On the local space-time structure of non-equilibrium steady states

TL;DR

This paper develops a local space-time framework for non-equilibrium steady states in Hamiltonian lattice systems, deriving temperature profiles and thermal conductivities, with explicit formulas for harmonic cases and applications to anharmonic models.

Contribution

It introduces a local action functional for non-equilibrium dynamics, establishes a link between local conductivity and Green-Kubo formula, and provides explicit formulas for stationary measures and thermal conductivities.

Findings

Derived the shape of temperature profiles in non-equilibrium states.

Established the equivalence of local conductivity with Green-Kubo formula.

Provided explicit formulas for harmonic systems and applications to anharmonic models.

Abstract

We consider the Gibbs representation over space-time of non-equilibrium dynamics of Hamiltonian systems defined on a lattice with local interactions. We first write the corresponding action functional as a sum of local terms, defining a local action functional. We replace the local system by a translation-invariant system whose dynamics has an identical space-time characterization. We study in details the irreversible properties of the new dynamics, define the local conductivity and show its equivalence with the Green-Kubo formula. Given the definition of the local heat conductivity and using conservation of energy, we derive the shape of the temperature profile. Next, we find an explicit formula for the non-equilibrium stationary measure of harmonic systems. Finally, we apply our scheme to various approximations of anharmonic Hamiltonian models, show how to compute their thermal conductivity and recover results confirmed in numerical simulations.