Non-equilibrium Statistical Mechanics of Anharmonic Crystals with Self-consistent Stochastic Reservoirs

TL;DR

This paper develops a formalism to analyze correlation functions in anharmonic crystals with stochastic reservoirs, demonstrating Fourier's law and deriving thermal conductivity in a one-dimensional weakly coupled system.

Contribution

It introduces an integral formalism for correlation functions and provides a perturbative analysis showing Fourier's law and thermal conductivity in anharmonic crystals.

Findings

Correlation functions formalism for anharmonic crystals

Fourier's law holds in weakly coupled 1D systems

Explicit expression for thermal conductivity

Abstract

We consider a d-dimensional crystal with an arbitrary harmonic interaction and an anharmonic on-site potential, with stochastic Langevin heat bath at each site. We develop an integral formalism for the correlation functions that is suitable for the study of their relaxation (time decay) as well as their behavior in space. Furthermore, in a perturbative analysis, for the one-dimensional system with weak coupling between the sites and small quartic anharmonicity, we investigate the steady state and show that the Fourier's law holds. We also obtain an expression for the thermal conductivity (for arbitrary next-neighbor interactions) and give the temperature profile in the steady state.