Energy Transport in an Ising Disordered Model

TL;DR

This paper introduces a versatile microcanonical dynamics for Ising systems, enabling the study of energy transport and local temperature definitions in disordered and inhomogeneous models, with results on steady states and heat conductivity.

Contribution

It presents a new general dynamics for Ising models, applicable to disordered and out-of-equilibrium systems, and analyzes heat transport and local temperature concepts.

Findings

Steady states are achieved under the new dynamics.

Heat transport follows a Fourier law on average.

Disorder reduces thermal conductivity, especially at low temperatures.

Abstract

We introduce a new microcanonical dynamics for a large class of Ising systems isolated or maintained out of equilibrium by contact with thermostats at different temperatures. Such a dynamics is very general and can be used in a wide range of situations, including disordered and topologically inhomogenous systems. Focusing on the two-dimensional ferromagnetic case, we show that the equilibrium temperature is naturally defined, and it can be consistently extended as a local temperature when far from equilibrium. This holds for homogeneous as well as for disordered systems. In particular, we will consider a system characterized by ferromagnetic random couplings $J_{ij} \in [ 1 - \epsilon, 1 + \epsilon ]$. We show that the dynamics relaxes to steady states, and that heat transport can be described on the average by means of a Fourier equation. The presence of disorder reduces the conductivity, the effect being especially appreciable for low temperatures. We finally discuss a possible singular behaviour arising for small disorder, i.e. in the limit $\epsilon \to 0$.