A.N. Gorban, I.V. Karlin, Invariant Manifolds for Physical and Chemical Kinetics,
Lect. Notes Phys. 660, Springer, Berlin, Heidelberg, 200
Dedication: To our parents
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 ({\it the
invariance equation}). The equation of motion for immersed manifolds is obtained ({\it the film
extension of the dynamics}). Invariant manifolds are fixed points for this equation, and slow
invariant manifolds are Lyapunov stable fixed points, thus {\it 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.