The large deviation approach to statistical mechanics

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.

These lecture notes give a short review of methods such as the matrix ansatz, the additivity principle or the macroscopic fluctuation theory, developed recently in the theory of non-equilibrium phenomena. They show how these methods allow us to calculate the fluctuations and large deviations of the density and the current in non-equilibrium steady states of systems like exclusion processes. The properties of these fluctuations and large deviation functions in non-equilibrium steady states (for example, non-Gaussian fluctuations of density or non-convexity of the large deviation function which generalizes the notion of free energy) are compared with those of systems at equilibrium.

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Non-equilibrium steady states: fluctuations and large deviations of the density and of the current

The question of how reversible microscopic equations of motion can lead to irreversible macroscopic behaviour has been one of the central issues in statistical mechanics for more than a century. The basic issues were known to Gibbs. Boltzmann conducted a very public debate with Loschmidt and others without a satisfactory resolution. In recent decades there has been no real change in the situation. In1993 we discovered a relation, subsequently known as the Fluctuation Theorem (FT), which gives an analytical expression for the probability of observing Second Law violating dynamicaluctuations in thermostatted dissipa- tive non-equilibrium systems. The relation was derived heuristically and applied to the special case of dissipative non-equilibrium systems subject to constant energy `thermostatting'. These restrictions meant that the full importance of the Theorem was not immediately apparent. Within a few years, derivations of the Theorem were improved but it has only been inthe last few of years that the generality ofthe Theorem has been appreciated. We now know that the Second Law of Thermo- dynamics can be derived assuming ergodicity at equilibrium, and causality. We take the assumption of causality to be axiomatic. It is causality which ultimately is responsible for breaking time reversal symmetry and which leads to the possibility of irreversible macroscopic behaviour. The Fluctuation Theorem does much more than merely prove that in large systems observed for long periods of time, the Second Law is overwhelmingly likely to be valid. The Fluctuation Theorem quanti®es the probability of observing Second Law violations in small systems observed for a short time. Unlike the Boltzmann equation, the FT is completely consistent with Loschmidt's observa- tion that for time reversible dynamics, every dynamical phase space trajectory and its conjugate time reversedanti-trajectory', are both solutions of the underlying equations of motion. Indeed the standard proofs of the FT explicitly consider conjugate pairs of phase space trajectories. Quantitative predictions made by the Fluctuation Theorem regarding the probability of Second Law violations have been con®rmed experimentally, both using molecular dynamics computer simula- tion and very recently in laboratory experiments.

http://inis.jinr.ru/sl/P_Physics/PT_Thermodynamics,%20statistical%20physics/Evans,%20Searles.%20Fluctuation%20theorem%20(2002)(57s).pdf

The Fluctuation Theorem

The Jarzynski equality is generalized to situations in which nonequilibrium systems are subject to a feedback control. The new terms that arise as a consequence of the feedback describe the mutual information content obtained by measurement and the efficacy of the feedback control. Our results lead to a generalized fluctuation-dissipation theorem that reflects the readout information, and they can be experimentally tested using small thermodynamic systems. We illustrate our general results by introducing an "information ratchet", which can transport a Brownian particle in one direction and extract positive work from the particle.

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Generalized Jarzynski equality under nonequilibrium feedback control.

We present a definition of entropy production rate for classes of deterministic and stochastic dynamics. The point of departure is a Gibbsian representation of the steady state path space measure for which “the density” is determined with respect to the time-reversed process. The Gibbs formalism is used as a unifying algorithm capable of incorporating basic properties of entropy production in nonequilibrium systems. Our definition is motivated by recent work on the Gallavotti–Cohen (local) fluctuation theorem and it is illustrated via a number of examples.

On the definition of entropy production, via examples

We show that the natural invariant state for Manneville-Pomeau maps can be characterized as a weakly Gibbsian state. In this way we make a connection between the study of intermittency via non-uniformly expanding maps and the thermodynamic formalism for non- uniformly convergent interactions. AMS classification scheme numbers: 37C40, 82B20

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Intermittency and weak Gibbs states

We present a unified approach to establishing the Gibbsian character of a wide class of non-Gibbsian states, arising in the Renormalisation Group theory. Inside the realm of the Pirogov--Sinai theory for lattice spin systems, we prove that RG transformations applied to low temperature phases give rise to weakly Gibbsian measures. In other words, we show that the Griffiths--Pearce--Israel scenario of RG pathologies is carried by atypical configurations. The renormalized measures are described by an effective interaction, with relative energies well-defined on a full measure set of configurations. In this way we complete the first part of the Dobrushin Restoration Program: to give a Gibbsian description to non-Gibbsian states. A disagreement percolation estimate is used in the proof to bound the decay of quenched correlations through which the interaction potential is constructed. The percolation is controlled via a novel type of pathwise large deviation theory.

Annelies Van Moffaert

Papers

On the definition of entropy production, via examples

Intermittency and weak Gibbs states

Percolation, Path Large Deviations and Weakly Gibbs States