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

Abstract

We consider the Gibbs representation over space–time of non-equilibrium dynamics ofHamiltonian systems defined on a lattice with local interactions. We first write thecorresponding 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 anidentical space–time characterization. We study in detail the irreversible properties of thenew dynamics, define the local conductivity and show its equivalence with the Green–Kuboformula. Given the definition of the local heat conductivity and using conservation ofenergy, we derive the shape of the temperature profile. Next, we apply our scheme tovarious approximations of anharmonic Hamiltonian models, show how to compute theirthermal conductivity and recover results confirmed in numerical simulations.