The gradient condition and the contribution of the dynamical part of Green-Kubo formula to the diffusion coefficient

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 non-linear diffusion equation is derived as the hydrodynamic equation. The bulk diffusion coefficient of the limiting equation is given by 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 Green-Kubo formula is 0. In this paper, we consider the converse, namely if the contribution of the dynamical part of 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. As an application of the result, we consider a class of stochastic models for energy transport studied by Gaspard and Gilbert in [1, 2], where the exact problem is discussed for this specific model.