Superdiffusion of Energy in a Chain of Harmonic Oscillators with Noise

Abstract

We consider a one dimensional infinite chain of harmonic oscillators whose dynamics is perturbed by a stochastic term conserving energy and momentum. We prove that in the unpinned case the macroscopic evolution of the energy converges to a fractional diffusion governed by −|∆| 3/4 . For a pinned system we prove that its energy evolves diffusively, generalizing some results of [4].

Diffusive Propagation of Energy in a Non-acoustic Chain

“…It turns out that this superdiffusive behavior of the energy is governed by a fractional laplacian heat equation. This picture can be mathematically rigorously proven in the case of a harmonic chain perturbed by a local random exchange of momentum, see [8,9]. In particular, it has been shown in [9] that in the models driven by the tension, there is a separation of the time evolution scales between the long modes (that evolve on a hyperbolic time scale) and the thermal short modes that evolve in a longer superdiffusive scale.…”

“…Instead, we develop a method already used in [8], based on Wigner distributions for the energy of the acoustic chain. Thanks to the energy conservation property of the dynamics we can easily conclude (see Section 5.4) that the Wigner distributions form a compact family of elements in the weak topology of an appropriate Banach space.…”

Fluctuating Hydrodynamics Approach to Equilibrium Time Correlations for Anharmonic Chains

“…For them the couplings can be more easily adjusted than for anharmonic chains, which offers the possibility to test the dynamical phase diagram. Also anharmonic chains with a stochastic collision mechanism, respecting the conservation laws, have been studied in considerable detail [67,68,42]. …”

Anomalous diffusion phenomena: A kinetic approach

“…See in particular Basile-Bernardin-Olla [3,4], Basile-Olla-Spohn [5], Jara-Komorowski-Olla [18,19], Olla [30], Bernardin-Gonçalves-Jara [8].…”