Diffusive Limit of the Asymmetric Simple Exclusion: The Navier-Stokes Correction

Abstract

We consider the simple exclusion process (SEP) on a cubic sublattice, Λ _L ⊂ Z ^d, of size 2L + 1, with periodic boundary conditions. The dynamics is as follows. Let e denote one of the 2d possible directions in Z ^d. A particle in the site x, independently of the others, waits for an exponential time and jumps, with probability proportional to p _e ≥ 0, to the site x + e, if it is empty; otherwise a particle stays in x and the process starts again. We denote by η _x (τ) = 0,1 the number of particles in the site x at time τ and by L the generator of the process: L f = ∑_b L _b f, the sum running on the set of all oriented bonds b = (x,y) in Z ^d such that y − x = e and

$$ {{L}_{b}}f = {{p}_{e}}{{\eta }_{x}}\left[ {f\left( {{{\eta }^{b}}} \right) - f\left( \eta \right)} \right] $$

((1))

with

$$ ({{n}^{b}})z = ({{n}^{{x,y}}})z = \left\{ {\begin{array}{*{20}{c}} {{{\eta }_{y}},} & {if} & {z = x} \\ {{{\eta }_{x}},} & {if} & {z = y} \\ {{{\eta }_{z}},} & {if} & {otherwise.} \\ \end{array} } \right. $$

It is convenient to choose the normalization p _−e + p _e = 2 for all e.