Second Order Boltzmann–Gibbs Principle for Polynomial Functions and Applications

Abstract

In this paper we give a new proof of the second order Boltzmann-Gibbs principle introduced in [6]. The proof does not impose the knowledge on the spectral gap inequality for the underlying model and it relies on a proper decomposition of the antisymmetric part of the current of the system in terms of polynomial functions. In addition, we fully derive the convergence of the equilibrium fluctuations towards 1) a trivial process in case of super-diffusive systems, 2) an Ornstein-Uhlenbeck process or the unique energy solution of the stochastic Burgers equation, as defined in [8,9], in case of weakly asymmetric diffusive systems. Examples and applications are presented for weakly and partial asymmetric exclusion processes, weakly asymmetric speed change exclusion processes and hamiltonian systems with exponential interactions.

Energy solutions of KPZ are unique

“…Following [GJ14] it has been shown for a variety of models that their fluctuations subsequentially converge to energy solutions of the KPZ equation or the SBE, for example for zero range processes and kinetically constrained exclusion processes in [GJS15], various exclusion processes in [GS15,FGS16,BGS16,GJ16], interacting Brownian motions in [DGP16], and Hairer-Quastel type SPDEs in [GP16]. This is coherent with the conjecture that the SBE/KPZ equation describes the universal behavior of a wide class of conservative dynamics or interface growth models in the particular limit where the asymmetry is "small" (depending on the spatial scale), the so called weak KPZ universality conjecture, see [Cor12,Qua14,QS15,Spo16].…”

Stationary directed polymers and energy solutions of the Burgers equation

“…Uniqueness for this weak formulation on the whole line was proved in [23]. Since [17], this approach was successfully applied to show the convergence of many discrete models to the KPZ/ Burgers equation [18,19,14,32]. One substantial advantage is that it requires very weak quantitative estimates.…”

Scaling of the Sasamoto-Spohn model in equilibrium

“…At the technical level, our approach relies on the techniques of [9] and avoids the use of any spectral gap estimate. The core of the proof consists in deriving certain dynamical estimates among which the so-called second order Boltzmann-Gibbs principle plays a major role.…”