  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.
