Microscopic versus macroscopic approaches to non-equilibrium systems

TL;DR

This paper reviews two main methods—microscopic and macroscopic—for deriving the large deviation functional of the density in the steady state of the one-dimensional symmetric simple exclusion process, a key model in non-equilibrium statistical physics.

Contribution

It compares and explains the microscopic matrix product approach and the macroscopic fluctuation theory for analyzing non-equilibrium steady states.

Findings

Both approaches yield the same large deviation functional.

The microscopic approach uses matrix product ansatz and additivity.

The macroscopic approach is based on macroscopic fluctuation theory.

Abstract

The one dimensional symmetric simple exclusion process (SSEP) is one of the very few exactly soluble models of non-equilibrium statistical physics. It describes a system of particles which diffuse with hard core repulsion on a one dimensional lattice in contact with two reservoirs of particles at unequal densities. The goal of this note is to review the two main approaches which lead to the exact expression of the large deviation functional of the density of the SSEP in its steady state: a microscopic approach (based on the matrix product ansatz and an additivity property) and a macroscopic approach (based on the macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim).