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

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 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.