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.

The large deviation approach to statistical mechanics

Statistical mechanics and dynamics of solvable models with long-range interactions

We discuss several pairing-related phenomena in nuclear systems, ranging from superfluidity in neutron stars to the gradual breaking of pairs in finite nuclei. We focus on the links between many-body pairing as it evolves from the underlying nucleon-nucleon interaction and the eventual experimental and theoretical manifestations of superfluidity in infinite nuclear matter and of pairing in finite nuclei. We analyse the nature of pair correlations in nuclei and their potential impact on nuclear structure experiments. We also describe recent experimental evidence that points to a relation between pairing and phase transitions (or transformations) in finite nuclear systems. Finally, we discuss recent investigations of ground-state properties of random two-body interactions where pairing plays little role although the interactions yield interesting nuclear properties such as 0+ ground states in even-even nuclei.

https://arxiv.org/pdf/nucl-th/0210033.pdf

Pairing in Nuclear Systems: From Neutron Stars to Finite Nuclei

Nuclear spinodal fragmentation

Roughly speaking, classical statistical physics is the branch of theoretical physics that aims to account for the thermal behaviour of macroscopic bodies in terms of a classical mechanical model of their microscopic constituents, with the help of probabilistic assumptions. In the last century and a half, a fair number of approaches have been developed to meet this aim. This study of their foundations assesses their coherence and analyzes the motivations for their basic assumptions, and the interpretations of their central concepts. The most outstanding foundational problems are the explanation of time-asymmetry in thermal behaviour, the relative autonomy of thermal phenomena from their microscopic underpinning, and the meaning of probability. A more or less historic survey is given of the work of Maxwell, Boltzmann and Gibbs in statistical physics, and the problems and objections to which their work gave rise. Next, we review some modern approaches to (i) equilibrium statistical mechanics, such as ergodic theory and the theory of the thermodynamic limit; and to (ii) non-equilibrium statistical mechanics as provided by Lanford's work on the Boltzmann equation, the so-called Bogolyubov-Born-Green-Kirkwood-Yvon approach, and stochastic approaches such as `coarse-graining' and the `open systems' approach. In all cases, we focus on the subtle interplay between probabilistic assumptions, dynamical assumptions, initial conditions and other ingredients used in these approaches.

/pdf/compendium-of-the-foundations-of-classical-statistical-1sd3mnxghs.pdf

D.H.E. Gross

Papers

Microcanonical thermodynamics and statistical fragmentation of dissipative systems. The topological structure of the N-body phase space