Superdiffusion transition for a phonon Boltzmann equation

Abstract

We consider an infinite harmonic chain of charged particles submitted to the action of a magnetic field of intensity $B$ and subject to the action of a stochastic noise conserving the energy. In arXiv:0809.0177 and arXiv:1402.2988 it has been proved that if $B=0$ the transport of energy is described by a $3/4$-fractional diffusion while it has been proved in arXiv:1808.01040 that if $B\ne 0$ it is described by a $5/6$-fractional diffusion. In arXiv:0809.0177 and in arXiv:1808.01040 the authors used a two step argument, i.e. they first proved that the kinetic limit of the Wigner distribution is the solution of a phonon Boltzmann equation and then proved that this solution converges to the solution of a fractional diffusion equation with exponent $3/4$ if $B = 0$ (see arXiv:0809.0177) and exponent $5/6$ if $B\ne 0$ (see arXiv:1808.01040). In this paper we quantify the intensity of the magnetic field required to switch from one macroscopic regime to the other one from the phonon Boltzmann equation. We also describe the transition mechanism to cross the two different phases.