Invariant manifolds for physical and chemical kinetics
type: Item
instance of: scholarly article
title: Invariant manifolds for physical and chemical kinetics
zbMATH Open document ID: 1086.82009
author: Alexander N. Gorban
author: I. V. Karlin
published in: Lecture Notes in Physics
publication date: 2005-03-15T00:00:00Z
review text: The book under review presents a collection of constructive methods for the study of slow (stable) positively invariant manifolds of finite and infinite dissipative (and partly conservative) dynamic systems that arise in kinetics. Kinetics is mainly described by the Boltzmann equation, a variety of kinetic models such as, e.g., the nonlinear Bhatnagar-Gross-Krook (BGK) model, Fokker-Planck, Enskog, and Vlasov equations, and equations of chemical kinetics which altogether are thoroughly discussed in Chapter 2 of the present book. A major problem that is raised by the Boltzmann equation is the problem of the reduced description that consists in answering the following questions: 1. What are the conditions of the validity of the macroscopic description? 2. Which macroscopic variables are relevant for this description? 3. What are the equations for the macroscopic variables and how they can be derived from the kinetic equations? The classical methods of the reduced description for the Boltzmann equation are the Hilbert method, the Chapman-Enskog method, and the Grad moment method. The reduction of description suggests the existence of a manifold of slow motions or shortly, a slow manifold in the phase space of the studied dynamic system. It is also assumed that this manifold is positively invariant, that is, if a motion starts on the manifold at \(t = t_o\), it retains therein at \(t > t_o\). The invariance equation (Chapter 3) that tells that the defect of invariance always vanishes and thus provides the necessary conditions of invariance of a manifold is the key tool, together with the concept of projector field, in the analysis conducted throughout this book. The definition of slowness of a positively invariant manifold is presented in Chapter 4: in fact, a slow invariant manifold is Lyapunov stable fixed point of the film extension of the dynamics. The next Chapter introduces the basic thermodynamic structures, the entropy, the entropic scalar product which is gener…
Mathematics Subject Classification ID: 82C03
Mathematics Subject Classification ID: 82C05
Mathematics Subject Classification ID: 82-01
zbMATH DE Number: 2145997
zbMATH Keywords: Boltzmann equation
zbMATH Keywords: Hilbert method
zbMATH Keywords: Chapman-Enskog method
zbMATH Keywords: Grad moment method
zbMATH Keywords: lattice gas
zbMATH Keywords: Fokker-Planck equation
zbMATH Keywords: Vlasov equation
zbMATH Keywords: reduced description
zbMATH Keywords: slowness
zbMATH Keywords: entropy
zbMATH Keywords: quasiequilibrium
zbMATH Keywords: projector field
zbMATH Keywords: Newton method
zbMATH Keywords: invariance equation
zbMATH Keywords: relaxation methods
zbMATH Keywords: method of invariant grids
zbMATH Keywords: method of natural projector
zbMATH Keywords: irreversibility
zbMATH Keywords: slow positively invariant manifold
zbMATH Keywords: open system
zbMATH Keywords: attractors
reviewed by: Eugene Kryachko
MaRDI profile type: Publication
full work available at URL: b98103
OpenAlex ID: W1581063030
Wikidata QID: Q57384877
DOI: 10.1007/B98103
description: scientific article; zbMATH DE number 2145997