Equilibrium Fluctuations for the Totally Asymmetric Zero-Range Process

Abstract

Abstract. We prove a Central Limit Theorem for the empirical measure in the one-dimensional Totally Asymmetric Zero-Range Process in the hyperbolic scaling N , starting from the equilibrium measure νρ. We also show that when taking the direction of the characteristics, the limit density fluctuation field does not evolve in time until N 4/3 , which implies the current across the characteristics to vanish in this longer time scale.

A hyperbolic conservation law and particle systems

“…The results we are going to present here hold for more general jump rates but in order to keep the presentation simple and clear, we opt to describe the results for this specific processes. For details on more general jump rates we refer the reader to [5] and [6]. Now, we define precisely these processes by means of their generators.…”

“…For a proof of last result for the TASEP or TAZRP we refer the interested reader to [5] or [6], respectively. In these two cases, the limit density fluctuation field at time t is a translation of the initial one, since dY t =W (σ)∇Y t dt i.e.…”

“…and π σ . Using the martingale approach one can derive the Boltzmann-Gibbs principle by using a multi-scale argument, see [5] for the TASEP and [6] for the TASEP. As a consequence, one can close the integral part of the martingale associated to (6.2) and state that: Theorem 6.2.…”

“…for the current over a fixed bond can be obtained. We state the result here, but for a proof for the TASEP we refer the reader to Theorem 4.2 of [5], while for the TAZRP we refer to section 3.2 of [6].…”

Convergence to the stochastic Burgers equation from a degenerate microscopic dynamics

“…Technically, this principle is obtained by a multi-scale argument which consists in replacing a local function by another function whose support increases at each step and whose variance decreases in order to make the errors eventually vanish. This argument had already been used in [5,6] to show the triviality of the fluctuations of the asymmetric simple exclusion and the asymmetric zero-range process. The BGP2 is shown to hold for a large class of weakly asymmetric exclusion processes in [7,9], and for zero-range, non-degenerate kinetically constrained models and other models starting from a state close to equilibrium in [9].…”