On Prigogine's approaches to irreversibility: a case study by the baker map
type: Item
instance of: scholarly article
title: On Prigogine's approaches to irreversibility: a case study by the baker map
zbMATH Open document ID: 1076.82029
DOI: 10.1155/S1026022604312069
author: Xianqiang Yang
published in: Discrete Dynamics in Nature and Society
publication date: 2005-03-21T00:00:00Z
full work available at URL: https://eudml.org/doc/125357
review text: One of the long standing problems in statistical mechanics is the consistent description of macroscopic irreversibility in terms of reversible microscopic dynamics. Two different approaches to irreversibility by I. Prigogine school are known: the \(\Lambda\)-transformation and complex spectral theories. The author studies the baker map on these two approaches and compares the results. It is shown that in both theories the evolution operator \(U^{+}\) becomes dissipative by restriction obvervables to an appropriate subspace \(\Phi\) of the Hilbert space \(L_2\), and its spectral set contains an annulus in the unit disc. But these theories are not equivalent. According to the first one a bijective map \(\Lambda^{\dag -1}:\Phi\rightarrow L^2\) is looked for and the evolution operator \(U\) of densities is transformed to a dissipative operator \(W=\Lambda U\Lambda^{-1}\). On the second one the class of densities is restricted further so that the most values in the interior of the annulus are removed from the spectrum. Here the relaxation of expectation values is described by the Pollicot-Ruelle decomposition.
Mathematics Subject Classification ID: 82C03
Mathematics Subject Classification ID: 37A60
Mathematics Subject Classification ID: 37N20
Mathematics Subject Classification ID: 37A05
zbMATH DE Number: 2148593
zbMATH Keywords: statistical mechanics
zbMATH Keywords: I.Prigogine theories of irreversibility
zbMATH Keywords: \(\Lambda\)-transformation
zbMATH Keywords: complex spectral theory
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
OpenAlex ID: W2027934965
description: scientific article; zbMATH DE number 2148593