ON THE GREEN-KUBO FORMULA AND THE GRADIENT CONDITION ON CURRENTS
成果类型：
Article
署名作者：
Sasada, Makiko
署名单位：
University of Tokyo
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
DOI：
10.1214/17-AAP1369
发表日期：
2018
页码：
2727-2739
关键词：
摘要：
In the diffusive hydrodynamic limit for a symmetric interacting particle system (such as the exclusion process, the zero range process, the stochastic Ginzburg-Landau model, the energy exchange model), a possibly nonlinear diffusion equation is derived as the hydrodynamic equation. The bulk diffusion coefficient of the limiting equation is given by the Green-Kubo formula and it can be characterized by a variational formula. In the case the system satisfies the gradient condition, the variational problem is explicitly solved and the diffusion coefficient is given from the Green-Kubo formula through a static average only. In other words, the contribution of the dynamical part of the Green-Kubo formula is 0. In this paper, we consider the converse, namely if the contribution of the dynamical part of the Green-Kubo formula is 0, does it imply the system satisfies the gradient condition or not. We show that if the equilibrium measure mu is product and L-2 space of its single site marginal is separable, then the converse also holds. The result gives a new physical interpretation of the gradient condition. As an application of the result, we consider a class of stochastic models for energy transport studied by Gaspard and Gilbert in [J. Stat. Mech. Theory Exp. 2008 (2008) P11021; J. Stat. Mech. Theory Exp. 2009 (2009) P08020], where the exact problem is discussed for this specific model.