[RUSSIAN]  
Selected Publications and Preprints  
2008  
[EN] A.N. Gorban and O. Radulescu
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. 

[EN] A. N. Gorban
The problem of finitedimensional asymptotics of infinitedimensional dynamic systems is studied. A nonlinear kinetic system with conservation of supports for distributions has generically finitedimensional 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 finitedimensional 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 selfsynchronization of cell division are studied, as an example of selection in systems with additional symmetry. Appropriate construction of the notion of typicalness in infinitedimensional space is discussed, and the notion of “completely thin” sets is introduced. 

[EN] R. A. Brownlee, A. N. Gorban, and J. Levesley
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 liddriven cavity for Reynolds numbers between 2000 and 7500 on a coarse 100?100 grid. All limiter constructions are applicable both for entropic and for nonentropic equilibria. 

2007  
[EN] A.N. Gorban, O. Radulescu
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. 

[EN]
R.A. Brownlee,
A.N. Gorban,
J. Levesley
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 liddriven cavity for Reynolds numbers Re between 2000 and 7500 on a coarse 100*100 grid. All limiter constructions are applicable for both entropic and nonentropic quasiequilibria. 

[EN] R. A. Brownlee, A. N. Gorban, and J. Levesley
We revisit the classical stability versus accuracy dilemma for the lattice Boltzmann methods (LBM). Our goal is a stable method of secondorder accuracy for fluid dynamics based on the lattice BhatnagerGrossKrook 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 secondorder accurate fluid dynamics modeling. In particular, it is proven that in order to guarantee secondorder 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 secondorder 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 onedimensional (1D) shock tube and the unsteady 2D flow around a square cylinder up to Reynolds number Re~20,000. 

[EN] 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 MichaelisMenten 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. 

[EN] A.N. Gorban
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 ensemblebased point of view to a statebased 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 (accelerationdominated 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. 

2006  
[EN] R. A. Brownlee, A. N. Gorban, and J.
Levesley
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' coarsegraining idea. 

[EN] A.N. Gorban,
B.M. Kaganovich, S.P. Filippov, A.V. Keiko, V.A. Shamansky, I.A. Shirkalin


[EN] Ed. by Alexander N. Gorban, Nikolaos Kazantzis, Ioannis G. Kevrekidis, Hans
Christian Ottinger, Constantinos Theodoropoulos


[EN] A. Gorban, I. Karlin, A. Zinovyev
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 (MichaelisMenten mechanism) and the hydrogen oxidation. 

[EN] A.N.
Gorban
42 pgs, 11 figs. A talk given at the research
workshop: "Model
Reduction and CoarseGraining Approaches for Multiscale
Phenomena," 

[EN] A.N.
Gorban, I.V. Karlin
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. 

2005  
[EN] A.N.
Gorban, I.V. Karlin
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 finitemoment 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. 

[EN] A.N. Gorban, I.V. Karlin
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 quasichemical 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; FokkerPlanck Equation; NavierStokes Equation; Burnett Equation; Quasichemical Approximation; Oldroyd Equation; Polymer Dynamics; Molecular Individualism; Accuracy Estimation; Postprocessing. PACS codes: 05.20.Dd Kinetic theory, 02.30.Mv Approximations and expansions, 02.70.Dh Finiteelement and Galerkin methods, 05.70.Ln Nonequilibrium and irreversible thermodynamics. 

[EN] A.N.
Gorban
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 ensemblebased point of view to a statebased one, at least, partially. In this paper, the orderdisorder problem is represented as a problem of relation between distance and measure. The effect of strong orderdisorder 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 orderdisorder separation has high average acceleration squared, which can be interpreted as evolution through a series of collisions (accelerationdominated dynamics). The time arrow direction from order to disorder follows from the strong orderdisorder 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. 

[EN] S. Ansumali, S. Archidiacono, S. Chikatamarla, A.N.
Gorban, I.V. Karlin
A new approach to model hydrodynamics at the level of oneparticle distribution function is presented. The construction is based on the choice of quasiequilibria 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. 

[EN] A.N.
Gorban, G.S. Yablonsky
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. 

[EN] Gorban, A.N.
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 MorseSmale systems. 

[EN] Gorban, A.N.;Gorban, P.A.;Karlin, I.V.
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 postprocessing, and the thermodynamic projector. The thermodynamic projector is the tool to transform almost any anzatz to a thermodynamically consistent model. The postprocessing 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 FENEP model of polymer kinetic theory. The multipeak model of polymer dynamics is developed. 

[EN] Gorban, A.N.;Karlin, I.V.
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 coarsegraining 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 FokkerPlanck 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. 

[EN] Gorban, A.N.;Karlin, I.V.;Zinovyev, A.Y.
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. 

[EN] Gorban, A.N.;Karlin, I.V.;Zinovyev, A.Y.
In
this paper, we review the construction of lowdimensional 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 

2003  
[EN] Gorban, A.N.;Karlin, I.V.
In this paper, we review the construction
of lowdimensional 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 

[EN] Gorban A. N., Karlin I.
V.
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 ChurchTuring 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 nonequilibrium 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 macrokinetics even when there are no autonomous equations of macrokinetics 

[EN] Gorban A. N. , Karlin I. V., Zinovyev A. Yu. In this paper, we review the construction of lowdimensional 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 gridbased 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. 

[EN] Gorban A. N. , Karlin I. V. [EN] Explicit method of constructing of approximations (Triangle Entropy
Method) is developed for strongly nonequilibrium problems of Boltzmann'stype 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. 

[EN] Iliya V. Karlin, Larisa L. Tatarinova, Alexander N. Gorban, Hans Christian Ottinger A recently introduced systematic approach to derivations of the macroscopic dynamics from the underlying microscopic equations of motions in the shortmemory 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 coarsegraining, realized via a matched expansion of both the microscopic and the macroscopic motions. Applications of this method to a derivation of the nonlinear VlasovFokkerPlanck equation, diffusion equation and hydrodynamic equations of the uid with a longrange mean field interaction are presented in full detail. The advantage of the method is illustrated by the computation of the postNavierStokes approximation of the hydrodynamics which is shown to be stable unlike the Burnett hydrodynamics. 

[EN] Alexander N. Gorban, Iliya V. Karlin We derive a oneparametric 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 oneparametric family is different from the Tsallis entropies, and is a convex combination of the Boltzmann GibbsShannon 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. 

[RU] Горбань А.Н., Карлин И.В. Конспект лекции, прочитанной на V Всероссийском семинаре "Моделирование неравновесных систем", Красноярск, 1820 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 какойто 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. 

[EN] Gorban A. N., Karlin I. V. A general geometrical setting of nonequilibrium thermodynamics is developed. The approach is based on the notion of the natural projection which generalizes Ehrenfests' coarsegraining. It is demonstrated how derivations of irreversible macroscopic dynamics from the microscopic theories can be addressed through a study of stability of quasiequilibrium manifolds. 

[EN] Gorban A.N., Karlin I.V. 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: 

[EN] Gorban A.N., Karlin I.V. In this paper, we review the construction of lowdimensional 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 lowdimensional manifolds. A gridbased version of MIM is developed, and model extensions of lowdimensional dynamics are described. Generalizations to open systems are suggested. The set of methods covered makes it possible to effectively reduce description in chemical kinetics. 

[EN] Gorban A.N., Karlin I.V. Starting with the additivity condition for Lyapunov functions of master equation, we derive a oneparametric 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 oneparametric family is different from Tsallis entropies, and is essentially a convex combination of the BoltzmannGibbsShannon 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. 

[EN] Gorban A.N., Karlin I.V. A detailed treatment of the classical ChapmanEnskog 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 ChapmanEnskog method can be studied exactly, thereby providing the basis to compare various approximations in extending the hydrodynamic description beyond the NavierStokes approximation. Various techniques, such as the method of partial summation, Pad_e approximants, and invariance principle are compared both in linear and nonlinear situations. 

[EN] Karlin I.V., Grmela M., Gorban A.N. 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. 

[EN] Gorban A.N., Karlin I.V., Ottinger H.C. There exists only one generalization of the classical BoltzmannGibbsShannon entropy functional to a oneparametric 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. 

[EN] Gorban A.N., Karlin I.V. The recently derived fluctuationdissipation 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 ChapmanEnskog 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. 

[EN] Gorban' A., Braverman M., and Silantyev V. 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. 

[EN] Gorban' A., Silantyev V. 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. 

[EN] Gorban' A.N., Gorlov A.N., Silantyev V.M. 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 zerohead 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 twodimensional, partially penetrable plate in an incompressible fluid. It is nearly ideal for twodimensional propellers and less suitable for threedimensional crossflow 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 threedimensional helical turbine is more efficient than the twodimensional propeller, at least in water applications. Moreover, welldocumented tests have shown that the helical turbine has an efficiency of 35 percent, making it preferable for use in free water currents. 

[EN] Gorban A.N., Karlin I.V., Ilg P., Ottinger H.C.
We give a compact nontechnical presentation of two basic principles for reducing the description of nonequilibrium systems based on the quasiequilibrium approximation. These two principles are: construction of invariant manifolds for the dissipative microscopic dynamics, and coarsegraining for the entropyconserving 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 nonlinear dumbbell models; second, Ehrenfest’s coarsegraining is extended to general quasiequilibrium approximations, which gives the simplest way to derive dissipative equations from the Liouville equation in the short memory approximation. 

[EN] Gorban A.N., Karlin I.V., Ottinger H.C., Tatarinova
L.L. 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. 

[EN] Gorban A.N., Karlin I.V., Zmievskii V.B., Dymova
S.V. Models of complex reactions in thermodynamically isolated systems often demonstrate evolution towards lowdimensional 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. 

[EN] Zmievski V.B., Karlin I. V., Deville M.
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. 

[EN] Gorban A.N., Karlin I.V., Dukek G., Nonnenmacher
T. F. Considering the Grad moment ansatz as a suitable first approximation to a closed finitemoment dynamics, the correction is derived from the Boltzmann equation. The correction consists of two parts, local and nonlocal. Locally corrected thirteenmoment 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. 

[EN] Karlin I.V., Gorban A.N., Succi S., Boffi V.
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 BhatnagarGrossKrook models, and develop a systematic method to account for additional constraints. 

[EN] Gorban A.N., Shokin Yu.I., Verbitskii V.I. 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.) 

[EN] Gorban A.N., Karlin I.V. The ChapmanEnskog 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 shortwave domain is obtained. 

[EN] Gorban A.N., Karlin I.V. Nonnenmacher T. F.,
Zmievskii V.B. 

[EN] Gorban A. N., Karlin I. V. 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 Gradlike 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. 

[EN] Alexander N. Gorban' , Iliya V. Karlin 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 Htheorem 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 Htheorem. Kinetic models for Grad moment approximations and for the TammMottSmith approximation are constructed explicitly. In particular, the problem of concordance of the ESmodel with the Htheorem is solved. 

[EN] Alexander N. Gorban' , Iliya V. Karlin 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 Htheorem at every step. In particular, for the TammMottSmith approximation, dynamic equations are obtained, an expansion for the strong shock is introduced, and a linear equation for the first correction is found. 

[EN] Gorban A.N., Bykov V.I. The aim of this paper is to show that association reactions can result in the appearance of autooscillations in nonlinear systems. 

[EN] Gorban A.N., Bykov V.I., Yablonskii G.S.
Function similar to Lyapunov’s function has been constructed for reactions with α_{i}A_{i}→∑_{j}α_{j}A_{j} stages. This provides for the quasithermodynamics of the appropriate kinetic model, which implies steadystate 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. 

[EN] Gorban A.N., Bykov V.I. 

[EN] Gorban A.N. 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 εneighbourhood of the limit set, the time of final entering in this neighbourhood and the time of stay of the motion outside the εneighbourhood of the limit set. The singularities of relaxation times as functions of (x_{0}; 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 twodimensional structural stable systems. 
