Homogenization results for a linear dynamics in random Glauber type environment

Abstract

We consider an energy conserving linear dynamics that we perturb by a Glauber dynamics with random site dependent intensity. We prove hydrodynamic limits for this non-reversible system in random media. The diffusion coefficient turns out to depend on the random field only by its statistics. The diffusion coefficient defined through the Green-Kubo formula is also studied and its convergence to some homogenized diffusion coefficient is proved

Hydrodynamic limit for a disordered harmonic chain

“…Averaging lemma for the field N . The method of the corrected empirical measure is an elegant method to deal with the randomness on the masses in deriving the hydrodynamic limit for N and N [12,14,6]. However, in our case, it seems more convenient to use the following lemma:…”

“…On the one hand, deriving the macroscopic evolution of the fields r x and p x becomes less obvious because some homogenization over the masses is required. This difficulty can be solved by the elegant method of the "corrected empirical measure", see [12,14,6] (though we will actually solve it another way). On the other hand, and this is the main point in considering random masses, the evolution of the energy e x is now much better approximated by Euler equation.…”