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

2008/09/30 by Julien Tailleur, Jorge Kurchan, Vivien Lecomte · 123 citations
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Applied mathematics #Boltzmann distribution #Detailed balance #Hamiltonian (control theory) #Limit (mathematics) #Mathematical analysis #Mathematical optimization #Mathematics #Physics #Quantum #Quantum mechanics #Semiclassical physics #Sequence (biology) #Statistical physics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech

paper · pdf · doi:10.1088/1751-8113/41/50/505001

published in Journal of Physics A Mathematical and Theoretical 41(50), 505001 (Institute of Physics)

openalex publication_date 2008/11/03 · arxiv created 2008/11/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 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 (the driven symmetric exclusion process and the 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.

Citations

• Fluctuation statistics in networks: A stochastic path integral approach
• Free Energy Functional for Nonequilibrium Systems: An Exactly Solvable Case
• Large Deviations for a Stochastic Model of Heat Flow
• Macroscopic Fluctuation Theory for Stationary Non-Equilibrium States
• Mapping Nonequilibrium onto Equilibrium: The Macroscopic Fluctuations of Simple Transport Models

Cited by

• Stochastic dissipative quantum spin chains (I) : Quantum fluctuating discrete hydrodynamics
• Langevin Dynamics, Large Deviations and Instantons for the Quasi-Geostrophic Model and Two-Dimensional Euler Equations
• Current Fluctuations in One Dimensional Diffusive Systems with a Step Initial Density Profile
• Activated Escape of a Self-Propelled Particle from a Metastable State
• Finite Size Corrections to the Large Deviation Function of the Density in the One Dimensional Symmetric Simple Exclusion Process

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