4,994 research outputs found
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.Comment: v1: 89 pages, 18 figures, pdflatex. v2: 95 pages, 20 figures, text,
figures and appendices added, many references cut, close to published versio
Extending the definition of entropy to nonequilibrium steady states
We study the nonequilibrium statistical mechanics of a finite classical
system subjected to nongradient forces and maintained at fixed kinetic
energy (Hoover-Evans isokinetic thermostat). We assume that the microscopic
dynamics is sufficiently chaotic (Gallavotti-Cohen chaotic hypothesis) and that
there is a natural nonequilibrium steady state . When is
replaced by one can compute the change of
(linear response) and define an entropy change based on
energy considerations. When is varied around a loop, the total change of
need not vanish: outside of equilibrium the entropy has curvature. But at
equilibrium (i.e. if is a gradient) we show that the curvature is zero,
and that the entropy near equilibrium is well defined to
second order in .Comment: plain TeX, 10 pagesemacs ded
From the Theory of Chaos to Nonequilibrium Statistical Mechanics
AbstractThe theory of chaos has emerged from a multidisciplinary encounter of mathematics, physics, and other sciences. The study of chaotic (microscopic) dynamics is now contributing in a major way to our understanding of nonequilibrium statistical mechanics. In particular one does not want to be restricted to situations close to equilibrium. We shall discuss some of the successes and challenges that have been met in this new development
From dynamical systems to statistical mechanics: the case of the fluctuation theorem
This viewpoint relates to an article by Jorge Kurchan (1998 J. Phys. A: Math.
Gen. 31, 3719) as part of a series of commentaries celebrating the most
influential papers published in the J. Phys. series, which is celebrating its
50th anniversary
Chaotic dynamics, fluctuations, nonequilibrium ensembles
A review of some recent results and ideas about the expected behaviour of
large chaotic systems and fluids.Comment: 10 pages: LaTeX-REVTe
Fluctuations in Nonequilibrium Statistical Mechanics: Models, Mathematical Theory, Physical Mechanisms
The fluctuations in nonequilibrium systems are under intense theoretical and
experimental investigation. Topical ``fluctuation relations'' describe
symmetries of the statistical properties of certain observables, in a variety
of models and phenomena. They have been derived in deterministic and, later, in
stochastic frameworks. Other results first obtained for stochastic processes,
and later considered in deterministic dynamics, describe the temporal evolution
of fluctuations. The field has grown beyond expectation: research works and
different perspectives are proposed at an ever faster pace. Indeed,
understanding fluctuations is important for the emerging theory of
nonequilibrium phenomena, as well as for applications, such as those of
nanotechnological and biophysical interest. However, the links among the
different approaches and the limitations of these approaches are not fully
understood. We focus on these issues, providing: a) analysis of the theoretical
models; b) discussion of the rigorous mathematical results; c) identification
of the physical mechanisms underlying the validity of the theoretical
predictions, for a wide range of phenomena.Comment: 44 pages, 2 figures. To appear in Nonlinearity (2007
Fluctuation Theorem and Chaos
The heat theorem (i.e. the second law of thermodynamics or the existence of
entropy) is a manifestation of a general property of hamiltonian mechanics and
of the ergodic Hypothesis. In nonequilibrium thermodynamics of stationary
states the chaotic hypothesis plays a similar role: it allows a unique
determination of the probability distribution (called {\rm SRB} distribution on
phase space providing the time averages of the observables. It also implies an
expression for a few averages concrete enough to derive consequences of
symmetry properties like the fluctuation theorem or to formulate a theory of
coarse graining unifying the foundations of equilibrium and of nonequilibrium.Comment: Basis for the plenary talk at StatPhys23 (Genova July 2007
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