Mapping out of equilibrium into equilibrium in one-dimensional transport models

TL;DR

This paper demonstrates how certain one-dimensional driven transport models can be transformed into equilibrium systems satisfying detailed balance, enabling explicit solutions for their distribution functions through a series of variable changes.

Contribution

It introduces a non-local transformation that maps out-of-equilibrium models to equilibrium systems, simplifying their analysis and solution.

Findings

Models can be transformed into equilibrium systems with Gibbs-Boltzmann distribution.

Transformation applies to driven exclusion and energy exchange models.

Method simplifies solving for distribution functions in these models.

Abstract

Systems with conserved currents driven by reservoirs at the boundaries offer an opportunity for a general analytic study that is unparalleled in more general out of equilibrium systems. The evolution of coarse-grained variables is governed by stochastic {\em hydrodynamic} equations in the limit of small noise.} As such it is amenable to a treatment formally equal to the semiclassical limit of quantum mechanics, which reduces the problem of finding the full distribution functions to the solution of a set of Hamiltonian equations. It is in general not possible to solve such equations explicitly, but for an interesting set of problems (driven Symmetric Exclusion Process and Kipnis-Marchioro-Presutti model) it can be done by a sequence of remarkable changes of variables. We show that at the bottom of this `miracle' is the surprising fact that these models can be taken through a non-local transformation into isolated systems satisfying detailed balance, with probability distribution given by the Gibbs-Boltzmann measure. This procedure can in fact also be used to obtain an elegant solution of the much simpler problem of non-interacting particles diffusing in a one-dimensional potential, again using a transformation that maps the driven problem into an undriven one.