60 research outputs found
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
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
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
Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive
Phase space can be constructed for equal and distinguishable subsystems
that could be (probabilistically) either {\it weakly} (or {\it "locally"})
correlated (e.g., independent, i.e., uncorrelated), or {\it strongly} (or {\it
globally}) correlated. If they are locally correlated, we expect the
Boltzmann-Gibbs entropy to be {\it
extensive}, i.e., for . In particular, if
they are independent, is {\it strictly additive}, i.e., . However, if the subsystems are globally correlated, we
expect, for a vast class of systems, the entropy (with ) for some special value of to be the
one which extensive (i.e., for ).Comment: 15 pages, including 9 figures and 8 Tables. The new version is
considerably enlarged with regard to the previous ones. New examples and new
references have been include
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
Numerical indications of a q-generalised central limit theorem
We provide numerical indications of the -generalised central limit theorem
that has been conjectured (Tsallis 2004) in nonextensive statistical mechanics.
We focus on binary random variables correlated in a {\it scale-invariant}
way. The correlations are introduced by imposing the Leibnitz rule on a
probability set based on the so-called -product with . We show
that, in the large limit (and after appropriate centering, rescaling, and
symmetrisation), the emerging distributions are -Gaussians, i.e., , with , and
with coefficients approaching finite values . The
particular case recovers the celebrated de Moivre-Laplace theorem.Comment: Minor improvements and corrections have been introduced in the new
version. 7 pages including 4 figure
Quasiclassical Coarse Graining and Thermodynamic Entropy
Our everyday descriptions of the universe are highly coarse-grained,
following only a tiny fraction of the variables necessary for a perfectly
fine-grained description. Coarse graining in classical physics is made natural
by our limited powers of observation and computation. But in the modern quantum
mechanics of closed systems, some measure of coarse graining is inescapable
because there are no non-trivial, probabilistic, fine-grained descriptions.
This essay explores the consequences of that fact. Quantum theory allows for
various coarse-grained descriptions some of which are mutually incompatible.
For most purposes, however, we are interested in the small subset of
``quasiclassical descriptions'' defined by ranges of values of averages over
small volumes of densities of conserved quantities such as energy and momentum
and approximately conserved quantities such as baryon number. The
near-conservation of these quasiclassical quantities results in approximate
decoherence, predictability, and local equilibrium, leading to closed sets of
equations of motion. In any description, information is sacrificed through the
coarse graining that yields decoherence and gives rise to probabilities for
histories. In quasiclassical descriptions, further information is sacrificed in
exhibiting the emergent regularities summarized by classical equations of
motion. An appropriate entropy measures the loss of information. For a
``quasiclassical realm'' this is connected with the usual thermodynamic entropy
as obtained from statistical mechanics. It was low for the initial state of our
universe and has been increasing since.Comment: 17 pages, 0 figures, revtex4, Dedicated to Rafael Sorkin on his 60th
birthday, minor correction
Escort mean values and the characterization of power-law-decaying probability densities
Escort mean values (or -moments) constitute useful theoretical tools for
describing basic features of some probability densities such as those which
asymptotically decay like {\it power laws}. They naturally appear in the study
of many complex dynamical systems, particularly those obeying nonextensive
statistical mechanics, a current generalization of the Boltzmann-Gibbs theory.
They recover standard mean values (or moments) for . Here we discuss the
characterization of a (non-negative) probability density by a suitable set of
all its escort mean values together with the set of all associated normalizing
quantities, provided that all of them converge. This opens the door to a
natural extension of the well known characterization, for the instance,
of a distribution in terms of the standard moments, provided that {\it all} of
them have {\it finite} values. This question would be specially relevant in
connection with probability densities having {\it divergent} values for all
nonvanishing standard moments higher than a given one (e.g., probability
densities asymptotically decaying as power-laws), for which the standard
approach is not applicable. The Cauchy-Lorentz distribution, whose second and
higher even order moments diverge, constitutes a simple illustration of the
interest of this investigation. In this context, we also address some
mathematical subtleties with the aim of clarifying some aspects of an
interesting non-linear generalization of the Fourier Transform, namely, the
so-called -Fourier Transform.Comment: 20 pages (2 Appendices have been added
Linear instability and statistical laws of physics
We show that a meaningful statistical description is possible in conservative
and mixing systems with zero Lyapunov exponent in which the dynamical
instability is only linear in time. More specifically, (i) the sensitivity to
initial conditions is given by with
; (ii) the statistical entropy in the infinitely fine graining limit (i.e., {\it
number of cells into which the phase space has been partitioned} ),
increases linearly with time only for ; (iii) a nontrivial,
-generalized, Pesin-like identity is satisfied, namely the . These facts (which are
in analogy to the usual behaviour of strongly chaotic systems with ), seem
to open the door for a statistical description of conservative many-body
nonlinear systems whose Lyapunov spectrum vanishes.Comment: 7 pages including 2 figures. The present version is accepted for
publication in Europhysics Letter
Functional-differential equations for %-transforms of -Gaussians
In the paper the question - Is the q-Fourier transform of a q-Gaussian a
q'-Gaussian (with some q') up to a constant factor? - is studied for the whole
range of . This question is connected with applicability of
the q-Fourier transform in the study of limit processes in nonextensive
statistical mechanics. We prove that the answer is affirmative if and only if q
> 1, excluding two particular cases of q<1, namely, q = 1/2 and q = 2/3, which
are also out of the theory valid for q \ge 1. We also discuss some applications
of the q-Fourier transform to nonlinear partial differential equations such as
the porous medium equation.Comment: 14 pages A new section on a related solution of the porous medium
equation in comparison with the previous version has been introduc
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