Hydrodynamic Limit of Reversible Nongradient Systems

Abstract

We investigate in this chapter the hydrodynamic behavior of reversible nongradient systems. To fix ideas we consider one of the simplest examples, the so called symmetric generalized exclusion process. This is the Markov process introduced in section 2.4 that describes the evolution of particles on a lattice with an exclusion rule that allows at most k particles per site. Here K is a fixed positive integer greater or equal than 2. The generator of this Markov process acts on cylinder functions as

$$\left( {{L_N}f} \right)\left( \eta \right) = \left( {1/2} \right)\sum\limits_{\mathop {x,y \in {T^d}}\limits_{|x - y| = 1} } {r\left( {\eta \left( x \right),\eta \left( y \right)} \right)} \left[ {f\left( {{\eta ^{x,y}}} \right) - f\left( \eta \right)} \right],$$

(0.1)

where r(a, b) = 1{a > 0, b < k} and η ^{x, y} is the configuration obtained from η moving a particle from x to y.