12,080 research outputs found
Coarse-grained distributions and superstatistics
We show an interesting connexion between the coarse-grained distribution
function arising in the theory of violent relaxation for collisionless stellar
systems (Lynden-Bell 1967) and the notion of superstatistics introduced
recently by Beck & Cohen (2003). We also discuss the analogies and differences
between the statistical equilibrium state of a multi-components
self-gravitating system and the metaequilibrium state of a collisionless
stellar system. Finally, we stress the important distinction between mixing
entropies, generalized entropies, H-functions, generalized mixing entropies and
relative entropies
On the average uncertainty for systems with nonlinear coupling
The increased uncertainty and complexity of nonlinear systems have motivated
investigators to consider generalized approaches to defining an entropy
function. New insights are achieved by defining the average uncertainty in the
probability domain as a transformation of entropy functions. The Shannon
entropy when transformed to the probability domain is the weighted geometric
mean of the probabilities. For the exponential and Gaussian distributions, we
show that the weighted geometric mean of the distribution is equal to the
density of the distribution at the location plus the scale, i.e. at the width
of the distribution. The average uncertainty is generalized via the weighted
generalized mean, in which the moment is a function of the nonlinear source.
Both the Renyi and Tsallis entropies transform to this definition of the
generalized average uncertainty in the probability domain. For the generalized
Pareto and Student's t-distributions, which are the maximum entropy
distributions for these generalized entropies, the appropriate weighted
generalized mean also equals the density of the distribution at the location
plus scale. A coupled entropy function is proposed, which is equal to the
normalized Tsallis entropy divided by one plus the coupling.Comment: 24 pages, including 4 figures and 1 tabl
Generalized isotropic Lipkin-Meshkov-Glick models: ground state entanglement and quantum entropies
We introduce a new class of generalized isotropic Lipkin-Meshkov-Glick models
with su spin and long-range non-constant interactions, whose
non-degenerate ground state is a Dicke state of su type. We evaluate in
closed form the reduced density matrix of a block of spins when the whole
system is in its ground state, and study the corresponding von Neumann and
R\'enyi entanglement entropies in the thermodynamic limit. We show that both of
these entropies scale as when tends to infinity, where the
coefficient is equal to in the ground state phase with
vanishing su magnon densities. In particular, our results show that none
of these generalized Lipkin-Meshkov-Glick models are critical, since when
their R\'enyi entropy becomes independent of the parameter
. We have also computed the Tsallis entanglement entropy of the ground state
of these generalized su Lipkin-Meshkov-Glick models, finding that it can
be made extensive by an appropriate choice of its parameter only when
. Finally, in the su case we construct in detail the phase
diagram of the ground state in parameter space, showing that it is determined
in a simple way by the weights of the fundamental representation of su.
This is also true in the su case; for instance, we prove that the region
for which all the magnon densities are non-vanishing is an -simplex in
whose vertices are the weights of the fundamental representation
of su.Comment: Typeset with LaTeX, 32 pages, 3 figures. Final version with
corrections and additional reference
Generalized thermodynamics and Fokker-Planck equations. Applications to stellar dynamics, two-dimensional turbulence and Jupiter's great red spot
We introduce a new set of generalized Fokker-Planck equations that conserve
energy and mass and increase a generalized entropy until a maximum entropy
state is reached. The concept of generalized entropies is rigorously justified
for continuous Hamiltonian systems undergoing violent relaxation. Tsallis
entropies are just a special case of this generalized thermodynamics.
Application of these results to stellar dynamics, vortex dynamics and Jupiter's
great red spot are proposed. Our prime result is a novel relaxation equation
that should offer an easily implementable parametrization of geophysical
turbulence. This relaxation equation depends on a single key parameter related
to the skewness of the fine-grained vorticity distribution. Usual
parametrizations (including a single turbulent viscosity) correspond to the
infinite temperature limit of our model. They forget a fundamental systematic
drift that acts against diffusion as in Brownian theory. Our generalized
Fokker-Planck equations may have applications in other fields of physics such
as chemotaxis for bacterial populations. We propose the idea of a
classification of generalized entropies in classes of equivalence and provide
an aesthetic connexion between topics (vortices, stars, bacteries,...) which
were previously disconnected.Comment: Submitted to Phys. Rev.
Fractal geometry, information growth and nonextensive thermodynamics
This is a study of the information evolution of complex systems by
geometrical consideration. We look at chaotic systems evolving in fractal phase
space. The entropy change in time due to the fractal geometry is assimilated to
the information growth through the scale refinement. Due to the incompleteness
of the state number counting at any scale on fractal support, the incomplete
normalization is applied throughout the paper, where is the
fractal dimension divided by the dimension of the smooth Euclidean space in
which the fractal structure of the phase space is embedded. It is shown that
the information growth is nonadditive and is proportional to the trace-form
which can be connected to several nonadditive
entropies. This information growth can be extremized to give power law
distributions for these non-equilibrium systems. It can also be used for the
study of the thermodynamics derived from Tsallis entropy for nonadditive
systems which contain subsystems each having its own . It is argued that,
within this thermodynamics, the Stefan-Boltzmann law of blackbody radiation can
be preserved.Comment: Final version, 10 pages, no figures, Invited talk at the
international conference NEXT2003, 21-28 september 2003, Villasimius
(Cagliari), Ital
A note on bounded entropies
The aim of the paper is to study the link between non additivity of some
entropies and their boundedness. We propose an axiomatic construction of the
entropy relying on the fact that entropy belongs to a group isomorphic to the
usual additive group. This allows to show that the entropies that are additive
with respect to the addition of the group for independent random variables are
nonlinear transforms of the R\'enyi entropies, including the particular case of
the Shannon entropy. As a particular example, we study as a group a bounded
interval in which the addition is a generalization of the addition of
velocities in special relativity. We show that Tsallis-Havrda-Charvat entropy
is included in the family of entropies we define. Finally, a link is made
between the approach developed in the paper and the theory of deformed
logarithms.Comment: 10 pages, 1 figur
Group entropies, correlation laws and zeta functions
The notion of group entropy is proposed. It enables to unify and generalize
many different definitions of entropy known in the literature, as those of
Boltzmann-Gibbs, Tsallis, Abe and Kaniadakis. Other new entropic functionals
are presented, related to nontrivial correlation laws characterizing
universality classes of systems out of equilibrium, when the dynamics is weakly
chaotic. The associated thermostatistics are discussed. The mathematical
structure underlying our construction is that of formal group theory, which
provides the general structure of the correlations among particles and dictates
the associated entropic functionals. As an example of application, the role of
group entropies in information theory is illustrated and generalizations of the
Kullback-Leibler divergence are proposed. A new connection between statistical
mechanics and zeta functions is established. In particular, Tsallis entropy is
related to the classical Riemann zeta function.Comment: to appear in Physical Review
New Class of Generalized Extensive Entropies for Studying Dynamical Systems in Highly Anisotropic Phase Space
Starting from the geometrical interpretation of the R\'enyi entropy, we
introduce further extensive generalizations and study their properties. In
particular, we found the probability distribution function obtained by the
MaxEnt principle with generalized entropies. We prove that for a large class of
dynamical systems subject to random perturbations, including particle transport
in random media, these entropies play the role of Liapunov functionals. Some
physical examples, which can be treated by the generalized R\'enyi entropies
are also illustrated.Comment: 13 pages, 0 figure
Generalized entropies and logarithms and their duality relations
For statistical systems that violate one of the four Shannon-Khinchin axioms,
entropy takes a more general form than the Boltzmann-Gibbs entropy. The
framework of superstatistics allows one to formulate a maximum entropy
principle with these generalized entropies, making them useful for
understanding distribution functions of non-Markovian or non-ergodic complex
systems. For such systems where the composability axiom is violated there exist
only two ways to implement the maximum entropy principle, one using escort
probabilities, the other not. The two ways are connected through a duality.
Here we show that this duality fixes a unique escort probability, which allows
us to derive a complete theory of the generalized logarithms that naturally
arise from the violation of this axiom. We then show how the functional forms
of these generalized logarithms are related to the asymptotic scaling behavior
of the entropy.Comment: 4 pages, 1 page supporting informatio
A survey of uncertainty principles and some signal processing applications
The goal of this paper is to review the main trends in the domain of
uncertainty principles and localization, emphasize their mutual connections and
investigate practical consequences. The discussion is strongly oriented
towards, and motivated by signal processing problems, from which significant
advances have been made recently. Relations with sparse approximation and
coding problems are emphasized
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