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 $\rho(t,\cdot)$. This equation is a hyperbolic conservation law of type $\partial_{t}\rho(t,u)+\nabla F(\rho(t,u))=0$, 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 on 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.