Energy Diffusion and Superdiffusion in Oscillators Lattice Networks

Abstract

Abstract. I review here some recent result on the thermal conductivity of chains of oscillators whose hamiltonian dynamics is perturbed by a noise conserving energy and momentum.

Diffusive behavior from a quantum master equation

“…The questions and the applied techniques are however quite similar to what has been studied starting from classical Boltzmann-type equations, see e.g. [3,20,23]. A quantum example that is very related to ours is in [15].…”

“…We explain that analogy here. Among other things it relates our results to the recent interest and work on diffusive behavior in systems of coupled oscillators where energy transport can be understood as a wave scattered by anharmonicities, [20].…”

Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law

“…The rigorous proof of such a statement was given in [16,17] in the case of the slab geometry and provided that θ + − θ − is sufficiently small (uniformly in the Knudsen number K n ). This is a special case of a problem which has received recently a large attention in the Statistical Mechanics community, the derivation of the Fourier law from the microscopic deterministic evolutions ruled by the Newton or Schrödinger equation or from stochastic models [8,30,6,1]. The aim of this paper is to analyze the thermal conduction phenomena in the kinetic regime.…”

Anomalous Energy Transport in FPU- $$\beta $$ β Chain

“…In this approach, the Hamiltonian dynamics of the microscopic system is described by an harmonic potential but it is perturbed by a stochastic noise conserving momentum and energy. In the weak noise limit, one also recovers fractional diffusion phenomena (see [4], [5], [17], [6] and the review paper [29]). More recently, a similar approach has been developed without the weak noise assumption to derive fractional heat equations of order 3/4 (see [16] and [9]).…”