Conservation of Local Equilibrium for Attractive Systems

In Chapter 1 we introduced the concept of local equilibrium and proved the conservation of local equilibrium for a superposition of independent random walks. Then, from Chapter 4 to Chapter 8, we proved a weaker version of local equilibrium for a large class of interacting particle systems: we showed that the empirical measure π _t N converges in probability to an absolutely continuous measure whose density is the solution of some partial differential equation. The purpose of this chapter is to to show that in the case of attractive processes, the conservation of local equilibrium may be deduced from a law of large numbers for local fields, i.e., from the convergence in probability of the averages $${N^{ - d}}\sum\limits_x {H(x/N){\tau _x}} \Psi ({\eta _t})to\int_{{T^d}} {H(u)\tilde \Psi } (\rho (t,u))du$$ for every t ≥ 0, every continuous function H and every bounded cylinder function ψ. Here ρ(t, u) is the solution of the hydrodynamic equation. This statement is slightly stronger than the convergence of the empirical measures since it involves all local fields.