Heat flow in anharmonic crystals with internal and external stochastic baths: A convergent polymer expansion for a model with discrete time and long range interparticle interaction

TL;DR

This paper develops a rigorous mathematical framework using polymer expansion to analyze heat flow in anharmonic crystals with long-range interactions and stochastic baths, providing insights into thermal conductivity.

Contribution

It introduces a convergent polymer expansion method for a model with discrete time, long-range interactions, and stochastic baths, enabling rigorous analysis of heat flow in anharmonic crystals.

Findings

Proves convergence of the polymer expansion uniformly in volume.

Shows two-point correlation decay matches interparticle interaction decay.

Supports perturbative analysis of heat flow and thermal conductivity.

Abstract

We investigate a chain of oscillators with anharmonic on-site potentials, with long range interparticle interactions, and coupled both to external and internal stochastic thermal reservoirs of Ornstein-Uhlenbeck type. We develop an integral representation, a la Feynman-Kac, for the correlations and the heat current. We assume the approximation of discrete times in the integral formalism (together with a simplification in a subdominant part of the harmonic interaction) in order to develop a suitable polymer expansion for the model. In the regime of strong anharmonicity, strong harmonic pinning, and for the interparticle interaction with integrable polynomial decay, we prove the convergence of the polymer expansion uniformly in volume (number of sites and time). We also show that the two-point correlation decays in space such as the interparticle interaction. The existence of a convergent polymer expansion is of practical interest: it establishes a rigorous support for a perturbative analysis of the heat flow problem and for the computation of the thermal conductivity in related anharmonic crystals, including those with inhomogeneous potentials and long range interparticle interactions.