1,113 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 $q$-sum $x \oplus_q y \equiv x+y+(1-q) xy$ ($x \oplus_1 y=x+y$) and the
$q$-product $x\otimes_q y \equiv [x^{1-q} +y^{1-q}-1]^{\frac{1}{1-q}}$
($x\otimes_1 y=x y$) emerge naturally within nonextensive statistical
mechanics. We show here how they lead to two-parameter (namely, $q$ and
$q^\prime$) generalizations of the logarithmic and exponential functions (noted
respectively $\ln_{q,q^\prime}x$ and $e_{q,q^\prime}^{x}$), as well as of the
Boltzmann-Gibbs-Shannon entropy $S_{BGS}\equiv -k \sum_{i=1}^Wp_i \ln p_i$
(noted $S_{q,q^\prime}$). The remarkable properties of the
$(q,q^\prime)$-generalized logarithmic function make the entropic form
$S_{q,q^\prime} \equiv k \sum_{i=1}^W p_i \ln_{q,q^\prime}(1/p_i)$ to satisfy,
for large regions of $(q,q^\prime)$, 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 $N$ 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
$\lambda_{q}$ and the rate of entropy production $K_{q}$ given by the
nonextensive statistical mechanics. Our results advocate the validity of the
$q$-generalized Pesin identity at critical points of one-dimensional nonlinear
dissipative maps.Comment: Revtex, 5 pages, 3 figure

### Strictly and asymptotically scale-invariant probabilistic models of $N$ correlated binary random variables having {\em q}--Gaussians as $N\to \infty$ limiting distributions

In order to physically enlighten the relationship between {\it
$q$--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 $\nu=1,2,3,...$,
unifying the Leibnitz triangle ($\nu=1$) and the case of independent variables
($\nu\to\infty$); (ii) two slightly different discretizations of
$q$--Gaussians; (iii) a special family, characterized by the parameter $\chi$,
which generalizes the usual case of independent variables (recovered for
$\chi=1/2$). Models (i) and (iii) are in fact strictly scale-invariant. For
models (i), we analytically show that the $N \to\infty$ probability
distribution is a $q$--Gaussian with $q=(\nu -2)/(\nu-1)$. Models (ii) approach
$q$--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} $q$--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)
$q$--independence, which, in turn, mandates $q$--Gaussian attractors.Comment: The present version is accepted for publication in JSTA

### 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

### Risk aversion in economic transactions

Most people are risk-averse (risk-seeking) when they expect to gain (lose).
Based on a generalization of ``expected utility theory'' which takes this into
account, we introduce an automaton mimicking the dynamics of economic
operations. Each operator is characterized by a parameter q which gauges
people's attitude under risky choices; this index q is in fact the entropic one
which plays a central role in nonextensive statistical mechanics. Different
long term patterns of average asset redistribution are observed according to
the distribution of parameter q (chosen once for ever for each operator) and
the rules (e.g., the probabilities involved in the gamble and the indebtedness
restrictions) governing the values that are exchanged in the transactions.
Analytical and numerical results are discussed in terms of how the sensitivity
to risk affects the dynamics of economic transactions.Comment: 4 PS figures, to appear in Europhys. Let

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