Fluctuations in the Weakly Asymmetric Exclusion Process with Open Boundary Conditions

Abstract

We investigate the fluctuations around the average density profile in the weakly asymmetric exclusion process with open boundaries in the steady state. We show that these fluctuations are given, in the macroscopic limit, by a centered Gaussian field and we compute explicitly its covariance function. We use two approaches. The first method is dynamical and based on fluctuations around the hydrodynamic limit. We prove that the density fluctuations evolve macroscopically according to an autonomous stochastic equation, and we search for the stationary distribution of this evolution. The second approach, which is based on a representation of the steady state as a sum over paths, allows one to write the density fluctuations in the steady state as a sum over two independent processes, one of which is the derivative of a Brownian motion, the other one being related to a random path in a potential.

Open ASEP in the Weakly Asymmetric Regime

“…In particular, for system size N , we scale q D 1=2 1=. When started out of equilibrium, [41] and [58] prove convergence in the (respectively) weakly asymmetric and symmetric cases to generalized Ornstein-Uhlenbeck processes (see also [22] for an earlier physics prediction of these results). In both cases, the boundary parameters˛;ˇ; ; ı are fixed.…”

“…In the stationary state, the (single time) spatial fluctuations for the weakly asymmetric system are Gaussian [22], while those of the totally asymmetric case are known and non-Gaussian [23]. In both cases, the stationary state fluctuations are given in terms of the sum of two processes; however, there seems to be no way to scale the weakly asymmetric limit process to the totally asymmetric limit process.…”

A Diffusive System Driven by a Battery or by a Smoothly Varying Field

“…The steady state correlations can be predicted from a macroscopic approach based on "fluctuating hydrodynamics" [Sp1,DELO,OS]. Alternatively, one can derive the density large deviation functional for the steady state and then recover the correlations by expanding the functional near the steady state [BDGJL1,BDGJL4,D1,BDLW].…”

On the scaling limits of weakly asymmetric bridges

“…De Masi, Presutti and Scacciatelli [11] and Dittrich and Gärtner [14] prove convergence of the fluctuations of the weakly asymmetric simple exclusion process (WASEP) on the full line when the asymmetry scales like ǫ and time is sped up by ǫ −2 : the limiting fluctuations are then given by an equation similar to (7). Let us also cite the work of Derrida, Enaud, Landim and Olla [13] on a related model interacting with reservoirs. An important ingredient in the proof of this theorem is the Boltzmann-Gibbs principle, which is adapted to the present setting in Proposition 18.…”