5/6-Superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field

TL;DR

This paper studies energy transport in a 1D chain of charged oscillators under magnetic field and stochastic perturbation, showing it follows a fractional diffusion equation with exponent 5/6.

Contribution

It proves the energy distribution evolves according to a phonon Boltzmann equation and its scaled limit is a fractional diffusion equation with exponent 5/6.

Findings

Energy density follows a linear phonon Boltzmann equation.

Scaled solutions converge to a fractional diffusion equation with exponent 5/6.

Demonstrates superdiffusive energy transport in the system.

Abstract

We consider a one-dimensional infinite chain of coupled charged har- monic oscillators in a magnetic field with a small stochastic perturbation of order $\epsilon$. We prove that for a space-time scale of order $\epsilon$^{-1} the density of energy distribution (Wigner distribution) evolves according to a linear phonon Boltzmann equation. We also prove that an appropriately scaled limit of solutions of the lin- ear phonon Boltzmann equation is a solution of the fractional diffusion equation with exponent 5/6.