Totally asymmetric limit for models of heat conduction

TL;DR

This paper analyzes weakly asymmetric heat conduction models in one dimension, computing quasi-potentials via macroscopic fluctuation theory, and explores the large external field limit where these functionals become local and independent of reservoirs.

Contribution

It extends the computation of quasi-potentials to convex mobility models like KMP and KMPd, and investigates their totally asymmetric limits both microscopically and macroscopically.

Findings

Quasi-potentials become local in the large external field limit.

In the limit, dependence on external reservoirs disappears.

Some asymmetric limits of the KMP model have product invariant measures.

Abstract

We consider one dimensional weakly asymmetric boundary driven models of heat conduction. In the cases of a constant diffusion coefficient and of a quadratic mobility we compute the quasi-potential that is a non local functional obtained by the solution of a variational problem. This is done using the dynamic variational approach of the macroscopic fluctuation theory \cite{MFT}. The case of a concave mobility corresponds essentially to the exclusion model that has been discussed in \cite{Lag,CPAM,BGLa,ED}. We consider here the convex case that includes for example the Kipnis-Marchioro-Presutti (KMP) model and its dual (KMPd) \cite{KMP}. This extends to the weakly asymmetric regime the computations in \cite{BGL}. We consider then, both microscopically and macroscopically, the limit of large external fields. Microscopically we discuss some possible totally asymmetric limits of the KMP model. In one case the totally asymmetric dynamics has a product invariant measure. Another possible limit dynamics has instead a non trivial invariant measure for which we give a duality representation. Macroscopically we show that the quasi-potentials of KMP and KMPd, that for any fixed external field are non local, become local in the limit. Moreover the dependence on one of the external reservoirs disappears. For models having strictly positive quadratic mobilities we obtain instead in the limit a non local functional having a structure similar to the one of the boundary driven asymmetric exclusion process.