One-dimensional classical massive particle in the ideal gas
type: Item
instance of: scholarly article
title: One-dimensional classical massive particle in the ideal gas
zbMATH Open document ID: 0589.60084
DOI: 10.1007/BF01210949
author: M. R. Soloveichik
author: Ya. G. Sinai
published in: Communications in Mathematical Physics
publication date: 1986-01-01T00:00:00Z
review text: Consider a massive particle (m.p.) of mass \(M\) moving in one direction under the action of elastic collisions with equal particles of mass 1. Assume that the equal particles have no interaction. Denote by \(q_ 0\) and \(v_ 0\) the coordinate and velocity of the m.p., respectively. A collection of equal particles is denoted by \(X=\{x\}\), \(x=(q,v)\) and put \(Y=(x_ 0,X)\) where \(x_ 0=(q_ 0,v_ 0)\). The set of all possible \(Y\) is denoted by \(\Omega\). The dynamics of the system is described by the equilibrium measure \(\mu\) on \(\Omega\) : \[ d\mu =dq_ 0\sqrt{\beta M/2\pi}e^{-\beta Mv^ 2_ 0/2}dv_ 0dP(x) \] where \(P\) is the limit Gibbs measure on the phase space of all possible \(X\), i.e. the equal particles distributed on the line according to a Poisson distribution with density \(\rho\) and their velocities have independent Gaussian distribution with the density \(\sqrt{\beta /2\pi}\exp [-\beta v^ 2/2].\) The authors obtain the following results: (1) For every \(\epsilon >0\) and \(\mu\)-almost every \(Y\) one can find \(t_ 0(Y,\epsilon)\) such that for all \(t>t_ 0(Y,\epsilon)\), \(| q_ 0(t)-q_ 0| \leq t^{+\epsilon}\); (2) For \(\mu\)-almost every \(Y\) each particle of the equal particles has finitely many collisions with m.p.; (3) Assume that \(q_ 0=0\) and denote by \(\mu_ 0\) the conditional measure of \(\mu\) under \(q_ 0=0\). Then with respect to \(\mu_ 0,\) \(q_ 0(t)=n^+(t)-n^- (t)+\xi (t)\) where the random variables \(n^{\pm}(t)/\sqrt{t}\) have the same Gaussian limit distribution for \(t\to \infty\), while \(\xi(t)/\sqrt{t}\) converges in probability to zero.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B05
zbMATH DE Number: 3947374
zbMATH Keywords: elastic collisions
zbMATH Keywords: equilibrium measure
zbMATH Keywords: Gibbs measure
zbMATH Keywords: Gaussian limit distribution
reviewed by: Mu-Fa Chen
MaRDI profile type: Publication
cites work: Q5822308
cites work: Q4189915
cites work: Diffusion with “collisions” between particles
cites work: On the effect of collisions on the motion of an atom in \(R^1\)
cites work: Bounds for the limiting variance of the ''heavy particle'' in \(R^ 1\)
description: scientific article; zbMATH DE number 3947374