Mapping out of equilibrium into equilibrium in one-dimensional transport 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.
Cite
@article{arxiv.0809.0709,
title = {Mapping out of equilibrium into equilibrium in one-dimensional transport models},
author = {Julien Tailleur and Jorge Kurchan and Vivien Lecomte},
journal= {arXiv preprint arXiv:0809.0709},
year = {2008}
}