Asymptotic analysis of a particle system with mean-field interaction

We study a system of $N$ interacting particles on $\bf{Z}$. The stochastic dynamics consists of two components: a free motion of each particle (independent random walks) and a pair-wise interaction between particles. The interaction belongs to the class of mean-field interactions and models a rollback synchronization in asynchronous networks of processors for a distributed simulation. First of all we study an empirical measure generated by the particle configuration on $\bf{R}$. We prove that if space, time and a parameter of the interaction are appropriately scaled (hydrodynamical scale), then the empirical measure converges weakly to a deterministic limit as $N$ goes to infinity. The limit process is defined as a weak solution of some partial differential equation. We also study the long time evolution of the particle system with fixed number of particles. The Markov chain formed by individual positions of the particles is not ergodic. Nevertheless it is possible to introduce relative coordinates and to prove that the new Markov chain is ergodic while the system as a whole moves with an asymptotically constant mean speed which differs from the mean drift of the free particle motion.