Fluctuations for some non-stationary interacting particle systems via Boltzmann-Gibbs Principle

Abstract

Conjecture II.3.6 of Spohn in [Spohn '91] and Lecture 7 of Jensen-Yau in [Jensen-Yau '99] ask for a general derivation of universal fluctuations of hydrodynamic limits in large-scale stochastic interacting particle systems. However, the past few decades have witnessed only minimal progress according to [Goncalves-Landim-Milanes '17]. In this paper, we develop a general method for deriving the so-called Boltzmann-Gibbs principle for a general family of non-integrable and non-stationary interacting particle systems, thereby responding to Spohn and Jensen-Yau. Most importantly, our method depends mostly on local and dynamical, and thus more general/universal, features of the model. This contrasts with previous works, which rely on global and non-universal assumptions on invariant measures or initial measures of the model. As a concrete application of the method, we derive the KPZ equation as a large-scale limit of the height functions for a family of non-stationary and non-integrable exclusion processes with an environment-dependent asymmetry. This establishes a first result to Big Picture Question 1.6 in [AimPLKPZ] for non-stationary and non-integrable "speed-change" models that have also been of interest beyond KPZ [De Masi-Presutti-Spohn-Wick '86, Funaki '18, Funaki-Handa-Uchiyama '91, Komoriya '98].