Toward the Fourier law for a weakly interacting anharmonic crystal

Toward the Fourier Law for a Weakly Interacting Anharmonic Crystal Carlangelo Liverani and Stefano Olla

Anomalous energy transport in FPU-β chain

This paper is devoted to the derivation of a macroscopic diffusion equation (Fourier’s law) describing the transport of energy in an anharmonic chain. More precisely, we study here the so-called FPU-β chain, which is a very simple model for a one-dimensional crystal in which atoms are coupled to their nearest neighbors by a harmonic potential, weakly perturbed by a quartic potential. The starti...

On the derivation of Fourier’s law for coupled anharmonic oscillators

We study the Hamiltonian system made of weakly coupled anharmonic oscillators arranged on a three dimensional lattice Z2N × Z2, and subjected to a stochastic forcing mimicking heat baths of temperatures T1 and T2 on the hyperplanes at 0 and N . We introduce a truncation of the Hopf equations describing the stationary state of the system which leads to a nonlinear equation for the two-point stat...

Normal heat conductivity in a periodic chain of anharmonic oscillators

We consider a periodic chain of coupled and pinned anharmonic oscillators subject to a nonequilibrium random forcing. Assuming that the six-point correlation functions of the stationary state factorize in terms of the two-point correlation functions, we show that the heat current going through the system is consistent with Fourier law and that the conductivity behaves as κ ∼ 1 λ2T2 where λ is t...

Approach to equilibrium for the phonon Boltzmann equation

We study the asymptotics of solutions of the Boltzmann equation describing the kinetic limit of a lattice of classical interacting anharmonic oscillators. We prove that, if the initial condition is a small perturbation of an equilibrium state, and vanishes at infinity, the dynamics tends diffusively to equilibrium. The solution is the sum of a local equilibrium state, associated to conserved qu...