Nonequilibrium fluctuations in particle systems modelling reaction- diffusion equations
type: Item
instance of: scholarly article
title: Nonequilibrium fluctuations in particle systems modelling reaction- diffusion equations
zbMATH Open document ID: 0758.60107
DOI: 10.1016/0304-4149(92)90023-J
author: Anna De Masi
author: C. Boldrighini
author: A. Pellegrinotti
published in: Stochastic Processes and their Applications
publication date: 1992-09-27T00:00:00Z
review text: This paper gives a new example of derivation of a reaction-diffusion equation from a discrete particle model with stochastic dynamics. The basic model (considered by the same authors in a previous paper) consists of particles performing independent random walks on the lattice \(\mathbb{Z}\), the jump times being independently distributed with Poisson distribution of intensity \(\varepsilon^{-2}\), where \(\varepsilon\to 0\) is the parameter that controls the macroscopic continuum limit. At each site \(x\in\mathbb{Z}\) particles are created and destroyed with rates \(q_ +(\eta_ t(x))\), \(q_ -(\eta_ t(x))\), depending on the occupation number \(\eta_ t(x)\) of the site at time \(t\). The functions \(q_ +\) and \(q_ -\) are nonnegative polynomials in the occupation number such that the degree of the annihilation rate \(q_ -\) is larger than that of the creation rate \(q_ +\). As \(\varepsilon\to 0\), the process approximates the solution \(\rho_ t(r)\) (interpreted as the particle density) of a reaction-diffusion equation. In the present paper the process is defined on a periodic lattice \(\mathbb{Z}_ \varepsilon\) with \(\varepsilon^{-2}\) sites. The main result is that the limiting fluctuation field of the system is a generalized Ornstein-Uhlenbeck process whose covariance kernel is given by the solution of a linear equation with coefficients depending on the particle density \(\rho_ t(r)\).
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60J70
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 64562
zbMATH Keywords: interacting particle systems
zbMATH Keywords: reaction-diffusion equation
zbMATH Keywords: discrete particle model with stochastic dynamics
zbMATH Keywords: macroscopic continuum limit
zbMATH Keywords: annihilation rate
zbMATH Keywords: fluctuation field
zbMATH Keywords: Ornstein-Uhlenbeck process
reviewed by: Luis G. Gorostiza
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 64562