767 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
Thermodynamic Limit in Statistical Physics
The thermodynamic limit in statistical thermodynamics of many-particle
systems is an important but often overlooked issue in the various applied
studies of condensed matter physics. To settle this issue, we review tersely
the past and present disposition of thermodynamic limiting procedure in the
structure of the contemporary statistical mechanics and our current
understanding of this problem. We pick out the ingenious approach by N. N.
Bogoliubov, who developed a general formalism for establishing of the limiting
distribution functions in the form of formal series in powers of the density.
In that study he outlined the method of justification of the thermodynamic
limit when he derived the generalized Boltzmann equations. To enrich and to
weave our discussion, we take this opportunity to give a brief survey of the
closely related problems, such as the equipartition of energy and the
equivalence and nonequivalence of statistical ensembles. The validity of the
equipartition of energy permits one to decide what are the boundaries of
applicability of statistical mechanics. The major aim of this work is to
provide a better qualitative understanding of the physical significance of the
thermodynamic limit in modern statistical physics of the infinite and "small"
many-particle systems.Comment: 28 pages, Refs.180. arXiv admin note: text overlap with
arXiv:1011.2981, arXiv:0812.0943 by other author
Asymptotic equivalence of probability measures and stochastic processes
Let and be two probability measures representing two different
probabilistic models of some system (e.g., an -particle equilibrium system,
a set of random graphs with vertices, or a stochastic process evolving over
a time ) and let be a random variable representing a 'macrostate' or
'global observable' of that system. We provide sufficient conditions, based on
the Radon-Nikodym derivative of and , for the set of typical values
of obtained relative to to be the same as the set of typical values
obtained relative to in the limit . This extends to
general probability measures and stochastic processes the well-known
thermodynamic-limit equivalence of the microcanonical and canonical ensembles,
related mathematically to the asymptotic equivalence of conditional and
exponentially-tilted measures. In this more general sense, two probability
measures that are asymptotically equivalent predict the same typical or
macroscopic properties of the system they are meant to model.Comment: v1: 16 pages. v2: 17 pages, precisions, examples and references
added. v3: Minor typos corrected. Close to published versio
Symmetry, Entropy, Diversity and (why not?) Quantum Statistics in Society
We describe society as a nonequilibrium probabilistic system: N individuals
occupy W resource states in it and produce entropy S over definite time
periods. Resulting thermodynamics is however unusual because a second entropy,
H, measures a typically social feature, inequality or diversity in the
distribution of available resources. A symmetry phase transition takes place at
Gini values 1/3, where realistic distributions become asymmetric. Four
constraints act on S: expectedly, N and W, and new ones, diversity and
interactions between individuals; the latter result from the two coordinates of
a single point in the data, the peak. The occupation number of a job is either
zero or one, suggesting Fermi-Dirac statistics for employment. Contrariwise, an
indefinite nujmber of individuals can occupy a state defined as a quantile of
income or of age, so Bose-Einstein statistics may be required.
Indistinguishability rather than anonymity of individuals and resources is thus
needed. Interactions between individuals define define classes of equivalence
that happen to coincide with acceptable definitions of social classes or
periods in human life. The entropy S is non-extensive and obtainable from data.
Theoretical laws are compared to data in four different cases of economical or
physiological diversity. Acceptable fits are found for all of them.Comment: 13 pages, 2 figure
Nonequivalent ensembles and metastability
This paper reviews a number of fundamental connections that exist between
nonequivalent microcanonical and canonical ensembles, the appearance of
first-order phase transitions in the canonical ensemble, and thermodynamic
metastable behavior.Comment: 4 pages, RevTeX, 1 figure. Contribution to the Proceedings of the
31st Workshop of the International School of Solid State Physics
``Complexity, Metastability and Nonextensivity'', held at the Ettore Majorana
Foundation and Centre for Scientific Culture, Erice, Sicily, Italy, July
2004. Edited by C. Tsallis, A. Rapisarda and C. Beck. To be published by
World Scientific, 200
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