Fluctuation relations: a pedagogical overview

Abstract

The fluctuation relations have received considerable attention since their emergence and development in the 1990s. We present a summary of the main results and suggest ways to interpret this material. Starting with a consideration of the under-determined time evolution of a simple open system, formulated using continuous Markovian stochastic dy- namics, an expression for the entropy generated over a time interval is developed in terms of the probability of observing a trajectory associated with a prescribed driving protocol, and the probability of its time-reverse. This forms the basis for a general theoretical description of non-equilibrium thermodynamic pro- cesses. Having established a connection between entropy production and an inequivalence in probability for forward and time-reversed events, we proceed in the manner of Sekimoto and Seifert, in particular, to derive results in stochastic thermodynamics: a description of the evolution of a system between equilibrium states that ties in with well-established thermodynamic expectations. We derive fluctuation relations, state conditions for their validity, and illustrate their op- eration in some simple cases, thereby providing some introductory insight into the various celebrated symmetry relations that have emerged in this field.

Chapter contributed to R.Klages, W.Just, C.Jarzynski (Eds.), Nonequilibrium Statistical Physics of Small Systems: Fluctuation Relations and Beyond (Wiley-VCH, Weinheim, 2012; ISBN 978-3-527-41094-1)

References in corpus (11)

• The large deviation approach to statistical mechanics
• Three detailed fluctuation theorems
• The Three Faces of the Second Law: I. Master Equation Formulation
• An Extension of the Fluctuation Theorem
• The Three Faces of the Second Law: II. Fokker-Planck Formulation
• Path-integral analysis of fluctuation theorems for general Langevin processes
• Comparison of far-from-equilibrium work relations
• Integral fluctuation theorem for the housekeeping heat
• Comparison of work fluctuation relations
• Fluctuation relations for a driven Brownian particle
• Entropy production for mechanically or chemically driven biomolecules

Cited by in corpus (5)

• Local non-equilibrium thermodynamics
• Symmetries of cyclic work distributions for an isolated harmonic oscillator
• Contributions to the Theory of Thermostated Systems
• Roles of Local Non-equilibrium Free Energy in the Description of Biomolecules
• Stochastic entropy production in diffusive systems