаре "Моделирование неравновесных систем", Красноярск, 18-20 октября 2002 г.

 Дан новый систематический подход к проблеме необратимости. Построена конструктивная схема вывода уравнений кинетики из микроописания.

 Систематически используются квазиравновесные ансамбли, реализующие условный максимум энтропии. Вводится понятие "макроскопически определимых ансамблей". Они строятся в результате применения двух операций: (а) приведения системы в квазиравновесное состояние по всему набору макроскопических переменных M или по какой-то его части; (б) изменения ансамбля в силу микроскопической динамики (например, уравнения Лиувилля) в течение некоторого времени.

 Подробно описан метод натурального проектора, использующий проектирование с помощью отрезков траекторий (или их тейлоровских приближений – струй) в фазовом пространстве. Он используется для построения диссипативной макрокинетики на основе консервативной микродинамики.

 Показано, что неравновесное состояние системы, соответствующее квазиравновесным начальным условиям, всегда принадлежит некоторому инвариантному многообразию в фазовом пространстве – пленке неравновесных состояний. Получены дифференциальные уравнения, определяющие пленку. Даны методы их приближенного решения.

 Выделены два принципиально различных случая перехода от микро- к макроописанию. В одном из них уравнения на макропеременные при движении по пленке стремятся к некоторой предельной системе уравнений. Этот случай соответствует, в частности, применимости метода статистического оператора Зубарева. Во втором случае предела нет, необходимо введение дополнительных переменных, которые могут быть выбраны в виде разности микроскопической и квазиравновесных энтропий.

 Показано, что классическое представление о разделении времен релаксации не соответствует процессам, начинающимся с квазиравновесных начальных условий. Для них, наоборот, есть иерархия "рождения диссипации": сначала диссипация отсутствует, далее она возникает (первоначально без разделения на процессы – один кинетический коэффициент), потом происходит ветвление на процессы, которое либо обрывается (существует предел макроскопических уравнений), либо ветвится до бесконечности.

 В приложении дан краткий обзор метода инвариантного многообразия для диссипативных систем.

Gorban A. N., Karlin I. V.
Geometry of irreversibility
Русское название: Геометрия необратимости
Recent Developments in Mathematical and Experimental Physics, Volume C: Hydrodynamics and Dynamical Systems, Ed. F. Uribe (Kluwer, Dordrecht, 2002), pp. 19-43. http://www.wkap.nl/prod/b/0-306-47401-8

 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.

Gorban A.N., Karlin I.V.
Methods of nonlinear kinetics.
Русское название: Методы нелинейной кинетики
Contribution to the "Encyclopedia of Life Support Systems" (EOLSS Publishers, Oxford).
On-line version: http://mat.ethz.ch/d-werk/oettinger/pi25/abstract.html (2002)

 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.

Contents:
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.
Method of invariant manifold for chemical kinetics
Русское название: Метод инвариантного многообразия в химической кинетике
arXiv:cond-mat/0207231 v1 9 Jul 2002

  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 Newton iterations. A review of existing alternative methods is extended by a thermodynamically consistent version of the method of intrinsic low-dimensional manifolds. A grid-based version of MIM is developed, and model extensions of low-dimensional dynamics are described. Generalizations to open systems are suggested. The set of methods covered makes it possible to effectively reduce description in chemical kinetics.

Gorban A.N., Karlin I.V.
Family of additive entropy functions out of thermodynamic limit
Русское название: Семейство аддитивных функций энтропии за рамками термодинамического предела
Physical Review E, 2003, in print. On-line: arXiv:cond-mat/0205511 v1 24 May 2002

 Starting with the additivity condition for Lyapunov functions of master equation, we derive a one-parametric family of entropy functions which may be appropriate for a description of certain effects of finiteness of statistical systems, in particular, distribution functions with long tails. This one-parametric family is different from Tsallis entropies, and is essentially a convex combination of the Boltzmann-Gibbs-Shannon entropy and the entropy function introduced by Burg. An example of how longer tails are described within the present approach is worked out for the canonical ensemble. In addition, we discuss a possible origin of a hidden statistical dependence, and give explicit recipes how to construct corresponding generalizations of master equation.

Gorban A.N., Karlin I.V.
Hydrodynamics from Grad's equations: What can we learn from exact solutions?
Русское название: Гидродинамика из уравнений Грэда: чему мы можем научиться из точных решений.
Annalen der Physics, 2002. On-line version: arXiv:cond-mat/0209560 v1 24 Sep 2002

  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.
Русское название: Двойственность в неэкстенсивной статистической механике
Physical Review E, 2002, Volume 65, 036128. P.1-4.

 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
Русское название: Аддитивные обобщения энтропии Больцмана
arXiv:cond-mat/0209319 v1 13 Sep 2002

 There exists only one generalization of the classical Boltzmann-Gibbs-Shannon entropy functional to a one-parametric family of additive entropy functionals. We find analytical solution to the corresponding extension of the classical ensembles, and discuss in some detail the example of the deformation of the uncorrelated state.

Gorban A.N., Karlin I.V.
Macroscopic dynamics through coarse-graining: A solvable example
Русское название: Получение макроскопической динамики с помощью огрубления: точно решаемый пример
Physical Review E, 2002, Volume 65, 026116, P.1-5.

 The recently derived fluctuation-dissipation formula (A. N. Gorban et al., Phys. Rev. E 63, 066124. 2001) is illustrated by the explicit computation for McKean’s 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., Braverman M., and Silantyev V.
Modified Kirchhoff flow with a partially penetrable obstacle and its application to the efficiency of free flow turbines
Mathematical and Computer Modelling, Volume 35, Issue 13, June 2002, P. 1371-1375

 An explicitly solvable analog of the Kirchhoff flow for the case of a semipenetrable obstacle is considered. Its application to estimating the efficiency of free flow turbines is discussed.

Gorban' A., Silantyev V.
Riabouchinsky flow with partially penetrable obstacle
Mathematical and Computer Modelling, Volume 35, Issue 13, June 2002, P. 1365-1370.

 An explicitly solvable Riabouchinsky model with a partially penetrable obstacle is introduced. This model applied to the estimation of the efficiency of free flow turbines allows us to take into account the pressure drop past the lamina.

Gorban' A.N., Gorlov A.N., Silantyev V.M.
Limits of the Turbine Efficiency for Free Fluid Flow
Journal of Energy Resources Technology - December 2001 - Volume 123, Issue 4, pp. 311-317.

 An accurate estimate of the theoretical power limit of turbines in free fluid flows is important because of growing interest in the development of wind power and zero-head water power resources. The latter includes the huge kinetic energy of ocean currents, tidal streams, and rivers without dams. Knowledge of turbine efficiency limits helps to optimize design of hydro and wind power farms. An explicitly solvable new mathematical model for estimating the maximum efficiency of turbines in a free (nonducted) fluid is presented. This result can be used for hydropower turbines where construction of dams is impossible (in oceans) or undesirable (in rivers), as well as for wind power farms. The model deals with a finite two-dimensional, partially penetrable plate in an incompressible fluid. It is nearly ideal for two-dimensional propellers and less suitable for three-dimensional cross-flow Darrieus and helical turbines. The most interesting finding of our analysis is that the maximum efficiency of the plane propeller is about 30 percent for free fluids. This is in a sharp contrast to the 60 percent given by the Betz limit, commonly used now for decades. It is shown that the Betz overestimate results from neglecting the curvature of the fluid streams. We also show that the three-dimensional helical turbine is more efficient than the two-dimensional propeller, at least in water applications. Moreover, well-documented tests have shown that the helical turbine has an efficiency of 35 percent, making it preferable for use in free water currents.

Gorban A.N., Karlin I.V., Ilg P., Ottinger H.C.
Corrections and enhancements of quasi-equilibrium states
Русское название: Уточнения и улучшения квазиравновесных состояний
J. Non-Newtonian Fluid Mech. 2001, 96, P. 203–219.

 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, Ehrenfest’s 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’s argument extended to a formalism of nonequilibrium thermodynamics
Русское название: Аргументация Эренфестов, расширенная до формализма неравновесной термодинамики
Physical Review E, 2001. Volume 63, 066124, P.1-6.

  A general method of constructing dissipative equations is developed, following Ehrenfest’s idea 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
Русское название: Сокращение описания в химической кинетике
Physica A, 2000, 275, P.361-379.

  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.

Zmievski V.B., Karlin I. V., Deville M.
The universal limit in dynamics of dilute polymeric solutions.
Русское название: Универсальный предел в динамике разбавленных растворов полимеров
Physica A 2000, 275 P. 152-177.

 The method of invariant manifold is developed for a derivation of reduced description in kinetic equations of dilute polymeric solutions. It is demonstrated that this reduced description becomes universal in the limit of small Deborah and Weissenberg numbers, and it is represented by the (revised) Oldroyd 8 constants constitutive equation for the polymeric stress tensor. Coe_cients of this constitutive equation are expressed in terms of the microscopic parameters. A systematic procedure of corrections to the revised Oldroyd 8 constants equations is developed. Results are tested with simple flows.

Gorban A.N., Karlin I.V., Dukek G., Nonnenmacher T. F.
Dynamic correction to moment approximations.
Русское название: Динамические поправки к моментным приближениям
Physical Review E, February 1998 Volume 57, Number 2, P.1668-1672.

 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.
Русское название: Принцип максимума энтропии для решеточных кинетических уравнений
Physical Review Letters Volume 81, Number 1, 6 July 1998, P.6-9.

 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.

Gorban A.N., Shokin Yu.I., Verbitskii V.I.
Simultaneously dissipative operators and the infinitesimal moore effect in interval spaces
Русское название: Совместно диссипативные операторы и инфинитиземальный эффект Мура в интервальных пространствах
arXiv:physics/9702021, 1997.

 In solving a system of ordinary differential equations by an interval method the approximate solution at any considered moment of time t represents a set (called interval) containing the exact solution at the moment t. The intervals determining the solution of a system are often expanded in the course of time irrespective of the method and step used.

 The phenomenon of interval expansion, called the Moore sweep effect, essentially decreases the efficiency of interval methods. In the present work the notions of the interval and the Moore effect are formalized and the Infinitesimal Moore Effect (IME) is studied for autonomous systems on positively invariant convex compact. With IME the intervals expand along any trajectory for any small step, and that means that when solving a system by a stepwise interval numerical method with any small step the interval expansion takes place for any initial data irrespective of the applied method. The local conditions of absence of IME in terms of Jacoby matrices field of the system are obtained. The relation between the absence of IME and simultaneous dissipativity of the Jacoby matrices is established, and some sufficient conditions of simultaneous dissipativity are obtained. (The family of linear operators is simultaneously dissipative, if there exists a norm relative to which all the operators are dissipative.)

Gorban A.N., Karlin I.V.
Short-Wave Limit of Hydrodynamics: A Soluble Example.
Русское название: Коротковолновой предел гидродинамики: точно решаемый пример
Physical Review Letters, Volume 77, Number 2, 8 July 1996. P. 282-285.

  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.

Gorban A.N., Karlin I.V. Nonnenmacher T. F., Zmievskii V.B.
Relaxation Trajectories: Global approximation.
Русское название: Траектории релаксации: глобальная аппроксимация
Physica A, 1996, 231, P.648-672.

Gorban A. N., Karlin I. V.
Scattering rates versus moments: Alternative Grad equations
Русское название: Скорости рассеяния вместо моментов. Альтернативные уравнения Грэда
Physical Review E October 1996 Volume 54, Number 4, P. 3109-3112

 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.

Alexander N. Gorban' , Iliya V. Karlin
General approach to constructing models of the Boltzmann equation
Русское название: Общий подход к построению моделей уравнения Больцмана
Physica A, 1994, 206, P.401-420

 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
Русское название:Термодинамическая параметризация
Physica A, 1992, 190, P.393-404

 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.

Gorban A.N., Bykov V.I.
A model of autooscillations in association reactions
Русское название: Модель автоколебаний в реакциях ассоциации
Chemical Engineering Science. 1987, Vol. 42, No. 5. P. 1249-1251.

 The aim of this paper is to show that association reactions can result in the appearance of autooscillations in nonlinear systems.

Gorban A.N., Bykov V.I., Yablonskii G.S.
Thermodynamic function analogue for reactions proceeding without interaction of various substances
Русское название: Аналог термодинамических функций для реакций без взаимодействия различных веществ
ChemicaI Engineering Science, 1986. Vol. 41, No. 11. P. 2739-2745.

  Function similar to Lyapunov’s function has been constructed for reactions with α_iA_i→∑_jα_jA_j stages. This provides for the quasi-thermodynamics of the appropriate kinetic model, which implies steady-state uniqueness and global stability in the reaction polyhedron. The kinetic law generalizing the Marcelin –de Donder kinetics has been written for a separate stage. Explicit Lyapunov thermodynamic functions have been written for various conditions of the reaction proceeding in closed systems. The matrix of linear approximation close to equilibrium is expressed by means of the introduced scalar product. Particularly, the absence of damped oscillations as equilibrium is approached as shown.

Gorban A.N., Bykov V.I.
Macroscopic clusters induced by diffusion in a catalytic oxidation reactions
Русское название: Макроскопические кластеры, индуцированные диффузией в каталитических реакциях окисления
ChemicaI Engineering Science, 1980. Vol. 35, P. 2351-2352

Gorban A.N.
Singularities of Transition Processes In Dynamical Systems.
Русское название: Особенности переходных процессов в динамических системах
arXiv:chao-dyn/9703010 v1 18 Mar 1997, Translation of Candidat (Ph.D) Thesis, 1980

 The paper gives the systematic analysis of singularities of transition processes in general dynamical systems. Dynamical systems depending on parameter are studied. A system of relaxation times is constructed. Each relaxation time depends on three variables: initial conditions, parameters k of the system and accuracy ε of relaxation. This system of times contains: the time before the first entering of the motion into ε-neighbourhood of the limit set, the time of final entering in this neighbourhood and the time of stay of the motion outside the ε-neighbourhood of the limit set. The singularities of relaxation times as functions of (x₀; k) under fixed ε are studied. A classification of different bifurcations (explosions) of limit sets is performed. The bifurcations fall into those with appearance of new limit points and bifurcations with appearance of new limit sets at finite distance from the existing ones. The relations between the singularities of relaxation times and bifurcations of limit sets are studied. The peculiarities of dynamics which entail singularities of transition processes without bifurcations are described as well. The peculiarities of transition processes under perturbations are studied. It is shown that the perturbations simplify the situation: the interrelations between the singularities of relaxation times and other peculiarities of dynamics for general dynamical system under small perturbations are the same as for smooth two-dimensional structural stable systems.