1,116 research outputs found
Nonextensive statistics: Theoretical, experimental and computational evidences and connections
The domain of validity of standard thermodynamics and Boltzmann-Gibbs
statistical mechanics is discussed and then formally enlarged in order to
hopefully cover a variety of anomalous systems. The generalization concerns
{\it nonextensive} systems, where nonextensivity is understood in the
thermodynamical sense. This generalization was first proposed in 1988 inspired
by the probabilistic description of multifractal geometries, and has been
intensively studied during this decade. In the present effort, after
introducing some historical background, we briefly describe the formalism, and
then exhibit the present status in what concerns theoretical, experimental and
computational evidences and connections, as well as some perspectives for the
future. In addition to these, here and there we point out various (possibly)
relevant questions, whose answer would certainly clarify our current
understanding of the foundations of statistical mechanics and its
thermodynamical implicationsComment: 15 figure
Two-parameter generalization of the logarithm and exponential functions and Boltzmann-Gibbs-Shannon entropy
The -sum () and the
-product
() emerge naturally within nonextensive statistical
mechanics. We show here how they lead to two-parameter (namely, and
) generalizations of the logarithmic and exponential functions (noted
respectively and ), as well as of the
Boltzmann-Gibbs-Shannon entropy
(noted ). The remarkable properties of the
-generalized logarithmic function make the entropic form
to satisfy,
for large regions of , important properties such as {\it
expansibility}, {\it concavity} and {\it Lesche-stability}, but not necessarily
{\it composability}.Comment: 9 pages, 4 figure
A note on q-Gaussians and non-Gaussians in statistical mechanics
The sum of sufficiently strongly correlated random variables will not in
general be Gaussian distributed in the limit N\to\infty. We revisit examples of
sums x that have recently been put forward as instances of variables obeying a
q-Gaussian law, that is, one of type (cst)\times[1-(1-q)x^2]^{1/(1-q)}. We show
by explicit calculation that the probability distributions in the examples are
actually analytically different from q-Gaussians, in spite of numerically
resembling them very closely. Although q-Gaussians exhibit many interesting
properties, the examples investigated do not support the idea that they play a
special role as limit distributions of correlated sums.Comment: 17 pages including 3 figures. Introduction and references expande
Note on a q-modified central limit theorem
A q-modified version of the central limit theorem due to Umarov et al.
affirms that q-Gaussians are attractors under addition and rescaling of certain
classes of strongly correlated random variables. The proof of this theorem
rests on a nonlinear q-modified Fourier transform. By exhibiting an invariance
property we show that this Fourier transform does not have an inverse. As a
consequence, the theorem falls short of achieving its stated goal.Comment: 10 pages, no figure
Nonextensive Pesin identity. Exact renormalization group analytical results for the dynamics at the edge of chaos of the logistic map
We show that the dynamical and entropic properties at the chaos threshold of
the logistic map are naturally linked through the nonextensive expressions for
the sensitivity to initial conditions and for the entropy. We corroborate
analytically, with the use of the Feigenbaum renormalization group(RG)
transformation, the equality between the generalized Lyapunov coefficient
and the rate of entropy production given by the
nonextensive statistical mechanics. Our results advocate the validity of the
-generalized Pesin identity at critical points of one-dimensional nonlinear
dissipative maps.Comment: Revtex, 5 pages, 3 figure
Derivation of Tsallis statistics from dynamical equations for a granular gas
In this work we present the explicit calculation of Probability Distribution
Function for a model system of granular gas within the framework of Tsallis
Non-Extensive Statistical Mechanics, using the stochastic approach by Beck [C.
Beck, Phys. Rev. Lett. 87, 180601 (2001)], further generalized by Sattin and
Salasnich [F. Sattin and L. Salasnich, Phys. Rev. E 65, 035106(R) (2002)]. The
calculation is self-consistent in that the form of Probability Distribution
Function is not given as an ansatz but is shown to necessarily arise from the
known microscopic dynamics of the system.Comment: 14 pages. An appendix adde
Is Tsallis thermodynamics nonextensive?
Within the Tsallis thermodynamics' framework, and using scaling properties of
the entropy, we derive a generalization of the Gibbs-Duhem equation. The
analysis suggests a transformation of variables that allows standard
thermodynamics to be recovered. Moreover, we also generalize Einstein's formula
for the probability of a fluctuation to occur by means of the maximum
statistical entropy method. The use of the proposed transformation of variables
also shows that fluctuations within Tsallis statistics can be mapped to those
of standard statistical mechanics.Comment: 4 pages, no figures, revised version, new title, accepted in PR
Strictly and asymptotically scale-invariant probabilistic models of correlated binary random variables having {\em q}--Gaussians as limiting distributions
In order to physically enlighten the relationship between {\it
--independence} and {\it scale-invariance}, we introduce three types of
asymptotically scale-invariant probabilistic models with binary random
variables, namely (i) a family, characterized by an index ,
unifying the Leibnitz triangle () and the case of independent variables
(); (ii) two slightly different discretizations of
--Gaussians; (iii) a special family, characterized by the parameter ,
which generalizes the usual case of independent variables (recovered for
). Models (i) and (iii) are in fact strictly scale-invariant. For
models (i), we analytically show that the probability
distribution is a --Gaussian with . Models (ii) approach
--Gaussians by construction, and we numerically show that they do so with
asymptotic scale-invariance. Models (iii), like two other strictly
scale-invariant models recently discussed by Hilhorst and Schehr (2007),
approach instead limiting distributions which are {\it not} --Gaussians. The
scenario which emerges is that asymptotic (or even strict) scale-invariance is
not sufficient but it might be necessary for having strict (or asymptotic)
--independence, which, in turn, mandates --Gaussian attractors.Comment: The present version is accepted for publication in JSTA
Nonextensive aspects of self-organized scale-free gas-like networks
We explore the possibility to interpret as a 'gas' the dynamical
self-organized scale-free network recently introduced by Kim et al (2005). The
role of 'momentum' of individual nodes is played by the degree of the node, the
'configuration space' (metric defining distance between nodes) being determined
by the dynamically evolving adjacency matrix. In a constant-size network
process, 'inelastic' interactions occur between pairs of nodes, which are
realized by the merger of a pair of two nodes into one. The resulting node
possesses the union of all links of the previously separate nodes. We consider
chemostat conditions, i.e., for each merger there will be a newly created node
which is then linked to the existing network randomly. We also introduce an
interaction 'potential' (node-merging probability) which decays with distance
d_ij as 1/d_ij^alpha; alpha >= 0). We numerically exhibit that this system
exhibits nonextensive statistics in the degree distribution, and calculate how
the entropic index q depends on alpha. The particular cases alpha=0 and alpha
to infinity recover the two models introduced by Kim et al.Comment: 7 pages, 5 figure
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