The hydrodynamical limit of asymmetric simple exclusion processes is given by an inviscid Burgers equation and its next-order correction is given by the viscous Burgers equation. The diffusivity can be characterized by an abstract formulation in a Hilbert space with the inverse of the diffusivity characterized by a variational formula. Alternatively, it can be described by the Green-Kubo formula. We give arguments that these two formulations are equivalent. We also derive two other variational formulas, one for the inverse of the diffusivity and one for the diffusivity itself, characterizing diffusivity as a supremum and as an infimum. These two formulas also provide an analytic criterion for deciding whether the diffusivity as defined by the linear response theory is symmetric. Furthermore, we prove the continuity of the diffusivity and a few other relations concerning diffusivity and solutions of the Euler-Lagrange equations of these variational problems.
"Some properties of the diffusion coefficient for asymmetric simple exclusion processes." Ann. Probab. 24 (4) 1779 - 1808, October 1996. https://doi.org/10.1214/aop/1041903206