Heat Conduction and Entropy Production in Anharmonic Crystals with Self-Consistent Stochastic Reservoirs

TL;DR

This paper studies heat conduction and entropy production in anharmonic crystals with self-consistent stochastic reservoirs, proving existence of stationary temperature profiles, their entropy-minimizing property, and deriving bounded heat conductivity consistent with Green-Kubo formulas.

Contribution

It establishes the existence and properties of self-consistent temperature profiles in anharmonic crystals and links the non-equilibrium heat conductivity to Green-Kubo formulas, ensuring boundedness.

Findings

Existence of self-consistent temperature profiles in finite systems.

The temperature profile minimizes entropy production at leading order.

Heat conductivity remains bounded and matches Green-Kubo predictions.

Abstract

We investigate a class of anharmonic crystals in $d$ dimensions, $d\ge 1$, coupled to both external and internal heat baths of the Ornstein-Uhlenbeck type. The external heat baths, applied at the boundaries in the 1-direction, are at specified, unequal, temperatures $\tlb$ and $\trb$. The temperatures of the internal baths are determined in a self-consistent way by the requirement that there be no net energy exchange with the system in the non-equilibrium stationary state (NESS). We prove the existence of such a stationary self-consistent profile of temperatures for a finite system and show it minimizes the entropy production to leading order in $(\tlb -\trb)$. In the NESS the heat conductivity $\kappa$ is defined as the heat flux per unit area divided by the length of the system and $(\tlb -\trb)$. In the limit when the temperatures of the external reservoirs goes to the same temperature $T$, $\kappa(T)$ is given by the Green-Kubo formula, evaluated in an equilibrium system coupled to reservoirs all having the temperature $T$. This $\kappa(T)$ remains bounded as the size of the system goes to infinity. We also show that the corresponding infinite system Green-Kubo formula yields a finite result. Stronger results are obtained under the assumption that the self-consistent profile remains bounded.