Thermal Conductivity for a Momentum Conserving Model

TL;DR

This paper studies a momentum-conserving oscillator model and shows that its thermal conductivity diverges in one and two dimensions but remains finite in three dimensions, with explicit calculations and bounds.

Contribution

It introduces a model with momentum and energy conservation, analyzing its thermal conductivity behavior across different dimensions and potentials.

Findings

Thermal conductivity diverges in 1D and 2D.

Finite conductivity in 3D and pinned cases.

Explicit formulas and bounds for conductivity.

Abstract

We introduce a model whose thermal conductivity diverges in dimension 1 and 2, while it remains finite in dimension 3. We consider a system of oscillators perturbed by a stochastic dynamics conserving momentum and energy. We compute thermal conductivity via Green-Kubo formula. In the harmonic case we compute the current-current time correlation function, that decay like $t^{-d/2}$ in the unpinned case and like $t^{-d/2-1}$ if a on-site harmonic potential is present. This implies a finite conductivity in $d\ge 3$ or in pinned cases, and we compute it explicitly. For general anharmonic strictly convex interactions we prove some upper bounds for the conductivity that behave qualitatively as in the harmonic cases.