A stochastic model of deposition processes with nucleation

Abstract

We study an interacting particle system on a one-dimensional infinite lattice and one-dimensional lattices with a periodic boundary. In this system, each site of the lattice may be either empty or occupied and initially all the lattice sites are empty. The evolution of the system is defined as follows: an empty site waits an exponential time with mean 1 and becomes occupied, and an occupied site becomes empty at a time which is distributed exponentially with meanμ _k, wherek is the number of occupied neighboring sites of this site in the current state of the system. We show that the mean number of the occupied sites of the lattice, considered as a function of time, may possess a convex part. A sufficient condition for this is thatμ ₀ is large andμ _k,k⩾1, are small. The studied system has been proposed recently as a mathematical model of certain deposition processes, in particular those which exhibit nucleation caused by lateral attractive interaction between the deposited molecules. Our research was motivated by the observation that the density of deposited molecules contains a convex part, over some time interval, if the attractive forces are strong, while this density is a concave function of time if these forces are weak or absent. Our result agrees with this observation.