Toward the Fourier law for a weakly interacting anharmonic crystal (Q2879893)

The paper deals with the problem of deriving the heat equation and Fourier's law for the macroscopic evolution of the energy starting from a microscopic dynamics of interacting atoms. The authors develop a weak coupling approach to the problem of energy diffusion that allows them to separate the time limit from the space one. They consider a finite system of anharmonic oscillators, having at least two degrees of freedom, whose Hamiltonian dynamics is perturbed by a noise conserving the kinetic energy of each oscillator. The noise could be thought of as modeling some chaotic internal degree of freedom that drives each atom towards (microcanonical) equilibrium. The oscillators are weakly coupled with a small parameter, consequently the exchange of energy between them is given by the currents multiplied by the parameter. The authors prove that at a large time scale, with respect to the coupling parameter, the macroscopic evolution is given by the so-called conservative (nongradient) Ginzburg-Landau model. Doing so, the authors accomplish the first step towards the derivation of the heat equation. The second would be that of extending some recently developed techniques to prove that under space-time diffusive rescaling, the energy evolves following a nonlinear heat equation. The proof of the first step exploits hypocoercivity and hypoellipticity properties of the uncoupled dynamics.