Stationary non-equilibrium properties for a heat conduction model

TL;DR

This paper analyzes a stochastic heat conduction model with two conserved quantities, establishing hydrodynamic limits, fluctuation fields, and large deviation principles in non-equilibrium steady states.

Contribution

It introduces a detailed study of large deviations for temperature profiles in a non-equilibrium setting, using macroscopic fluctuation theory.

Findings

Gaussian fluctuation fields in the limit

Explicit covariance matrix computed

Variational formula for large deviations derived

Abstract

We consider a stochastic heat conduction model for solids composed by N interacting atoms. The system is in contact with two heat baths at different temperature $T_\ell$ and $T_r$. The bulk dynamics conserve two quantities: the energy and the deformation between atoms. If $T_\ell \neq T_r$, a heat flux takes place in the system. For large $N$, the system adopts a linear temperature profile between $T_\ell$ and $T_r$. We establish the hydrodynamic limit for the two conserved quantities. We introduce the fluctuations field of the energy and of the deformation in the non-equilibrium steady state. As $N$ goes to infinity, we show that this field converges to a Gaussian field and we compute the limiting covariance matrix. The main contribution of the paper is the study of large deviations for the temperature profile in the non-equilibrium stationary state. A variational formula for the rate function is derived following the recent macroscopic fluctuation theory of Bertini et al.