Kinetic limits of the HPP cellular automaton
type: Item
instance of: scholarly article
title: Kinetic limits of the HPP cellular automaton
zbMATH Open document ID: 0925.82093
DOI: 10.1007/BF01060074
author: Anna De Masi
author: Raffaele Esposito
author: Errico Presutti
published in: Journal of Statistical Physics
publication date: 1993-10-27T00:00:00Z
review text: We study the Boltzmann-Grad limit in various versions of the two-dimensional HPP cellular automaton. In the completely deterministic case we prove convergence to an evolution that is not of kinetic type, a well-known phenomenon after Uchyiama's paper on the Broadwell gas, whereas the limiting equation becomes of kinetic type in the model with random collisions. The main part of the paper concerns the case where the collisions are deterministic and the randomness comes from inserting, between any two successive HPP updatings, \(\epsilon{}^-\){}\(^\nu{}\) stirring updatings, \(\nu{} <\) 1 being any fixed positive number and \(\epsilon{}\) a parameter which tends to 0. The initial measure is a product measure with average occupation numbers of the order of \(\epsilon{}\) (low-density limit) and varying on distances of the order of \(\epsilon{}^-\){}\(^1\). The limit as \(\epsilon{} \rightarrow{}\) 0 of the system evolved for times of the order of \(\epsilon{}^-\){}\(^1\){}\(^-\){}\(^\nu{}\) corresponds to the Boltzmann-Grad limit. We prove propagation of chaos and that the renormalized average occupation numbers (i.e., divided by \(\epsilon{}\)) converge to the solution of the Broadwell equation. Convergence is proven at all times for which the solution of the Broadwell equation is bounded.
Mathematics Subject Classification ID: 82B40
Mathematics Subject Classification ID: 82C20
Mathematics Subject Classification ID: 68Q80
Mathematics Subject Classification ID: 82C32
zbMATH DE Number: 111909
zbMATH Keywords: kinetic theory
zbMATH Keywords: stochastic processes
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 111909