Laws of large numbers for mesoscopic stochastic models of reacting and diffusing particles

Laws of large numbers for mesoscopic stochastic models of reacting and diffusing particles Christian Reichert ? Abstract We study the asymptotic behaviour of mesoscopic stochastic models for systems of reacting and diffusing particles (also known as density-dependent population pro- cesses) as the number of particles goes to infinity. Our approach is related to the variational approach to solving the parabolic partial differential equations that arise as limit dynamics. We first present a result for a model that converges to a system of reaction-diffusion equations. In addition, we discuss two models with nonlinear diffusion that give rise to quasilinear parabolic equations in the limit. Key words: reaction-diffusion model, interacting random processes, law of large numbers 1 Introduction In this paper we study the asymptotic behaviour of certain mesoscopic stochastic particle models (or density-dependent population processes) for reaction-diffusion systems as the number of particles goes to infinity. Mesoscopic stochastic particle models are informally defined as follows. We think of a chemical reactor as being composed of cells or compart- ments of mesoscopic size l. Each cell may contain up to about n particles of each species. Particles of species j jump randomly from a cell to an adjacent one in direction ±ek ? Rm according to rates dj,k± which may be functions of the particle densities in the cell (the particle numbers divided by n) and their discrete gradients.

• mesoscopic stochastic


• discrete finite-difference


• without chemical


• diffusion system


• chemical reactor


• vertex z ?


• stochastic particle