Continuum limits of discrete gases
type: Item
instance of: scholarly article
title: Continuum limits of discrete gases
zbMATH Open document ID: 0693.76078
DOI: 10.1007/BF00281496
author: N. E. Zubov
published in: Archive for Rational Mechanics and Analysis
publication date: 1989-01-01T00:00:00Z
review text: \textit{J. Greenberg} and \textit{G. Hedstrom} [(*) Arch. Ration. Mech. Anal. 40; 160-165 (1971; Zbl 0214.353)] investigated the limiting behavior of a discrete interacting particle system as the initial particle spacing and masses tend to zero. The model studied assumed a countable collection of particles, each with mass \({\tilde \rho}\)h, moving along the real axis according to \({\tilde \rho}\)hẍ\({}_ k=0\), \(\dot x{}_ k(0)=u_ 0(kh)\), and \(x_ k(0)=kh\), for \(k=0,\pm 1,\pm 2,..\). When particles collide, momenta and energy are conserved. This condition leads to the following interaction rule: if particles \(p,p+1,...,p+q\) collide at some instant and if \(v_ p^->v^-_{p+1}>...>v^-_{p+q}\) are the particle velocities before the collision, then the velocities after the collision, which we denote by \((v^+_ p,v^+_{p+1},...,v^+_{p+q})\), are given by \((v^+_ p,v^+_{p+1},...,v^+_{p+q})=(v^-_{p+q},v^-_{p+q- 1},...,v^-_ p).\) The principal motivation behind the Greenberg and Hedstrom's investigation (*) was to establish the existence of the limiting continuum motion \[ \chi (x,t)=^{def}\lim_{h\to 0,\quad kh=x\quad fixed}x_ k(t) \] and to examine regularity properties of this limit. No limiting equation of motion was obtained for \(\chi\) though it was shown that the limit motion did exhibit shock waves as boundaries of regions where the particle motions were highly oscillatory. In this note we again examine the problem discussed in (*). What is new is the simpler characterization of the limiting field quantities such as density (\({\hat \rho}\)), velocity \((\hat u)\), and pressure \((\hat p)\), as well as a characterization of the equations satisfied by these fields. In certain cases we obtain a closed system of conservation laws for \({\hat \rho}\), \(\hat u,\) and \(\hat p.\)
Mathematics Subject Classification ID: 76P05
Mathematics Subject Classification ID: 82B40
zbMATH DE Number: 4135610
zbMATH Keywords: discrete interacting particle system
zbMATH Keywords: initial particle spacing and masses tend to zero
zbMATH Keywords: limiting continuum motion
MaRDI profile type: Publication
cites work: A note on a model for the quasilinear wave equation
full work available at URL: https://doi.org/10.1007/bf00281496
OpenAlex ID: W2130069877
description: scientific article; zbMATH DE number 4135610