9,271 research outputs found
Statistical mechanics in the context of special relativity II
The special relativity laws emerge as one-parameter (light speed)
generalizations of the corresponding laws of classical physics. These
generalizations, imposed by the Lorentz transformations, affect both the
definition of the various physical observables (e.g. momentum, energy, etc), as
well as the mathematical apparatus of the theory. Here, following the general
lines of [Phys. Rev. E {\bf 66}, 056125 (2002)], we show that the Lorentz
transformations impose also a proper one-parameter generalization of the
classical Boltzmann-Gibbs-Shannon entropy. The obtained relativistic entropy
permits to construct a coherent and selfconsistent relativistic statistical
theory, preserving the main features of the ordinary statistical theory, which
recovers in the classical limit. The predicted distribution function is a
one-parameter continuous deformation of the classical Maxwell-Boltzmann
distribution and has a simple analytic form, showing power law tails in
accordance with the experimental evidence. Furthermore the new statistical
mechanics can be obtained as stationary case of a generalized kinetic theory
governed by an evolution equation obeying the H-theorem and reproducing the
Boltzmann equation of the ordinary kinetics in the classical limit.Comment: 14 pages, no figures, proof correction
Deformation Quantization: Twenty Years After
We first review the historical developments, both in physics and in
mathematics, that preceded (and in some sense provided the background of)
deformation quantization. Then we describe the birth of the latter theory and
its evolution in the past twenty years, insisting on the main conceptual
developments and keeping here as much as possible on the physical side. For the
physical part the accent is put on its relations to, and relevance for,
"conventional" physics. For the mathematical part we concentrate on the
questions of existence and equivalence, including most recent developments for
general Poisson manifolds; we touch also noncommutative geometry and index
theorems, and relations with group theory, including quantum groups. An
extensive (though very incomplete) bibliography is appended and includes
background mathematical literature.Comment: 39 pages; to be published with AIP Press in Proceedings of the 1998
Lodz conference "Particles, Fields and Gravitation". LaTeX (compatibility
mode) with aipproc styl
Thermostatistics of deformed bosons and fermions
Based on the q-deformed oscillator algebra, we study the behavior of the mean
occupation number and its analogies with intermediate statistics and we obtain
an expression in terms of an infinite continued fraction, thus clarifying
successive approximations. In this framework, we study the thermostatistics of
q-deformed bosons and fermions and show that thermodynamics can be built on the
formalism of q-calculus. The entire structure of thermodynamics is preserved if
ordinary derivatives are replaced by the use of an appropriate Jackson
derivative and q-integral. Moreover, we derive the most important thermodynamic
functions and we study the q-boson and q-fermion ideal gas in the thermodynamic
limit.Comment: 14 pages, 2 figure
Deformation Quantization: Genesis, Developments and Metamorphoses
We start with a short exposition of developments in physics and mathematics
that preceded, formed the basis for, or accompanied, the birth of deformation
quantization in the seventies. We indicate how the latter is at least a viable
alternative, autonomous and conceptually more satisfactory, to conventional
quantum mechanics and mention related questions, including covariance and star
representations of Lie groups. We sketch Fedosov's geometric presentation,
based on ideas coming from index theorems, which provided a beautiful frame for
developing existence and classification of star-products on symplectic
manifolds. We present Kontsevich's formality, a major metamorphosis of
deformation quantization, which implies existence and classification of
star-products on general Poisson manifolds and has numerous ramifications. Its
alternate proof using operads gave a new metamorphosis which in particular
showed that the proper context is that of deformations of algebras over
operads, while still another is provided by the extension from differential to
algebraic geometry. In this panorama some important aspects are highlighted by
a more detailed account.Comment: Latex file. 40 pages with 2 figures. To appear in: Proceedings of the
meeting between mathematicians and theoretical physicists, Strasbourg, 2001.
IRMA Lectures in Math. Theoret. Phys., vol. 1, Walter De Gruyter, Berlin
2002, pp. 9--5
Kappa-deformed random-matrix theory based on Kaniadakis statistics
We present a possible extension of the random-matrix theory, which is widely
used to describe spectral fluctuations of chaotic systems. By considering the
Kaniadakis non-Gaussian statistics, characterized by the index {\kappa}
(Boltzmann-Gibbs entropy is recovered in the limit {\kappa}\rightarrow0), we
propose the non-Gaussian deformations ({\kappa} \neq 0) of the conventional
orthogonal and unitary ensembles of random matrices. The joint eigenvalue
distributions for the {\kappa}-deformed ensembles are derived by applying the
principle maximum entropy to Kaniadakis entropy. The resulting distribution
functions are base invarient as they depend on the matrix elements in a trace
form. Using these expressions, we introduce a new generalized form of the
Wigner surmise valid for nearly-chaotic mixed systems, where a
basis-independent description is still expected to hold. We motivate the
necessity of such generalization by the need to describe the transition of the
spacing distribution from chaos to order, at least in the initial stage. We
show several examples about the use of the generalized Wigner surmise to the
analysis of the results of a number of previous experiments and numerical
experiments. Our results suggest the entropic index {\kappa} as a measure for
deviation from the state of chaos. We also introduce a {\kappa}-deformed
Porter-Thomas distribution of transition intensities, which fits the
experimental data for mixed systems better than the commonly-used
gamma-distribution.Comment: 18 pages, 8 figure
Extreme event statistics of daily rainfall: Dynamical systems approach
We analyse the probability densities of daily rainfall amounts at a variety
of locations on the Earth. The observed distributions of the amount of rainfall
fit well to a q-exponential distribution with exponent q close to q=1.3. We
discuss possible reasons for the emergence of this power law. On the contrary,
the waiting time distribution between rainy days is observed to follow a
near-exponential distribution. A careful investigation shows that a
q-exponential with q=1.05 yields actually the best fit of the data. A Poisson
process where the rate fluctuates slightly in a superstatistical way is
discussed as a possible model for this. We discuss the extreme value statistics
for extreme daily rainfall, which can potentially lead to flooding. This is
described by Frechet distributions as the corresponding distributions of the
amount of daily rainfall decay with a power law. On the other hand, looking at
extreme event statistics of waiting times between rainy days (leading to
droughts for very long dry periods) we obtain from the observed
near-exponential decay of waiting times an extreme event statistics close to
Gumbel distributions. We discuss superstatistical dynamical systems as simple
models in this context.Comment: 10 pages, 15 figures. Replaced by final version published in J.Phys.
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