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Second Order Boltzmann–Gibbs Principle for Polynomial Functions and Applications

2016; Springer Science+Business Media; Volume: 166; Issue: 1 Linguagem: Inglês

10.1007/s10955-016-1686-6

1572-9613

Patrícia Gonçalves, Milton Jara, Marielle Simon,

Theoretical and Computational Physics

In this paper we give a new proof of the second order Boltzmann–Gibbs principle introduced in Gonçalves and Jara (Arch Ration Mech Anal 212(2):597–644, 2014). 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 Gubinelli and Jara (SPDEs Anal Comput (1):325–350, 2013) and Gubinelli and Perkowski ( Arxiv:1508.07764 , 2015), 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.