Toward the Fourier law for a weakly interacting anharmonic crystal

Carlangelo Liverani (1) , Stefano Olla (2, 3)

1 DIPMAT - Dipartimento di Matematica [Roma II]
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 MICMAC - Methods and engineering of multiscale computing from atom to continuum

For a system of weakly interacting anharmonic oscillators, perturbed by an energy preserving stochastic dynamics, we prove an autonomous (stochastic) evolution for the energies at large time scale (with respect to the coupling parameter). It turn out that this macroscopic evolution is given by the so called conservative (non-gradient) Ginzburg-Landau system of stochastic differential equations. The proof exploits hypocoercivity and hypoellipticity properties of the uncoupled dynamics.

Mots clés: Weak coupling; scaling limits; hypoellipticity; hypocoercivity; Ginzburg-Landau dynamics; heat equation

Domaines: Probabilités [math.PR]

Carlangelo Liverani, Stefano Olla. Toward the Fourier law for a weakly interacting anharmonic crystal. Journal of the American Mathematical Society, 2011, 25 (2012), pp.555-583. ⟨10.1090/S0894-0347-2011-00724-8⟩. ⟨hal-00492016⟩