Onsager-Machlup Theory for Nonequilibrium Steady States and Fluctuation Theorems

Abstract. We consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model (among other things) biological transport and traffic flow. We begin with a broad overview of this class of driven diffusive systems-which includes exclusion processes-focusing on interesting physical properties, such as shocks and phase transitions. We then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form. In addition to a gentle introduction to this matrix product approach, how it works and how it relates to similar constructions that arise in other physical contexts, we present a unified, pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed. We also review a number of more advanced topics, including nonequilibrium free energy functionals, the classification of exclusion processes involving multiple particle species, existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled. We conclude with a brief discussion of open problems for future research.

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“…In sublattice updating the timestep is split into two halves: in the first half site 1, the even bonds (2, 3), (4,5) . .…”

Section: Sublattice Parallel Dynamicsmentioning

confidence: 99%

Nonequilibrium steady states of matrix-product form: a solver's guide

Blythe

Evans²

2007

J. Phys. A: Math. Theor.

638

1,014

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“…It was generalized to fluctuations in systems in a NESS in Refs. [8,9,10,11]. In this approach the transition probability f (v, t|v 0 , t 0 ) is formally expressed as a path integral, i.e., as an integral over all paths leading from the initial state (v 0 , t 0 ) to the final state (v, t).…”

Section: Path Integral Approachmentioning

confidence: 99%

A path integral approach to random motion with nonlinear friction

Baule¹,

Cohen²,

Touchette³

2009

J. Phys. A: Math. Theor.

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Using a path integral approach, we derive an analytical solution of a nonlinear and singular Langevin equation, which has been introduced previously by P.-G. de Gennes as a simple phenomenological model for the stick-slip motion of a solid object on a vibrating horizontal surface. We show that the optimal (or most probable) paths of this model can be divided into two classes of paths, which correspond physically to a sliding or slip motion, where the object moves with a non-zero velocity over the underlying surface, and a stick-slip motion, where the object is stuck to the surface for a finite time. These two kinds of basic motions underlie the behavior of many more complicated systems with solid/solid friction and appear naturally in de Gennes' model in the path integral framework.

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“…Jarzynski found an interesting relation between NEQ work and equilibrium free energy [12], which was later proved to be a special case of Crooks fluctuation theorems [7,13]. Since then, a number of stimulated studies have been published up to now regarding the fluctuation theorems and related phenomena [14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Nonequilibrium fluctuations for linear diffusion dynamics

2011

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We present the theoretical study on non-equilibrium (NEQ) fluctuations for diffusion dynamics in high dimensions driven by a linear drift force. We consider a general situation in which NEQ is caused by two conditions: (i) drift force not derivable from a potential function and (ii) diffusion matrix not proportional to the unit matrix, implying non-identical and correlated multi-dimensional noise. The former is a well-known NEQ source and the latter can be realized in the presence of multiple heat reservoirs or multiple noise sources. We develop a statistical mechanical theory based on generalized thermodynamic quantities such as energy, work, and heat. The NEQ fluctuation theorems are reproduced successfully. We also find the time-dependent probability distribution function exactly as well as the NEQ work production distribution P (W) in terms of solutions of nonlinear differential equations. In addition, we compute low-order cumulants of the NEQ work production explicitly. In two dimensions, we carry out numerical simulations to check out our analytic results and also to get P (W). We find an interesting dynamic phase transition in the exponential tail shape of P (W), associated with a singularity found in solutions of the nonlinear differential equation. Finally, we discuss possible realizations in experiments.

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Onsager-Machlup Theory for Nonequilibrium Steady States and Fluctuation Theorems

Cited by 89 publications

References 63 publications

Nonequilibrium steady states of matrix-product form: a solver's guide

Nonequilibrium steady states of matrix-product form: a solver's guide

A path integral approach to random motion with nonlinear friction

Nonequilibrium fluctuations for linear diffusion dynamics

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