In Chapter 1 we introduced in the context of interacting particle systems the physical concepts of local equilibrium and conservation of local equilibrium and we proved the persistence of local equilibrium in a model where particles evolve independently. Consider a particle system η t evolving on the torus T N d and possessing a family {υ α N , α ≥ 0} of product invariant measures indexed by the density. Fix a profile ρ 0: T d → ℝ+ and assume that the process η t has a hydrodynamic behavior described by the solution ρ(t, u) of some partial differential equation with initial condition ρ 0. Denote by μ N a sequence of initial states associated to the profile ρ 0 and by μ t N the state at the macroscopic time t of the process that started from μ N . The conservation of local equilibrium states that μ t N should be close to the product measure υ ρ(t,·) N with slowly varying parameter associated to ρ(t,·).