Towards a unified dynamical theory of the Brownian particle in an ideal gas
type: Item
instance of: scholarly article
title: Towards a unified dynamical theory of the Brownian particle in an ideal gas
zbMATH Open document ID: 0639.60106
DOI: 10.1007/BF01239014
author: D. Szász
author: Bálint Tóth
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: We consider the trajectory \(Q^ M(t)\) of a Brownian particle of mass M in an ideal gas of identical particles of mass 1 and of density 1 in equilibrium at inverse temperature 1 (the dynamics is uniform motion plus elastic collisions with the Brownian particle). Our theory, in dimension one, describes a variety of limiting processes - containing the Wiener process and the Ornstein-Uhlenbeck process - for \(A^{-1/2} Q^{M(A)} (At)\) depending on the asymptotic behaviour of M(A). Part of the theory is hypothetical while another part relies upon known results. We also prove that, if \(A^{+\epsilon}\ll M(A)\ll A\), then \(A^{-1/2} Q^{M(A)} (At)\) converges to a Wiener process whose variance is known from papers of Sinai-Soloveichik and of the present authors.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B05
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 4043020
zbMATH Keywords: Gibbs equilibrium state
zbMATH Keywords: Brownian particle
zbMATH Keywords: Wiener process
zbMATH Keywords: Ornstein- Uhlenbeck process
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01239014
OpenAlex ID: W2120312637
description: scientific article; zbMATH DE number 4043020