arXiv subjects

Constantino Tsallis

Publications and source records attributed to Constantino Tsallis.

At least 109 records · Page 6Linked to original sources

arXiv2010

A generalization of the cumulant expansion. Application to a scale-invariant probabilistic model

As well known, cumulant expansion is an alternative way to moment expansion to fully characterize probability distributions provided all the moments exist. If this is not the case, the so called escort mean values (or q-moments) have been proposed to characterize probability densities with divergent moments [C. Tsallis et al, J. Math. Phys 50, 043303 (2009)]. We introduce here a new mathematical object, namely the q-cumulants, which, in analogy to the cumulants, provide an alternative characterization to that of the q-moments for the probability densities. We illustrate this new scheme on a recently proposed family of scale-invariant discrete probabilistic models [A. Rodriguez et al, J. Stat. Mech. (2008) P09006; R. Hanel et al, Eur. Phys. J. B 72, 263268 (2009)] having q-Gaussians as limiting probability distributions.

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arXiv2010

Generalization of symmetric $α$-stable Lévy distributions for $q>1$

The $α$-stable distributions introduced by Lévy play an important role in probabilistic theoretical studies and their various applications, e.g., in statistical physics, life sciences, and economics. In the present paper we study sequences of long-range dependent random variables whose distributions have asymptotic power law decay, and which are called $(q,α)$-stable distributions. These sequences are generalizations of i.i.d. $α$-stable distributions, and have not been previously studied. Long-range dependent $(q,α)$-stable distributions might arise in the description of anomalous processes in nonextensive statistical mechanics, cell biology, finance. The parameter $q$ controls dependence. If $q=1$ then they are classical i.i.d. with $α$-stable Lévy distributions. In the present paper we establish basic properties of $(q,α)$-stable distributions, and generalize the result of Umarov, Tsallis and Steinberg (2008), where the particular case $α=2, q\in [1,3),$ was considered, to the whole range of stability and nonextensivity parameters $α\in (0,2]$ and $q \in [1,3),$ respectively. We also discuss possible further extensions of the results that we obtain, and formulate some conjectures.

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arXiv2009

Nonadditive entropy and nonextensive statistical mechanics - Some central concepts and recent applications

We briefly review central concepts concerning nonextensive statistical mechanics, based on the nonadditive entropy $S_q=k\frac{1-\sum_{i}p_i^q}{q-1} (q \in {\cal R}; S_1=-k\sum_{i}p_i \ln p_i)$. Among others, we focus on possible realizations of the $q$-generalized Central Limit Theorem, including at the edge of chaos of the logistic map, and for quasi-stationary states of many-body long-range-interacting Hamiltonian systems.

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arXiv2009

Limit distributions of scale-invariant probabilistic models of correlated random variables with the q-Gaussian as an explicit example

Extremization of the Boltzmann-Gibbs (BG) entropy under appropriate norm and width constraints yields the Gaussian distribution. Also, the basic solutions of the standard Fokker-Planck (FP) equation (related to the Langevin equation with additive noise), as well as the Central Limit Theorem attractors, are Gaussians. The simplest stochastic model with such features is N to infinity independent binary random variables, as first proved by de Moivre and Laplace. What happens for strongly correlated random variables? Such correlations are often present in physical situations as e.g. systems with long range interactions or memory. Frequently q-Gaussians become observed. This is typically so if the Langevin equation includes multiplicative noise, or the FP equation to be nonlinear. Scale-invariance, i.e. exchangeable binary stochastic processes, allow a systematical analysis of the relation between correlations and non-Gaussian distributions. In particular, a generalized stochastic model yielding q-Gaussians for all q (including q>1) was missing. This is achieved here by using the Laplace-de Finetti representation theorem, which embodies strict scale-invariance of interchangeable random variables. We demonstrate that strict scale invariance together with q-Gaussianity mandates the associated extensive entropy to be BG.

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arXiv2009

Comment on ``Proposed central limit behavior in deterministic dynamical systems''

In a recent Brief Report [Phys. Rev. E 79 (2009) 057201], Grassberger re-investigates probability densities of sums of iterates of the logistic map near the critical point and claims that his simulation results are inconsistent with previous results obtained by us [U. Tirnakli et al., Phys. Rev. E 75 (2007) 040106(R) and Phys Rev. E 79 (2009) 056209]. In this comment we point out several errors in Grassberger's paper. We clarify that Grassberger's numerical simulations were mainly performed in a parameter region that was explicitly excluded in our 2009 paper and that his number of iterations is insufficient for the region chosen. We also show that, contrary to what is claimed by the author, (i) Levy distributions are irrelevant for this problem, and that (ii) the probability distributions of sums that focus on transients are unlikely to be universal.

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arXiv2009

Nonextensivity at the edge of chaos of a new universality class of one-dimensional unimodal dissipative maps

We introduce a new universality class of one-dimensional unimodal dissipative maps. The new family, from now on referred to as the ($z_1,z_2$)-{\it logarithmic map}, corresponds to a generalization of the $z$-logistic map. The Feigenbaum-like constants of these maps are determined. It has been recently shown that the probability density of sums of iterates at the edge of chaos of the $z$-logistic map is numerically consistent with a $q$-Gaussian, the distribution which, under appropriate constraints, optimizes the nonadditive entropy $S_q$. We focus here on the presently generalized maps to check whether they constitute a new universality class with regard to $q$-Gaussian attractor distributions. We also study the generalized $q$-entropy production per unit time on the new unimodal dissipative maps, both for strong and weak chaotic cases. The $q$-sensitivity indices are obtained as well. Our results are, like those for the $z$-logistic maps, numerically compatible with the $q$-generalization of a Pesin-like identity for ensemble averages.

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arXiv2008

Nonadditive entropy: the concept and its use

The entropic form $S_q$ is, for any $q \neq 1$, {\it nonadditive}. Indeed, for two probabilistically independent subsystems, it satisfies $S_q(A+B)/k=[S_q(A)/k]+[S_q(B)/k]+(1-q)[S_q(A)/k][S_q(B)/k] \ne S_q(A)/k+S_q(B)/k$. This form will turn out to be {\it extensive} for an important class of nonlocal correlations, if $q$ is set equal to a special value different from unity, noted $q_{ent}$ (where $ent$ stands for $entropy$). In other words, for such systems, we verify that $S_{q_{ent}}(N) \propto N (N>>1)$, thus legitimating the use of the classical thermodynamical relations. Standard systems, for which $S_{BG}$ is extensive, obviously correspond to $q_{ent}=1$. Quite complex systems exist in the sense that, for them, no value of $q$ exists such that $S_q$ is extensive. Such systems are out of the present scope: they might need forms of entropy different from $S_q$, or perhaps -- more plainly -- they are just not susceptible at all for some sort of thermostatistical approach. Consistently with the results associated with $S_q$, the $q$-generalizations of the Central Limit Theorem and of its extended Lévy-Gnedenko form have been achieved. These recent theorems could of course be the cause of the ubiquity of $q$-exponentials, $q$-Gaussians and related mathematical forms in natural, artificial and social systems. All of the above, as well as presently available experimental, observational and computational confirmations -- in high energy physics and elsewhere --, are briefly reviewed. Finally, we address a confusion which is quite common in the literature, namely referring to distinct physical mechanisms {\it versus} distinct regimes of a single physical mechanism.

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arXiv2008

On the robustness of q-expectation values and Renyi entropy

We study the robustness of functionals of probability distributions such as the Rényi and nonadditive S_q entropies, as well as the q-expectation values under small variations of the distributions. We focus on three important types of distribution functions, namely (i) continuous bounded (ii) discrete with finite number of states, and (iii) discrete with infinite number of states. The physical concept of robustness is contrasted with the mathematically stronger condition of stability and Lesche-stability for functionals. We explicitly demonstrate that, in the case of continuous distributions, once unbounded distributions and those leading to negative entropy are excluded, both Renyi and nonadditive S_q entropies as well as the q-expectation values are robust. For the discrete finite case, the Renyi and nonadditive S_q entropies and the q-expectation values are robust. For the infinite discrete case, where both Renyi entropy and q-expectations are known to violate Lesche-stability and stability respectively, we show that one can nevertheless state conditions which guarantee physical robustness.

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arXiv2008

A closer look at time averages of the logistic map at the edge of chaos

The probability distribution of sums of iterates of the logistic map at the edge of chaos has been recently shown [see U. Tirnakli, C. Beck and C. Tsallis, Phys. Rev. E 75, 040106(R) (2007)] to be numerically consistent with a q-Gaussian, the distribution which, under appropriate constraints, maximizes the nonadditive entropy S_q, the basis of nonextensive statistical mechanics. This analysis was based on a study of the tails of the distribution. We now check the entire distribution, in particular its central part. This is important in view of a recent q-generalization of the Central Limit Theorem, which states that for certain classes of strongly correlated random variables the rescaled sum approaches a q-Gaussian limit distribution. We numerically investigate for the logistic map with a parameter in a small vicinity of the critical point under which conditions there is convergence to a q-Gaussian both in the central region and in the tail region, and find a scaling law involving the Feigenbaum constant delta. Our results are consistent with a large number of already available analytical and numerical evidences that the edge of chaos is well described in terms of the entropy S_q and its associated concepts.

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arXiv2008

Escort mean values and the characterization of power-law-decaying probability densities

Escort mean values (or $q$-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 $q=1$. 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 $q=1$ 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 $q$-Fourier Transform.

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arXiv2008

Nonadditive entropy reconciles the area law in quantum systems with classical thermodynamics

The Boltzmann-Gibbs-von Neumann entropy of a large part (of linear size L) of some (much larger) d-dimensional quantum systems follows the so-called area law (as for black holes), i.e., it is proportional to $L^{d-1}$. Here we show, for d=1,2, that the (nonadditive) entropy S_q satisfies, for a special value of $q \neq 1$, the classical thermodynamical prescription for the entropy to be extensive, i.e., $S_q \propto L^d$. Therefore, we reconcile with classical thermodynamics the area law widespread in quantum systems. Recently, a similar behavior was exhibited, by M. Gell-Mann, Y. Sato and one of us (C.T.), in mathematical models with scale-invariant correlations. Finally, we find that the system critical features are marked by a maximum of the special entropic index q.

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arXiv2008

Symmetric $(q,α)$-Stable Distributions. Part II: Second Representation

This paper is a continuation of papers \cite{UmarovTsallisSteinberg,UmarovTsallisGellmannSteinberg}. In Part I \cite{UmarovTsallisGellmannSteinberg} a description (representation) of $(q,α)$-stable distributions based on a $F_q$-transform was given. Here, in Part II, we present another description of these distributions. This approach generalizes results of \cite{UmarovTsallisSteinberg} (which corresponds to $α=2, Q\in [1,3)$) to the whole range of stability and nonextensivity parameters $α\in (0,2]$ and $Q \in [1,3),$ respectively. The present case $α=2$ recovers the $q$-Gaussian distributions. Similar to what is discussed in \cite{UmarovTsallisSteinberg}, a triplet $(q^{\ast},q,q_{\ast})$ arises for which the mapping $F_{q^{\ast}}: \mathcal{G}_{q} \to \mathcal{G}_{q_{\ast}}$ holds. Moreover, by unifying the two preceding descriptions, further possible extensions are discussed and some conjectures are formulated.

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arXiv2008

Symmetric $(q,α)$-Stable Distributions. Part I: First Representation

The classic central limit theorem and $α$-stable distributions play a key role in probability theory, and also in Boltzmann-Gibbs (BG) statistical mechanics. They both concern the paradigmatic case of probabilistic independence of the random variables that are being summed. A generalization of the BG theory, usually referred to as nonextensive statistical mechanics and characterized by the index $q$ ($q=1$ recovers the BG theory), introduces special (long range) correlations between the random variables, and recovers independence for $q=1$. Recently, a $q$-central limit theorem consistent with nonextensive statistical mechanics was established \cite{UmarovTsallisSteinberg} which generalizes the classic Central Limit Theorem. In the present paper we introduce and study symmetric $(q,α)$-stable distributions. The case $q=1$ recovers the Lévy $α$-stable distributions.

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arXiv2008

q-Gaussians in the porous-medium equation: stability and time evolution

The stability of $q$-Gaussian distributions as particular solutions of the linear diffusion equation and its generalized nonlinear form, $\pderiv{P(x,t)}{t} = D \pderiv{^2 [P(x,t)]^{2-q}}{x^2}$, the \emph{porous-medium equation}, is investigated through both numerical and analytical approaches. It is shown that an \emph{initial} $q$-Gaussian, characterized by an index $q_i$, approaches the \emph{final}, asymptotic solution, characterized by an index $q$, in such a way that the relaxation rule for the kurtosis evolves in time according to a $q$-exponential, with a \emph{relaxation} index $q_{\rm rel} \equiv q_{\rm rel}(q)$. In some cases, particularly when one attempts to transform an infinite-variance distribution ($q_i \ge 5/3$) into a finite-variance one ($q<5/3$), the relaxation towards the asymptotic solution may occur very slowly in time. This fact might shed some light on the slow relaxation, for some long-range-interacting many-body Hamiltonian systems, from long-standing quasi-stationary states to the ultimate thermal equilibrium state.

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arXiv2008

T. Dauxois' "Non-Gaussian distributions under scrutiny" under scrutiny

A recent paper by T. Dauxois entitled "Non-Gaussian distributions under scrutiny" is submitted to scrutiny. Several comments on its content are made, which constitute, at the same time, a brief state-of-the-art review of nonextensive statistical mechanics, a current generalization of the Boltzmann-Gibbs theory. Some inadvertences and misleading sentences are pointed out as well.

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arXiv2008

On a representation of the inverse Fq transform

A recent generalization of the Central Limit Theorem consistent with nonextensive statistical mechanics has been recently achieved through a generalized Fourier transform, noted $q$-Fourier transform. A representation formula for the inverse $q$-Fourier transform is here obtained in the class of functions $\mathcal{G}=\bigcup_{1\le q<3}\mathcal{G}_q,$ where $\mathcal{G}_{q}=\{f = a e_{q}^{-βx2}, \, a>0, \, β>0 \}$. This constitutes a first step towards a general representation of the inverse $q$-Fourier operation, which would enable interesting physical and other applications.

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arXiv2007

Extensive nonadditive entropy in quantum spin chains

We present details on a physical realization, in a many-body Hamiltonian system, of the abstract probabilistic structure recently exhibited by Gell-Mann, Sato and one of us (C.T.), that the nonadditive entropy $S_q=k [1- Tr \hatρ^q]/[q-1]$ ($\hatρ\equiv$ density matrix; $S_1=-k Tr \hatρ \ln \hatρ$) can conform, for an anomalous value of q (i.e., q not equal to 1), to the classical thermodynamical requirement for the entropy to be extensive. Moreover, we find that the entropic index q provides a tool to characterize both universal and nonuniversal aspects in quantum phase transitions (e.g., for a L-sized block of the Ising ferromagnetic chain at its T=0 critical transverse field, we obtain $\lim_{L\to\infty}S_{\sqrt{37}-6}(L)/L=3.56 \pm 0.03$). The present results suggest a new and powerful approach to measure entanglement in quantum many-body systems. At the light of these results, and similar ones for a d=2 Bosonic system discussed by us elsewhere, we conjecture that, for blocks of linear size L of a large class of Fermionic and Bosonic d-dimensional many-body Hamiltonians with short-range interaction at T=0, we have that the additive entropy $S_1(L) \propto [L^{d-1}-1]/(d-1)$ (i.e., $ \ln L$ for $d=1$, and $ L^{d-1}$ for d>1), hence it is not extensive, whereas, for anomalous values of the index q, we have that the nonadditive entropy $S_q(L)\propto L^d$ ($\forall d$), i.e., it is extensive. The present discussion neatly illustrates that entropic additivity and entropic extensivity are quite different properties, even if they essentially coincide in the presence of short-range correlations.

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arXiv2007

A generalization of the central limit theorem consistent with nonextensive statistical mechanics

The standard central limit theorem plays a fundamental role in Boltzmann-Gibbs statistical mechanics. This important physical theory has been generalized \cite{Tsallis1988} in 1988 by using the entropy $S_q = \frac{1-\sum_i p_i^q}{q-1}$ (with $q \in \mathcal{R}$) instead of its particular BG case $S_1=S_{BG}= -\sum_i p_i \ln p_i$. The theory which emerges is usually referred to as {\it nonextensive statistical mechanics} and recovers the standard theory for $q=1$. During the last two decades, this $q$-generalized statistical mechanics has been successfully applied to a considerable amount of physically interesting complex phenomena. A conjecture\cite{Tsallis2005} and numerical indications available in the literature have been, for a few years, suggesting the possibility of $q$-versions of the standard central limit theorem by allowing the random variables that are being summed to be strongly correlated in some special manner, the case $q=1$ corresponding to standard probabilistic independence. This is what we prove in the present paper for $1 \leq q<3$. The attractor, in the usual sense of a central limit theorem, is given by a distribution of the form $p(x) =C_q [1-(1-q) βx^2]^{1/(1-q)}$ with $β>0$, and normalizing constant $C_q$. These distributions, sometimes referred to as $q$-Gaussians, are known to make, under appropriate constraints, extremal the functional $S_q$ (in its continuous version). Their $q=1$ and $q=2$ particular cases recover respectively Gaussian and Cauchy distributions.

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