C. Landim, S. Olla and H.-T. Yau
Some properties of the diffusion coefficient
for asymmetric simple exclusion processes
(276K, ps)
ABSTRACT. Hydrodynamical limit of asymmetric simple exclusion
processes is given by a viscousless Burgers equation [R] and its next order correction
is given by the viscous Burgers equation [LOY]. The diffusivity can be characterized
by an abstract formulation in a Hilbert space with the inverse of the diffusivity
characterized by a variational formula ([EMY], [EMYp]). Alternatively
it can be described by the Green-Kubo formula [KLS].
We give arguments that these
two formulations are equivalent. We also derive
two other variational formulae,
one for the inverse of the diffusivity and
one for the diffusivity itself, characterizing diffusivity as a supremum
and as an infimum. These two formulae
also provide an analytic criterion whether
the diffusivity as defined by the linear
response theory is symmetric. Furthermore,
we prove continuity of the diffusivity and a few other relations concerning diffusivity
and solutions of the Euler-Lagrange equation of these variational problems.