arXiv subjects

C. Tsallis

Publications and source records attributed to C. Tsallis.

At least 19 recordsLinked to original sources

arXiv2024

Dynamics of Cities

This study investigates city dynamics employing a nonextensive diffusion equation suited for addressing diffusion within a fractal medium, where the nonadditive parameter, $q$, plays a relevant role. The findings demonstrate the efficacy of this approach in determining the relation between the fractal dimension of the city, the allometric exponent and $q$, and elucidating the stationary phase of urban evolution. The dynamic methodology facilitates the correlation of the fractal dimension with both the entropic index and the urban scaling exponent identified in data analyses. The results reveal that the scaling behaviour observed in cities aligns with the fractal dimension measured through independent methods. Moreover, the interpretation of these findings underscores the intimate connection between the fractal dimension and social interactions within the urban context. This research contributes to a deeper comprehension of the intricate interplay between human behaviour, urban dynamics, and the underlying fractal nature of cities.

physics.soc-ph↗

arXiv2022

Thermodynamically consistent entropic late-time cosmological acceleration

Entropic-force cosmology provides, in contrast with dark energy descriptions, a concrete physical understanding of the accelerated expansion of the universe. The acceleration appears to be a consequence of the entropy associated with the information storage in the universe. Since these cosmological models are unable of explaining the different periods of acceleration and deceleration unless a correction term is considered, we study the effects of including a subdominant power-law term within a thermodynamically admissible entropic-force model. The temperature of the universe horizon is obtained by a clear physical principle, i.e., requiring that the Legendre structure of thermodynamics is preserved. We analyze the various types of behaviors, and we compare the performance of thermodynamically consistent entropic-force models with regard to available supernovae data by providing appropriate constraints for optimizing alternative entropies and temperatures of the Hubble screen. The novelty of our work is that the analysis is based on a entropy scaling with an arbitrary power of the Hubble radius, instead of a specific entropy. This allows us to conclude on various models at once, compare them, and conserve the scaling exponent as a parameter to be fitted with observational data, thus providing information about the form of the actual cosmological entropy and temperature. We show that the introduced correction term is capable of explaining different periods of acceleration and deceleration in the late-time universe.

gr-qc↗

arXiv2022

Thermodynamically consistent entropic-force cosmology

We analyze the thermodynamical consistency of entropic-force cosmological models. Our analysis is based on a generalized entropy scaling with an arbitrary power of the Hubble radius. The Bekenstein-Hawking entropy, proportional to the area, and the nonadditive $S_{δ=3/2}$-entropy, proportional to the volume, are particular cases. One of the points to be solved by entropic-force cosmology for being taken as a serious alternative to mainstream cosmology is to provide a physical principle that points out what entropy and temperature have to be used. We determine the temperature of the universe horizon by requiring that the Legendre structure of thermodynamics is preserved. We compare the performance of thermodynamically consistent entropic-force models with regard to the available supernovae data by providing appropriate constraints for optimizing alternative entropies and temperatures of the Hubble screen. Our results point out that the temperature differs from the Hawking one.

gr-qc↗

arXiv2016

Curl Forces and the Nonlinear Fokker-Planck Equation

Nonlinear Fokker-Planck equations endowed with curl drift forces are investigated. The conditions under which these evolution equations admit stationary solutions, which are $q$-exponentials of an appropriate potential function, are determined. It is proved that when these stationary solutions exist, the nonlinear Fokker-Planck equations satisfy an $H$-theorem in terms of a free-energy like quantity involving the $S_q$ entropy. A particular two dimensional model admitting analytical, time-dependent, $q$-Gaussian solutions is discussed in detail. This model describes a system of particles with short-range interactions, performing overdamped motion under drag effects, due to a rotating resisting medium. It is related to models that have been recently applied to the study of type-II superconductors. The relevance of the present developments to the study of complex systems in physics, astronomy, and biology, is discussed.

cond-mat.stat-mech↗

arXiv2015

Dissipative Effects in Nonlinear Klein-Gordon Dynamics

We consider dissipation in a recently proposed nonlinear Klein-Gordon dynamics that admits soliton-like solutions of the power-law form $e_q^{i(kx-wt)}$, involving the $q$-exponential function naturally arising within the nonextensive thermostatistics [$e_q^z \equiv [1+(1-q)z]^{1/(1-q)}$, with $e_1^z=e^z$]. These basic solutions behave like free particles, complying, for all values of $q$, with the de Broglie-Einstein relations $p=\hbar k$, $E=\hbar ω$ and satisfying a dispersion law corresponding to the relativistic energy-momentum relation $E^2 = c^2p^2 + m^2c^4 $. The dissipative effects explored here are described by an evolution equation that can be regarded as a nonlinear version of the celebrated telegraphists equation, unifying within one single theoretical framework the nonlinear Klein-Gordon equation, a nonlinear Schroedinger equation, and the power-law diffusion (porous media) equation. The associated dynamics exhibits physically appealing soliton-like traveling solutions of the $q$-plane wave form with a complex frequency $ω$ and a $q$-Gaussian square modulus profile.

cond-mat.stat-mech↗

arXiv2011

q-Moments remove the degeneracy associated with the inversion of the q-Fourier transform

It was recently proven [Hilhorst, JSTAT, P10023 (2010)] that the q-generalization of the Fourier transform is not invertible in the full space of probability density functions for q > 1. It has also been recently shown that this complication disappears if we dispose of the q-Fourier transform not only of the function itself, but also of all of its shifts [Jauregui and Tsallis, Phys. Lett. A 375, 2085 (2011)]. Here we show that another road exists for completely removing the degeneracy associated with the inversion of the q-Fourier transform of a given probability density function. Indeed, it is possible to determine this density if we dispose of some extra information related to its q-moments.

math-ph↗

arXiv2011

Time--Evolving Statistics of Chaotic Orbits of Conservative Maps in the Context of the Central Limit Theorem

We study chaotic orbits of conservative low--dimensional maps and present numerical results showing that the probability density functions (pdfs) of the sum of $N$ iterates in the large $N$ limit exhibit very interesting time-evolving statistics. In some cases where the chaotic layers are thin and the (positive) maximal Lyapunov exponent is small, long--lasting quasi--stationary states (QSS) are found, whose pdfs appear to converge to $q$--Gaussians associated with nonextensive statistical mechanics. More generally, however, as $N$ increases, the pdfs describe a sequence of QSS that pass from a $q$--Gaussian to an exponential shape and ultimately tend to a true Gaussian, as orbits diffuse to larger chaotic domains and the phase space dynamics becomes more uniformly ergodic.

nlin.CD↗

arXiv2011

q-Generalization of the inverse Fourier transform

A wide class of physical distributions appears to follow the q-Gaussian form, which plays the role of attractor according to a Central Limit Theorem generalized in the presence of specific correlations between the relevant random variables. In the realm of this theorem, a q-generalized Fourier transform plays an important role. We introduce here a method which univocally determines a distribution from the knowledge of its q-Fourier transform and some supplementary information. This procedure involves a recently q-generalized Dirac delta and the class of functions on which it acts. The present method conveniently extends the inverse of the standard Fourier transform, and is therefore expected to be very useful in the study of many complex systems.

math-ph↗

arXiv2010

Pathway Model and Nonextensive Statistical Mechanics

The established technique of eliminating upper or lower parameters in a general hypergeometric series is profitably exploited to create pathways among confluent hypergeometric functions, binomial functions, Bessel functions, and exponential series. One such pathway, from the mathematical statistics point of view, results in distributions which naturally emerge within nonextensive statistical mechanics and Beck-Cohen superstatistics, as pursued in generalizations of Boltzmann-Gibbs statistics.

cond-mat.stat-mech↗

arXiv2010

New representations of pi and Dirac delta using the nonextensive-statistical-mechanics q-exponential function

We present a generalization of the representation in plane waves of Dirac delta, $δ(x)=(1/2π)\int_{-\infty}^\infty e^{-ikx}\,dk$, namely $δ(x)=(2-q)/(2π)\int_{-\infty}^\infty e_q^{-ikx}\,dk$, using the nonextensive-statistical-mechanics $q$-exponential function, $e_q^{ix}\equiv[1+(1-q)ix]^{1/(1-q)}$ with $e_1^{ix}\equiv e^{ix}$, being $x$ any real number, for real values of $q$ within the interval $[1,2[$. Concomitantly with the development of these new representations of Dirac delta, we also present two new families of representations of the transcendental number $π$. Incidentally, we remark that the $q$-plane wave form which emerges, namely $e_q^{ikx}$, is normalizable for $1<q<3$, in contrast with the standard one, $e^{ikx}$, which is not.

math-ph↗

arXiv2010

Time evolution towards q-Gaussian stationary states through unified Ito-Stratonovich stochastic equation

We consider a class of single-particle one-dimensional stochastic equations which include external field, additive and multiplicative noises. We use a parameter $θ\in [0,1]$ which enables the unification of the traditional Itô and Stratonovich approaches, now recovered respectively as the $θ=0$ and $θ=1/2$ particular cases to derive the associated Fokker-Planck equation (FPE). These FPE is a {\it linear} one, and its stationary state is given by a $q$-Gaussian distribution with $q = \frac{τ+ 2M (2 - θ)}{τ+ 2M (1 - θ)}<3$, where $τ\ge 0$ characterizes the strength of the confining external field, and $M \ge 0$ is the (normalized) amplitude of the multiplicative noise. We also calculate the standard kurtosis $κ_1$ and the $q$-generalized kurtosis $κ_q$ (i.e., the standard kurtosis but using the escort distribution instead of the direct one). Through these two quantities we numerically follow the time evolution of the distributions. Finally, we exhibit how these quantities can be used as convenient calibrations for determining the index $q$ from numerical data obtained through experiments, observations or numerical computations.

cond-mat.stat-mech↗

arXiv2009

On "Ergodicity and Central Limit Theorem in Systems with Long-Range Interactions" by Figueiredo et al

In the present paper we refute the criticism advanced in a recent preprint by Figueiredo et al [1] about the possible application of the $q$-generalized Central Limit Theorem (CLT) to a paradigmatic long-range-interacting many-body classical Hamiltonian system, the so-called Hamiltonian Mean Field (HMF) model. We exhibit that, contrary to what is claimed by these authors and in accordance with our previous results, $q$-Gaussian-like curves are possible and real attractors for a certain class of initial conditions, namely the one which produces nontrivial longstanding quasi-stationary states before the arrival, only for finite size, to the thermal equilibrium.

cond-mat.stat-mech↗

arXiv2008

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 $ν=1,2,3,...$, unifying the Leibnitz triangle ($ν=1$) and the case of independent variables ($ν\to\infty$); (ii) two slightly different discretizations of $q$--Gaussians; (iii) a special family, characterized by the parameter $χ$, which generalizes the usual case of independent variables (recovered for $χ=1/2$). 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 $q=(ν-2)/(ν-1)$. 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.

cond-mat.stat-mech↗

arXiv2008

A closer look at the indications of q-generalized Central Limit Theorem behavior in quasi-stationary states of the HMF model

We give a closer look at the Central Limit Theorem (CLT) behavior in quasi-stationary states of the Hamiltonian Mean Field model, a paradigmatic one for long-range-interacting classical many-body systems. We present new calculations which show that, following their time evolution, we can observe and classify three kinds of long-standing quasi-stationary states (QSS) with different correlations. The frequency of occurrence of each class depends on the size of the system. The different microsocopic nature of the QSS leads to different dynamical correlations and therefore to different results for the observed CLT behavior.

cond-mat.stat-mech↗

arXiv2007

Nonergodicity and Central Limit Behavior for Long-range Hamiltonians

We present a molecular dynamics test of the Central Limit Theorem (CLT) in a paradigmatic long-range-interacting many-body classical Hamiltonian system, the HMF model. We calculate sums of velocities at equidistant times along deterministic trajectories for different sizes and energy densities. We show that, when the system is in a chaotic regime (specifically, at thermal equilibrium), ergodicity is essentially verified, and the Pdfs of the sums appear to be Gaussians, consistently with the standard CLT. When the system is, instead, only weakly chaotic (specifically, along longstanding metastable Quasi-Stationary States), nonergodicity (i.e., discrepant ensemble and time averages) is observed, and robust $q$-Gaussian attractors emerge, consistently with recently proved generalizations of the CLT.

cond-mat.stat-mech↗

arXiv2007

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}.

cond-mat.stat-mech↗

arXiv2006

Comment on "Entropy of Classical Systems with Long-Range Interactions" by T.M. Rocha Filho et al, PRL 95, 190601 (2005)

In a recent letter (PRL 95, 190601 (2005)), T.M. Rocha Filho and coworkers address the very interesting issue of the entropic form to be used for Hamiltonians with long-range interactions. In our opinion the letter misses several points quite debated in the recent literature which are of fundamental importance for a complete discussion of this problem. Moreover it contains several statements which are not true or not corroborated by any evidence. In this comment we discuss these arguments and severely question the generality of the conclusions of this letter.

cond-mat.stat-mech↗

arXiv2003

Nonextensivity of the cyclic Lattice Lotka Volterra model

We numerically show that the Lattice Lotka-Volterra model, when realized on a square lattice support, gives rise to a {\it finite} production, per unit time, of the nonextensive entropy $S_q= \frac{1- \sum_ip_i^q}{q-1}$ $(S_1=-\sum_i p_i \ln p_i)$. This finiteness only occurs for $q=0.5$ for the $d=2$ growth mode (growing droplet), and for $q=0$ for the $d=1$ one (growing stripe). This strong evidence of nonextensivity is consistent with the spontaneous emergence of local domains of identical particles with fractal boundaries and competing interactions. Such direct evidence is for the first time exhibited for a many-body system which, at the mean field level, is conservative.

cond-mat↗