On the Green–Kubo formula and the gradient condition on currents

Abstract

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.