**Gorban**** & Karlin **

**Abstracts & E-prints**

**2010**

E. Chiavazzo, I.V. Karlin, A.N. Gorban, K. Boulouchos,

**Coupling of the model reduction technique
with the lattice Boltzmann method**, Combustion and Flame, 2010 doi:10.1016/j.combustflame.2010.06.009

A new framework of
simulation of reactive flows is proposed based on a coupling between accurate
reduced reaction mechanism and the lattice Boltzmann representation of the flow
phenomena. The model reduction is developed in the setting of slow invariant
manifold construction, and the simplest lattice Boltzmann equation is used in
order to work out the procedure of coupling of the reduced model with the flow
solver. Practical details of constructing slow invariant manifolds of a
reaction system under various thermodynamic conditions are reported. The
proposed method is validated with the two-dimensional simulation of a premixed counterflow flame in the hydrogen-air mixture.

E. Chiavazzo, I.V. Karlin, and A.N. Gorban,

**The Role of Thermodynamics
in Model Reduction when Using Invariant Grids****, ***Commun**.** Comput. Phys.,* Vol. **8**, No. 4 (2010), pp. 701-734.

In the present work, we develop in detail the process leading to reduction of models in chemical kinetics when using the Method of Invariant Grids (MIG). To this end, reduced models (invariant grids) are obtained by refining initial approximations of slow invariant manifolds, and used for integrating smaller and less stiff systems of equations capable to recover the detailed description with high accuracy. Moreover, we clarify the role played by thermodynamics in model reduction, and carry out a comparison between detailed and reduced solutions for a model hydrogen oxidation reaction.

**2009**

E. Chiavazzo, I. V. Karlin, A. N. Gorban and K Boulouchos,

**Combustion simulation via
lattice Boltzmann and reduced chemical kinetics****,** J. Stat.
Mech. (2009) P06013.

We present and validate a methodology for coupling reduced models of detailed combustion mechanisms within the lattice Boltzmann framework. A detailed mechanism (9 species, 21 elementary reactions) for modeling reacting mixtures of air and hydrogen is considered and reduced using the method of invariant grids (MIG). In particular, a 2D quasi-equilibrium grid is constructed, further refined via the MIG method, stored in the form of tables and used to simulate a 1D flame propagating freely through a homogeneous premixed mixture. Comparisons between the detailed and reduced models show that the technique presented enables one to achieve a remarkable speedup in the computations with excellent accuracy.

**2007**

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, pp. 964-992 CiCP2007vol2_n5_p964.pdf LOCAL COPY

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. Gorban,

**Invariant Grids: Method of Complexity Reduction in Reaction Networks,** Complexus, V.
2, 110–127. ComPlexUs2006.pdf LOCAL COPY

Complexity
in the description of big chemical reaction networks has both structural
(number of species and reactions) and temporal (very different reaction rates)
aspects. A consistent way to make model reduction is to construct the invariant
manifold which describes the asymptotic system behaviour. In this paper we
present a discrete analogue of this object: an invariant grid. The invariant
grid is introduced independently from the invariant manifold notion and can
serve to represent the dynamic system behaviour as well as to approximate the
invariant manifold after refinement. The method is designed for pure
dissipative systems and widely uses their thermodynamic properties but allows
also generalizations for some classes of open systems. The method is
illustrated by two examples: the simplest catalytic reaction (Michaelis-Menten mechanism) and the hydrogen oxidation.

A.N. Gorban, I.V. Karlin,**
Quasi-Equilibrium Closure Hierarchies for the Boltzmann Equation**, Physica A 360 (2006) 325–364 GKQEBoltzPhysA2006.pdf LOCAL COPY

In this paper, explicit method of constructing approximations (the Triangle Entropy Method) is developed for nonequilibrium problems. This method enables one to treat any complicated nonlinear functionals that fit best the physics of a problem (such as, for example, rates of processes) as new independent variables.

The work of the method was demonstrated on the Boltzmann's - type kinetics. New macroscopic variables are introduced (moments of the Boltzmann collision integral, or scattering rates). They are treated as independent variables rather than as infinite moment series. This approach gives the complete account of rates of scattering processes. Transport equations for scattering rates are obtained (the second hydrodynamic chain), similar to the usual moment chain (the first hydrodynamic chain). Various examples of the closure of the first, of the second, and of the mixed hydrodynamic chains are considered for the hard spheres model. It is shown, in particular, that the complete account of scattering processes leads to a renormalization of transport coefficients.

The method gives the explicit solution for the closure problem, provides thermodynamic properties of reduced models, and can be applied to any kinetic equation with a thermodynamic Lyapunov function

A.N. Gorban, I.V. Karlin,

**Invariance correction to Grad's equations: Where to go beyond
approximations?** Continuum Mechanics and Thermodynamics, 17(4) (2005), 311–335, GorKarCMT_05.pdf,
http://arxiv.org/abs/cond-mat/0504221
, LOCAL COPY

We review some recent developments of Grad's approach
to solving the Boltzmann equation and creating reduced description. The method
of invariant manifold is put forward as a unified principle to establish
corrections to Grad's equations. A consistent derivation of regularized Grad's
equations in the framework the method of invariant manifold is given. A new
class of kinetic models to lift the finite-moment description to a kinetic
theory in the whole space is established. Relations of Grad's approach to
modern mesoscopic integrators such as the entropic
lattice Boltzmann method are also discussed.

A.N. Gorban, I.V. Karlin,

**Invariant Manifolds for Physical and Chemical Kinetics,** Lect.
Notes Phys. 660, Springer, *Bull.** London Math. Soc**.** *38 (2006) (pdf)
LOCAL COPY] [Review
in Zentralblatt Math.

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 (

A collection of methods to derive analytically and to compute numerically the slow invariant manifolds is presented. Among them, iteration methods based on incomplete linearization, relaxation method and the method of invariant grids are developed. The systematic use of thermodynamic structures and of the quasi-chemical representation allows us to construct approximations which are in concordance with physical restrictions.

The following examples of applications are presented: Nonperturbative derivation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for Knudsen numbers Kn~1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of the list of variables) in order to gain more accuracy in description of highly nonequilibrium flows; kinetic theory of phonons; model reduction in chemical kinetics; derivation and numerical implementation of constitutive equations for polymeric fluids; the limits of macroscopic description for polymer molecules, cell division kinetics.

**Keywords:** Model Reduction;
Invariant Manifold; Entropy; Kinetics; Boltzmann Equation; Fokker--Planck
Equation; Navier-Stokes Equation; Burnett Equation;
Quasi-chemical Approximation; Oldroyd Equation;
Polymer Dynamics; Molecular Individualism; Accuracy Estimation;
Post-processing.

**PACS codes:** 05.20.Dd Kinetic
theory, 02.30.Mv Approximations and expansions, 02.70.Dh Finite-element and Galerkin methods, 05.70.Ln Nonequilibrium
and irreversible thermodynamics.

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

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.

**2004**

Gorban, A.N.;Gorban, P.A.;Karlin, I.V.

**Legendre integrators, post-processing and quasiequilibrium
**J. Non-Newtonian Fluid Mech. 120 (2004) 149-167GoGoKar2004.pdf LOCAL COPY
Online: http://arxiv.org/abs/cond-mat/0308488

A toolbox for the development and reduction of the dynamical models of nonequilibrium systems is presented. The main components of
this toolbox are: Legendre integrators, dynamical post-processing, and the
thermodynamic projector. The thermodynamic projector is the tool to transform
almost any anzatz to a thermodynamically consistent
model. The post-processing is the cheapestway to
improve the solution obtained by the Legendre integrators. Legendre integrators
give the opportunity to solve linear equations instead of nonlinear ones for quasiequilibrium (maximum entropy, MaxEnt)
approximations. The essentially new element of this toolbox, the method of
thermodynamic projector, is demonstrated on application to the FENE-P model of
polymer kinetic theory. The multi-peak model of polymer dynamics is developed.

Gorban, A.N.;Karlin,
I.V.

**Uniqueness of thermodynamic projector and kinetic basis of molecular
individualism** Physica A, 336, 2004, 391-432
UniMolIndRepr.pdf LOCAL COPY
Online: http://arxiv.org/abs/cond-mat/0309638

Three results are presented: First, we solve the problem of persistence of
dissipation for reduction of kinetic models. Kinetic equations with
thermodynamic Lyapunov functions are studied.
Uniqueness of the thermodynamic projector is proven: There exists only one
projector which transforms any vector field equipped with the given Lyapunov function into a vector field with the same Lyapunov function for a given anzatz
manifold which is not tangent to the Lyapunov function levels. Second, we use the thermodynamic
projector for developing the short memory approximation and coarse-graining for
general nonlinear dynamic systems. We prove that in this approximation the
entropy production increases. (The theorem about entropy
overproduction.) In example, we apply the thermodynamic projector to
derive the equations of reduced kinetics for the Fokker-Planck equation. A new
class of closures is developed, the kinetic multipeak
polyhedra. Distributions of this type are expected in
kinetic models with multidimensional instability as universally as the Gaussian
distribution appears for stable systems. The number of possible relatively
stable states of a nonequilibrium system grows as
2^m, and the number of macroscopic parameters is in order mn,
where n is the dimension of configuration space, and m is the number of
independent unstable directions in this space. The elaborated class of closures
and equations pretends to describe the effects of molecular individualism. This
is the third result.

Gorban, A.N.;Karlin, I.V.;Zinovyev, A.Y.

**Constructive methods of invariant manifolds for kinetic problems** Phys.
Rep., 396, 2004, 197-403 PhysRepCorr.pdf LOCAL COPY
Online: http://arxiv.org/abs/cond-mat/0311017

The concept of the slow invariant manifold is recognized as the central idea
underpinning a transition from micro to macro and model reduction in kinetic
theories. We present the Constructive Methods of Invariant Manifolds for model
reduction in physical and chemical kinetics, developed during last two decades.
The physical problem of reduced description is studied in the most general form
as a problem of constructing the slow invariant manifold. The invariance
conditions are formulated as the differential equation for a manifold immersed
in the phase space (the invariance equation). The equation of motion for
immersed manifolds is obtained (the film extension of the dynamics). Invariant
manifolds are fixed points for this equation, and slow invariant manifolds are Lyapunov stable fixed points, thus slowness is presented as
stability.

A collection of methods to derive analytically and to compute numerically the
slow invariant manifolds is presented. Among them, iteration methods based on
incomplete linearization, relaxation method and the method of invariant grids
are developed. The systematic use of thermodynamics
structures and of the quasi-chemical representation allow to construct
approximations which are in concordance with physical restrictions.

The following examples of applications are presented: nonperturbative
derivation of physically consistent hydrodynamics from the Boltzmann equation
and from the reversible dynamics, for Knudsen numbers Kn~1; construction of the
moment equations for nonequilibrium media and their
dynamical correction (instead of extension of list of variables) to gain more
accuracy in description of highly nonequilibrium
flows; determination of molecules dimension (as diameters of equivalent hard
spheres) from experimental viscosity data ; model reduction in chemical
kinetics; derivation and numerical implementation of constitutive equations for
polymeric fluids; the limits of macroscopic description for polymer molecules,
etc.

Gorban, A.N.;Karlin, I.V.;Zinovyev, A.Y.

**Invariant grids for reaction kinetics **Physica
A, 333, 2004 106-154 ChemGrPhA2004.pdf LOCAL COPY
Online: http://arxiv.org/abs/cond-mat/0307076

In this paper, we review the construction of low-dimensional manifolds of
reduced description for equations of chemical kinetics from the standpoint of
the method of invariant manifold (MIM). MIM is based on a formulation of the
condition of invariance as an equation, and its solution by

Gorban, A.N.;Karlin,
I.V.,**
Method of invariant manifold for chemical kinetics**, Chem.

In this paper, we review the construction of low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). The MIM is based on a formulation of the condition of invariance as an equation, and its solution by

**Geometry of irreversibility: The film of ****nonequilibrium**** states** E-print: http://arxiv.org/abs/cond-mat/0308331

A general geometrical framework of nonequilibrium
thermodynamics is developed. The notion of macroscopically definable ensembles
is developed. The thesis about macroscopically definable ensembles is
suggested. This thesis should play the same role in the nonequilibrium
thermodynamics, as the Church-Turing thesis in the theory of computability. The
primitive macroscopically definable ensembles are described. These are
ensembles with macroscopically prepared initial states. The method for
computing trajectories of primitive macroscopically definable nonequilibrium ensembles is elaborated. These trajectories
are represented as sequences of deformed equilibrium ensembles and simple
quadratic models between them. The primitive macroscopically definable
ensembles form the manifold in the space of ensembles. We call this manifold
the film of nonequilibrium states. The equation for
the film and the equation for the ensemble motion on the film are written down.
The notion of the invariant film of non-equilibrium states,
and the method of its approximate construction transform the the problem of nonequilibrium
kinetics into a series of problems of equilibrium statistical physics.
The developed methods allow us to solve the problem of macro-kinetics even when
there are no autonomous equations of macro-kinetics

Iliya V. Karlin, Larisa
L. Tatarinova, Alexander N. Gorban,
Hans Christian Ottinger

**Irreversibility in the short memory approximation** Physica
A, 327, 2003, 399-424 LOCAL COPY Online: http://arXiv.org/abs/cond-mat/0305419
v1 18 May 2003 KTGOe2003LANL.pdf LOCAL COPY

A recently introduced systematic approach to derivations of the macroscopic dynamics
from the underlying microscopic equations of motions in the short-memory
approximation [Gorban et al, Phys. Rev. E 63 , 066124
(2001)] is presented in detail. The essence of this method is a consistent
implementation of Ehrenfest's idea of coarse-graining,
realized via a matched expansion of both the microscopic and the macroscopic
motions. Applications of this method to a derivation of the nonlinear Vlasov-Fokker-Planck equation, diffusion equation and
hydrodynamic equations of the uid with a long-range
mean field interaction are presented in full detail. The advantage of the
method is illustrated by the computation of the post-Navier-Stokes
approximation of the hydrodynamics which is shown to be stable unlike the
Burnett hydrodynamics.

Alexander N. Gorban,
Iliya V. Karlin

**Family of additive entropy functions out of thermodynamic limit**,
Physical Review E 67, 016104, 2003. Online: http://arXiv.org/abs/cond-mat/0205511

We derive a one-parametric family of entropy functions
that respect the additivity condition, and which
describe effects of finiteness of statistical systems, in particular,
distribution functions with long tails. This one-parametric family is different
from the Tsallis entropies, and is a convex
combination of the Boltzmann- Gibbs-Shannon entropy and the entropy function
proposed by Burg. An example of how longer tails are
described within the present approach is worked out for the canonical ensemble.
We also discuss a possible origin of a hidden statistical dependence, and give
explicit recipes on how to construct corresponding generalizations of the
master equation.

Gorban A. N., Karlin I. V.

**Geometry of irreversibility**, in: Recent Developments in Mathematical and
Experimental Physics, Volume C: Hydrodynamics and Dynamical Systems, Ed. F. Uribe (Kluwer,

A general geometrical setting of nonequilibrium
thermodynamics is developed. The approach is based on the notion of the natural
projection which generalizes Ehrenfests'
coarse-graining. It is demonstrated how derivations of irreversible macroscopic
dynamics from the microscopic theories can be addressed through a study of
stability of quasiequilibrium manifolds.

Gorban A.N.,
Karlin I.V.

**Methods of nonlinear kinetics**, Contribution to the "Encyclopedia of Life Support Systems" (EOLSS
Publishers,

Nonlinear kinetic equations are reviewed for a wide
audience of specialists and postgraduate students in physics, mathematical
physics, material science, chemical engineering and interdisciplinary research.

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.

**Hydrodynamics from Grad's equations: What can we learn from exact solutions?
**Annalen der Physics, 2002. Online: http://arXiv.org/abs/cond-mat/0209560 v1 24 Sep 2002. annphys02.pdf LOCAL COPY

A detailed treatment of the classical Chapman-Enskog
derivation of hydrodynamics is given in the framework of Grad's moment
equations. Grad's systems are considered as the minimal kinetic models where
the Chapman-Enskog method can be studied exactly,
thereby providing the basis to compare various approximations in extending the
hydrodynamic description beyond the Navier-Stokes
approximation. Various techniques, such as the method of partial summation, Pad_e approximants, and invariance principle are compared
both in linear and nonlinear situations.

Karlin I.V., Grmela M., Gorban A.N.

**Duality in nonextensive statistical mechanics. **Physical Review E, 2002, Volume 65, 036128. P.1-4. PRE362002.pdf LOCAL COPY

We revisit recent derivations of kinetic equations based on Tsallis
entropy concept. The method of kinetic functions is introduced as a standard
tool for extensions of classical kinetic equations in the framework of Tsallis statistical mechanics. Our analysis of the
Boltzmann equation demonstrates a remarkable relation between thermodynamics
and kinetics caused by the deformation of macroscopic observables.

Gorban A.N., Karlin
I.V., Ottinger H.C.

**The additive generalization of the Boltzmann entropy, **Physical
Review E, 2003, Volume 67, 067104, LOCAL COPY. Online: http://arXiv.org/abs/cond-mat/0209319 v1

Gorban A.N., Karlin I.V.

**Macroscopic dynamics through coarse-graining: A solvable example,**
Physical Review E, 2002, Volume 65, 026116, P.1-5. PREEhr02.pdf LOCAL COPY

The recently derived fluctuation-dissipation formula (A. N. Gorban
et al., Phys. Rev. E 63, 066124. 2001) is illustrated by the explicit computation
for 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.N., Karlin I.V., Ilg P., Ottinger H.C.

**Corrections and enhancements of quasi-equilibrium states, **J.
Non-Newtonian Fluid Mech. 2001, 96, P. 203-219. NonNew01.pdf LOCAL COPY

We give a compact non-technical presentation of two
basic principles for reducing the description of nonequilibrium
systems based on the quasi-equilibrium approximation. These two principles are:
construction of invariant manifolds for the dissipative microscopic dynamics,
and coarse-graining for the entropy-conserving microscopic dynamics. Two new
results are presented: first, an application of the invariance principle to
hybridization of micro-macro integration schemes is introduced, and is illustrated
with non-linear dumbbell models; second, 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****’ argument extended to a
formalism of nonequilibrium thermodynamics,**
Physical Review E, 2001. Volume 63, 066124, P.1-6. PREEhr01.pdf
LOCAL COPY

A general method of constructing dissipative equations
is developed, following Ehrenfest’sidea of
coarse graining. The approach resolves the major issue of discrete time coarse
graining versus continuous time macroscopic equations. Proof of the H theorem for
macroscopic equations is given, several examples supporting the construction
are presented, and generalizations are suggested.

Gorban A.N., Karlin I.V., Zmievskii V.B., Dymova S.V.

**Reduced description in the reaction kinetics, **Physica
A, 2000, 275, P.361-379. GKZD2000.pdf LOCAL COPY

Models of complex reactions in thermodynamically isolated systems often
demonstrate evolution towards low-dimensional manifolds in the phase space. For
this class of models, we suggest a direct method to construct such manifolds,
and thereby to reduce the effective dimension of the problem. The approach
realizes the invariance principle of the reduced description, it is based on
iterations rather than on a small parameter expansion, it leads to tractable
linear problems, and is consistent with thermodynamic requirements. The
approach is tested with a model of catalytic reaction.

A. N. Gorban, I. V. Karlin

**Schrodinger operator in an overfull set**,
Europhys. Lett., 42 (2) (1998), 113-117. GK98Shro.pdf LOCAL COPY

Operational simplicity of an expansion of a wave function over a basis in the
Hilbert space is an undisputable advantage for many non-relativistic
quantum-mechanical computations. However, in certain cases, there are several
\natural" bases at one's disposal, and it is not easy to decide which is
preferable. Hence, it sounds attractive to use several bases simultaneously,
and to represent states as expansions over an overfull set obtained by a
junction of their elements. Unfortunately, as is well known, such a
representation is not unique, and lacks many convenient properties of full sets
(e.g., explicit formulae to compute coeffcients of
expansions). Because of this objection, overfull sets are used less frequently
than they, perhaps, deserve.

We introduce a variational principle which eliminates this ambiguity, and results in an expansion which provides “the best" representation to a given Schrodinger operator.

Karlin I.V., Gorban A.N., Dukek G., Nonnenmacher T. F.

**Dynamic correction to moment approximations.** Physical Review E, February
1998 Volume 57, Number 2, P.1668-1672. KGDN98.pdf LOCAL COPY

Considering the Grad moment ansatz as a suitable
first approximation to a closed finite-moment dynamics, the correction is
derived from the Boltzmann equation. The correction consists of two parts,
local and nonlocal. Locally corrected thirteen-moment equations are
demonstrated to contain exact transport coefficients. Equations resulting from
the nonlocal correction give a microscopic justification to some
phenomenological theories of extended hydrodynamics.

Karlin I.V., Gorban A.N., Succi S., Boffi V.

**Maximum Entropy Principle for Lattice Kinetic Equations.** Physical Review
Letters Volume 81, Number 1,

The entropy maximum approach to constructing equilibria
in lattice kinetic equations is revisited. For a suitable entropy function, we
derive explicitly the hydrodynamic local equilibrium, prove the H theorem for
lattice Bhatnagar-Gross-Krook
models, and develop a systematic method to account for additional constraints.

Gorban A.N.,
Karlin I.V.

**Short-Wave Limit of Hydrodynamics: A Soluble Example.** Physical Review
Letters, Volume 77, Number 2,

The Chapman-Enskog series for shear stress is summed
up in a closed form for a simple model of Grad moment equations. The resulting
linear hydrodynamics is demonstrated to be stable for all wavelengths, and the
exact asymptotic of the acoustic spectrum in the short-wave domain is
obtained.

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

**Relaxation Trajectories: Global approximation.** Physica
A, 1996, 231, P.648-672. GKZNPhA96.pdf LOCAL COPY

Gorban A.
N., Karlin I. V.

**Scattering rates versus moments: Alternative Grad equations,** Physical
Review E October 1996 Volume 54, Number 4, P. 3109-3112. pR3109_11996.pdf LOCAL COPY

Scattering rates (moments of collision integral) are treated as independent
variables, and as an alternative to moments of the distribution function, to describe
the rarefied gas near local equilibrium. A version of the entropy maximum
principle is used to derive the Grad-like description in terms of a finite
number of scattering rates. The equations are compared to the Grad moment
system in the heat nonconductive case. Estimations for hard spheres
demonstrate, in particular, some 10% excess of the viscosity coefficient
resulting from the scattering rate description, as compared to the Grad moment
estimation.

Gorban A. N., Karlin
I. V.

**On “Solid Liquid”
limit of Hydrodynamic Equations,** Transport theory and Statistical Physics
24 (9) (1995), 1419-1421. GKSolJet95s.pdf LOCAL COPY

An “infinitely viscid threshold” for compressible liquid is described. A rapid
compression of a flux amounts to a strong deceleration of particles (particles
loose velocity comparable to heat velocity on a distance compatible to the main
free path). Such a strong deceleration is described in the frames of
hydrodynamic equations by a divergency of viscosity.
A fluid becomes “solid”.

Alexander N. Gorban,
Iliya V. Karlin

**Method of invariant manifolds and
regularization of acoustic spectra,**
Transport Theory and Statistical Physics 23 (5) (1994), 559-632. GorbanKarlinTTSP94.pdf LOCAL COPY

A new approach to the problem of reduced description for Boltzmann-type systems
is developed. It involves a direct solution of two main problems: thermodynamicity and dynamic invariance of reduced
description. A universal construction is introduced, which gives a
thermodynamic parameterization of an almost arbitrary approximation.
Newton-type procedures of successive approximations are developed which correct
dynamic noninvariance. The method is applied to
obtain corrections to the local Maxwell manifold using parametrics
expansion instead of

Alexander N. Gorban'
, Iliya V. Karlin

**General approach to constructing models of the Boltzmann equation,** Physica A, 1994, 206, P.401-420. GKPhA94.pdf LOCAL COPY

The problem of thermodynamic parameterization of an arbitrary approximation of
reduced description is solved. On the base of this solution a new class of model
kinetic equations is constructed that gives a model extension of the chosen
approximation to a kinetic model. Model equations describe two processes: rapid
relaxation to the chosen approximation along the planes of rapid motions, and
the slow motion caused by the chosen approximation. The H-theorem is proved for
these models. It is shown, that the rapid process always leads to entropy
growth, and also a neighborhood of the approximation
is determined inside which the slow process satisfies the H-theorem. Kinetic
models for Grad moment approximations and for the Tamm-Mott-Smith approximation
are constructed explicitly. In particular, the problem of concordance of the
ES-model with the H-theorem is solved.

Alexander N. Gorban'
, Iliya V. Karlin

**Thermodynamic parameterization,** Physica A,
1992, 190, P.393-404 GKPhA92.pdf LOCAL COPY

A new method of successive construction of a solution is developed for problems
of strongly nonequilibrium Boltzmann kinetics beyond
normal solutions. Firstly, the method provides dynamic equations for any
manifold of distributions where one looks for an approximate solution.
Secondly, it gives a successive procedure of obtaining corrections to these
approximations. The method requires neither small parameters, nor strong
restrictions upon the initial approximation; it involves solutions of linear
problems. It is concordant with the H-theorem at every step. In particular, for
the Tamm-Mott-Smith approximation, dynamic equations are obtained, an expansion
for the strong shock is introduced, and a linear equation for the first
correction is found.

**1991 **

N. N. Bugaenko, A. N. Gorban', and I.
V. Karlin

**Universal expansion of three-particle
distribution function,** Theoretical and Mathematical Physics, Vol. 88, No. 3, 1991. Translated
from Teoreticheskaya i Matematicheskaya Fizika, Vol. 88,
No. 3, pp. 430-441, September, 1991.TMF1990.pdf LOCAL COPY

A universal, i.e., not dependent on the Hamiltonian of the two-particle
interaction, expansion of the equilibrium three-particle distribution function
with respect to the two-particle correlation functions is constructed. A
diagram technique that permits systematic calculation of the coefficients of
this expansion is proposed. In particular, it is established that allowance for
the first four orders in the absence of long-range correlations gives the