Convergence of the iterated Boltzmann map
type: Item
instance of: scholarly article
title: Convergence of the iterated Boltzmann map
zbMATH Open document ID: 0555.60058
author: Raymond F. Streater
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1984-01-01T00:00:00Z
review text: Consider the Boltzmann equation of degree \(d\geq 2\) for a discrete sample space \(\Omega\) and discrete time; in one step it defines a non-linear map \(\tau\) on the space of measures on \(\Omega\). We show that the entropy increases under \(\tau\). When the energy levels of \(\Omega\) are equally spaced and scattering conserves energy, the iterated map \(\{\tau^ m\mu \}_{m=1,2,...}\) converges in \(\ell_ 2\) to a canonical state \(\mu_{\beta}\) for any initial measure \(\mu\), if the scattering matrix mixes each energy shell. If \(\Omega\) is finite, \(\beta\) depends only on the energy of \(\mu\). Under other mixing conditions, \(\tau^ m\mu\) converges to the microcanonical or grand canonical ensemble.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B05
zbMATH DE Number: 3885086
zbMATH Keywords: Boltzmann equation
zbMATH Keywords: entropy
zbMATH Keywords: scattering matrix
zbMATH Keywords: mixing conditions
zbMATH Keywords: microcanonical or grand canonical ensemble
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195180872
OpenAlex ID: W2096474141
DOI: 10.2977/PRIMS/1195180872
description: scientific article; zbMATH DE number 3885086