Non-equilibrium Fluctuations of Interacting Particle Systems

Abstract

We obtain the large scale limit of the fluctuations around its hydrodynamic limit of the density of particles of a weakly asymmetric exclusion process in dimension d ≤ 3. The proof is based upon a sharp estimate on the relative entropy of the law of the process with respect to product reference measures associated to the hydrodynamic limit profile, which holds in any dimension and is of independent interest. As a corollary of this entropy estimate, we obtain some quantitative bounds on the speed of convergence of the aforementioned hydrodynamic limit.

Hydrodynamic limit of Exclusion Processes with slow boundaries on hypercubes

“…we can remove the absolute value symbol inside the exponential. By Hoeffding's lemma(see Lemma F.9 in [3], for instance), since η takes value in [0, 1] and ρ ∈ (ε 0 , 1 − ε 0 ) by Corollary 5.2, under the product measure ν n t , η t (x) − ρ(t, m n (x)) is subgaussian of order 1 4 . By Lemma F.12 in [3],…”

“…We do not claim the bound in (2.8) is sharp when the initial bound in (2.7) is small enough. It should be possible to apply the method introduced in the remarkable work of M. Jara and O.Menezes [3] to improve the bound. Nevertheless, the estimate of the relative entropy in the lemma is already good enough to derive the hydrodynamic limit of our model.…”

Sharp Convergence to Equilibrium for the SSEP with Reservoirs

“…A version of this estimate was obtained in [9] in the context of non-equilibrium fluctuations from the hydrodynamic limit. Our novelty is the exponential decay as a function of t, which in particular allows to use this estimate over divergent time windows.…”

“…If we assume that the law of the exclusion process is close to this product measure, then the log-Sobolev inequality would be more efficient. This strategy can be implemented as in [9], and it will give a sharp estimate for the relative entropy between the law of the process and this family of product measures. After that, the computation of the distance D n (t; ν n 0 ) is reduced to the computation of ν n t − νn ρ TV .…”

On the hydrodynamics of active matter models on a lattice

“…Some progress has been achieved making use of properties like duality or integrability, however such properties are strongly model-dependent and do not provide robust general methods to derive non-equilibrium fluctuations. Significant progress has been achieved on this front in [15], where the non-equilibrium fluctuations for the Weakly Asymmetric Simple Exclusion Process (WASEP) are obtained in dimension d ≤ 3, adapting Yau's so-called relative entropy method [26] together with refined entropy estimates.…”