Dynamical contribution to the heat conductivity in stochastic energy exchanges of locally confined gases

TL;DR

This paper computes the heat conductivity in a stochastic model of energy exchange in confined gases, revealing a small negative dynamic correction to static conductivity, supported by improved Monte Carlo simulations.

Contribution

It provides explicit bounds and a precise estimate of the dynamic correction to heat conductivity in a stochastic energy exchange model.

Findings

Dynamic contribution is approximately -0.000373 in dimensionless units.

Explicit upper bounds on conductivity show rapid convergence.

Monte Carlo simulations confirm theoretical predictions.

Abstract

We present a systematic computation of the heat conductivity of the Markov jump process modeling the energy exchanges in an array of locally confined hard spheres at the conduction threshold. Based on a variational formula [Sasada M. 2016, {\it Thermal conductivity for stochastic energy exchange models}, arXiv:1611.08866], explicit upper bounds on the conductivity are derived, which exhibit a rapid power-law convergence towards an asymptotic value. We thereby conclude that the ratio of the heat conductivity to the energy exchange frequency deviates from its static contribution by a small negative correction, its dynamic contribution, evaluated to be $-0.000\,373$ in dimensionless units. This prediction is corroborated by kinetic Monte Carlo simulations which were substantially improved compared to earlier results.