Temperature profile and boundary conditions in an anomalous heat transport model

TL;DR

This paper develops a framework to analyze how boundary conditions affect temperature profiles in models of anomalous heat conduction, applying it to harmonic chains and connecting it to Levy walker systems.

Contribution

It introduces a method to compute temperature profiles and conductivity exponents considering boundary effects in anomalous heat transport models.

Findings

Temperature profiles are singular at boundaries with exponents depending on boundary coupling.

The framework links harmonic chains to Levy walker systems with boundary-dependent reflection coefficients.

Anomalous conductivity exponent and integral equations for temperature are derived.

Abstract

A framework for studying the effect of the coupling to the heat bath in models exhibiting anomalous heat conduction is described. The framework is applied to the harmonic chain with momentum exchange model where the non-trivial temperature profile is calculated. In this approach one first uses the hydrodynamic (HD) equations to calculate the equilibrium current-current correlation function in large but finite chains, explicitly taking into account the BCs resulting from the coupling to the heat reservoirs. Making use of a linear response relation, the anomalous conductivity exponent $\alpha$ and an integral equation for the temperature profile are obtained. The temperature profile is found to be singular at the boundaries with an exponent which varies continuously with the coupling to the heat reservoirs expressed by the BCs. In addition, the relation between the harmonic chain and a system of noninteracting L{\'e}vy walkers is made explicit, where different BCs of the chain correspond to different reflection coefficients of the L{\'e}vy particles.