Convergence of a Kinetic Equation to a Fractional Diffusion Equation

The understanding of thermal conductance in both classical and quantum mechanical systems is one of the fundamental problems of non-equilibrium statistical mechanics. A particular aspect that has attracted much interest is the observation that autonomous translation invariant systems in dimensions one and two exhibit anomalously large conductivity. The canonical example here is a chain of anharmonic oscillators introduced by Fermi-Pasta-Ulam (FPU)[13], for which numerical evidence shows a super-diffusive spreading of energy (see [19] for a general review). However, the rigorous analysis of energy transport mechanism presents serious mathematical difficulties and few results are obtained starting from microscopic dynamics. The canonical approach to this problem, starting with the pioneering work of Peierls [23] for the case of weak non-linearity, is to derive a Boltzmann-type equation that will describe the energy transport in a kinetic limit. Recently, this approach was carried out rigorously for weakly anharmonic FPU chains [24, 1, 21]. A linear Boltzmann equation was derived in [20] for the harmonic chain with random masses. The same linear Boltzmann equation appears also as limit of a random Schrödinger equation (see for example [11], [10], [25], [2]). In [3] a kinetic limit was performed for a system of harmonic oscillators perturbed by a conservative stochastic noise and the following linear Boltzmann equation is deduced for the the energy density distribution of the normal modes, or phonons, characterized by a wave-number k ∈ [−1/2, 1/2]: