Energy-exchange stochastic models for non-equilibrium

TL;DR

This paper reviews stochastic energy-exchange models to understand non-equilibrium steady states and anomalous transport, highlighting duality methods for analyzing boundary-driven interacting systems.

Contribution

It introduces the duality approach as a rigorous mathematical tool for studying stochastic energy-exchange models in non-equilibrium systems.

Findings

Duality theory effectively analyzes Markov processes.

Boundary-driven models reveal universal properties.

Insights into anomalous transport in Hamiltonian systems.

Abstract

Non-equilibrium steady states are subject to intense investigations but still poorly understood. For instance, the derivation of Fourier law in Hamiltonian systems is a problem that still poses several obstacles. In order to investigate non-equilibrium systems, stochastic models of energy-exchange have been introduced and they have been used to identify universal properties of non-equilibrium. In these notes, after a brief review of the problem of anomalous transport in 1-dimensional Hamiltonian systems, some boundary-driven interacting random systems are considered and the "duality approach" to their rigorous mathematical treatment is reviewed. Duality theory, of which a brief introduction is given, is a powerful technique to deal with Markov processes and interacting particle systems. The content of these notes is mainly based on the papers [10, 11, 12].