Hydrodynamic Limit of Asymmetric Attractive Processes

Abstract

We examine in this chapter an alternative method to prove the hydrodynamic behavior of asymmetric interacting particle systems. This approach has the advantage over the one presented in Chapter 6 that it does not require the solution of the hydrodynamic equation to be smooth. On the other hand its main inconvenience is that it assumes the process to be attractive to permit the use of coupling arguments and the initial state to be a product measure. To illustrate this approach we consider an asymmetric attractive zero range process on the discrete d—dimensional torus T ^d _N . The generator of this Markov process, denoted by L _N , is given by

$$\left( {{L_N}f} \right)\left( \eta \right)\, = \,\sum\limits_{\begin{array}{*{20}{c}}{x \in {\text{T}}_N^d} \\{y \in {\mathbb{Z}^d}}\end{array}} {p\left( y \right)g\left( {\eta \left( x \right)} \right)\left[ {f\left( {{\eta ^{x,x + y}}} \right)\, - \,f\left( \eta \right)} \right]\,,} $$

(0.1)

where p(·) is a finite range irreducible transition probability on ℤ^d. Irreducible means here that for every z in ℤ^d, there exists a positive integer m and a sequence 0 = x ₀, x ₁,…, x _m , = z such that p(x _i+1 − x _i )+p(x _i − x _i+1) > 0 for 0 ≤ i ≤ m − 1. Throughout this chapter we assume the process to be attractive. We have seen in Chapter 2 that this hypothesis corresponds to assume that the rate at which a particle leaves a site is a non decreasing function of the total number of particles at that site:

(H) g(·) is a non decreasing function.