Self-diffusion in a non-uniform one dimensional system of point particles with collisions
type: Item
instance of: scholarly article
title: Self-diffusion in a non-uniform one dimensional system of point particles with collisions
zbMATH Open document ID: 0596.60056
DOI: 10.1007/BF00354038
author: Detlef Dürr
author: Sheldon Goldstein
author: Joel Lebowitz
publication date: 1987-01-01T00:00:00Z
review text: We generalize results of Spitzer, Jepsen and others on the motion of a tagged particle in a uniform one dimensional system of point particles undergoing elastic collisions to the case where there is also an external potential U(x). When U(x) is periodic or random (bounded and statistically translation invariant) then the scaled trajectory of a tagged particle \(y_ A(t)=y(At)/\sqrt{A}\) converges, as \(A\to \infty\), to a Brownian motion \(W_ D(t)\) with diffusion constant \(D=\rho_{\min}<| v| >/{\hat \rho}^ 2\), where \({\hat \rho}\) is the average density, \(<| v| >=\sqrt{2/\pi \beta m}\) is the mean absolute velocity and \(\beta^{-1}\) the temperature of the system. When U(x) is itself changing on a macroscopic scale, i.e. \(U_ A(x)=U(x/\sqrt{A})\), then the limiting process is a spatially dependent diffusion. The stochastic differential equation describing this process is now nonlinear, and is particularly simple in Stratonovich form. This lends weight to the belief that heuristics are best done in that form.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60J70
zbMATH DE Number: 3960688
zbMATH Keywords: motion of a tagged particle
zbMATH Keywords: elastic collisions
MaRDI profile type: Publication
cites work: Q5576048
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full work available at URL: https://doi.org/10.1007/bf00354038
OpenAlex ID: W2032822668
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 3960688