Fluctuations for Some Nonstationary Interacting Particle Systems via Boltzmann–Gibbs Principle

Abstract

Abstract Conjecture II.3.6 of Spohn in [47] and Lecture 7 of Jensen–Yau in [35] 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 [26]. In this paper, we develop a general method for deriving the so-called Boltzmann–Gibbs principle for a general family of nonintegrable and nonstationary 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 work [6, 8, 24, 34], all of which rely on global and nonuniversal 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 nonstationary and nonintegrable exclusion processes with an environment-dependent asymmetry. This establishes a first result to Big Picture Question 1.6 in [54] for nonstationary and nonintegrable ‘speed-change’ models that have also been of interest beyond KPZ [18, 22, 23, 38].