Nonequilibrium fluctuations in particle systems modelling reaction- diffusion equations
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 \(\rho_ t(r)\).
discrete particle model with stochastic dynamics
scientific article; zbMATH DE number 64562