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Journal ArticleDOI

Fluctuations and Irreversible Processes

15 Sep 1953-Physical Review (American Physical Society)-Vol. 91, Iss: 6, pp 1505-1512
TL;DR: In this paper, the probability of a given succession of (nonequilibrium) states of a spontaneously fluctuating thermodynamic system is calculated, on the assumption that the macroscopic variables defining a state are Gaussian random variables whose average behavior is given by the laws governing irreversible processes.
Abstract: The probability of a given succession of (nonequilibrium) states of a spontaneously fluctuating thermodynamic system is calculated, on the assumption that the macroscopic variables defining a state are Gaussian random variables whose average behavior is given by the laws governing irreversible processes.This probability can be expressed in terms of the dissipation function; the resulting relation, which is an extension of Boltzmann's principle, shows the statistical significance of the dissipation function. From the form of the relation, the principle of least dissipation of energy becomes evident by inspection.
Citations
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Journal ArticleDOI
TL;DR: Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production.
Abstract: Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation–dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production. (Some figures may appear in colour only in the online journal) This article was invited by Erwin Frey.

2,834 citations


Cites methods from "Fluctuations and Irreversible Proce..."

  • ...The path integral for such a multivariate process has been pioneered by Onsager and Machlup for linear processes [60, 61] and by Graham for non-linear processes including a spatially dependent diffusion constant [62, 63]....

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Book
01 Jan 1993
TL;DR: In this paper, the authors propose a theory which goes beyond the classical formulation of thermodynamics by enlarging the space of basic independent variables, through the introduction of non-equilibrium variables, such as the dissipative fluxes appearing in the balance equations.
Abstract: Our aim is to propose a theory which goes beyond the classical formulation of thermodynamics. This is achieved by enlarging the space of basic independent variables, through the introduction of non-equilibrium variables, such as the dissipative fluxes appearing in the balance equations. The next step is to find evolution equations for the dissipative fluxes. Whereas the evolution equations for the classical variables are given by the usual balance laws, no general criteria exist concerning the evolution equations of the dissipative fluxes, with the exception of the restrictions imposed on them by the second law of thermodynamics.

1,739 citations

Book ChapterDOI
Hannes Risken1
01 Jan 1984
TL;DR: In this paper, an equation for the distribution function describing Brownian motion was first derived by Fokker [11] and Planck [12] and it is shown that expectation values for nonlinear Langevin equations (367, 110) are much more difficult to obtain.
Abstract: As shown in Sects 31, 2 we can immediately obtain expectation values for processes described by the linear Langevin equations (31, 31) For nonlinear Langevin equations (367, 110) expectation values are much more difficult to obtain, so here we first try to derive an equation for the distribution function As mentioned already in the introduction, a differential equation for the distribution function describing Brownian motion was first derived by Fokker [11] and Planck [12]: many review articles and books on the Fokker-Planck equation now exist [15 – 15]

1,412 citations

Journal ArticleDOI
TL;DR: The theory of large deviations as mentioned in this paper is concerned with the exponential decay of probabilities of large fluctuations in random systems, and it provides exponential-order estimates of probabilities that refine and generalize Einstein's theory of fluctuations.

1,361 citations

Journal ArticleDOI
TL;DR: The theory of large deviations as discussed by the authors is concerned with the exponential decay of probabilities of large fluctuations in random systems, and it provides exponential-order estimates of probabilities that refine and generalize Einstein's theory of fluctuations.
Abstract: The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory. In the context of equilibrium statistical mechanics, the theory of large deviations provides exponential-order estimates of probabilities that refine and generalize Einstein's theory of fluctuations. This review explores this and other connections between large deviation theory and statistical mechanics, in an effort to show that the mathematical language of statistical mechanics is the language of large deviation theory. The first part of the review presents the basics of large deviation theory, and works out many of its classical applications related to sums of random variables and Markov processes. The second part goes through many problems and results of statistical mechanics, and shows how these can be formulated and derived within the context of large deviation theory. The problems and results treated cover a wide range of physical systems, including equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, as well as multifractals, disordered systems, and chaotic systems. This review also covers many fundamental aspects of statistical mechanics, such as the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations.

1,185 citations


Cites background or methods from "Fluctuations and Irreversible Proce..."

  • ...The topics discussed in the context of nonequilibrium systems are as varied, and include the study of large deviations in stochastic differential equations (Freidlin–Wentzell theory), dynamical models of equilibrium fluctuations (Onsager–Machlup theory), fluctuation relations, and systems of interacting particles....

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  • ...12 Onsager and Machlup [119] derived this result for linear equations as far back as 1953 (see also [120–122])....

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  • ...The minimum dissipation principle of Onsager and Machlup [119], for example, which states that the fluctuation and decay paths of equilibrium systems minimize some dissipation function, can be re-interpreted in terms of the variational principle of Eq....

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  • ...To find the expression of the quasipotential V (x), we follow Onsager and Machlup [119] and solve the Euler–Lagrange Eq....

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  • ...(227), and is what Onsager and Machlup refer to as theminimum dissipation principle [119]....

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