A hyperbolic conservation law and particle systems

Abstract

In these notes we consider two particle systems: the totally asymmetric simple exclusion process and the totally asymmetric zero-range process. We introduce the notion of hydrodynamic limit and describe the partial differential equation that governs the evolution of the conserved quantity – the density of particles p(t,.). This equation is a hyperbolic conservation law of type ətp(p, u) + vF(p(t, u)) = 0, where the flux F is a concave function. Taking these systems evolving on the Euler time scale tN, a central limit theorem for the empirical measure holds and the temporal evolution of the limit density field is deterministic. By taking the system in a reference frame with constant velocity, the limit density field does not evolve in time. In order to have a non-trivial limit, time needs to be speeded up and for time scales smaller than tN 4=3, there is still no temporal evolution. As a consequence, the current across a characteristic vanishes up to this longer time scale.