Searcharxiv⌕ Search

arXiv subjects

Constantino Tsallis

Publications and source records attributed to Constantino Tsallis.

At least 127 records · Page 7Linked to original sources
arXiv2007

Nonextensive statistical mechanics and central limit theorems II - Convolution of q-independent random variables

In this article we review recent generalisations of the central limit theorem for the sum of specially correlated (or q-independent) variables, focusing on q greater or equal than 1. Specifically, this kind of correlation turns the probability density function known as q-Gaussian, which emerges upon maximisation of the entropy Sq, into an attractor in probability space. Moreover, we also discuss a q-generalisation of a-stable Levy distributions which can as well be stable for this special kind of correlation.Within this context, we verify the emergence of a triplet of entropic indices which relate the form of the attractor, the correlation, and the scaling rate, similar to the q-triplet that connects the entropic indices characterising the sensitivity to initial conditions, the stationary state, and relaxation to the stationary state in anomalous systems.

cond-mat.stat-mech↗
arXiv2007

Nonextensive statistical mechanics and central limit theorems I - Convolution of independent random variables and q-product

In this article we review the standard versions of the Central and of the Levy-Gnedenko Limit Theorems, and illustrate their application to the convolution of independent random variables associated with the distribution known as q-Gaussian. This distribution emerges upon extremisation of the nonadditive entropy, basis of nonextensive statistical mechanics. It has a finite variance for q < 5/3, and an infinite one for q > 5/3. We exhibit that, in the case of (standard) independence, the q-Gaussian has either the Gaussian (if q < 5/3) or the a-stable Levy distributions (if q > 5/3) as its attractor in probability space. Moreover, we review a generalisation of the product, the q-product, which plays a central role in the approach of the specially correlated variables emerging within the nonextensive theory.

cond-mat.soft↗
arXiv2007

Central limit behavior of deterministic dynamical systems

We investigate the probability density of rescaled sums of iterates of deterministic dynamical systems, a problem relevant for many complex physical systems consisting of dependent random variables. A Central Limit Theorem (CLT) is only valid if the dynamical system under consideration is sufficiently mixing. For the fully developed logistic map and a cubic map we analytically calculate the leading-order corrections to the CLT if only a finite number of iterates is added and rescaled, and find excellent agreement with numerical experiments. At the critical point of period doubling accumulation, a CLT is not valid anymore due to strong temporal correlations between the iterates. Nevertheless, we provide numerical evidence that in this case the probability density converges to a $q$-Gaussian, thus leading to a power-law generalization of the CLT. The above behavior is universal and independent of the order of the maximum of the map considered, i.e. relevant for large classes of critical dynamical systems.

cond-mat.stat-mech↗
arXiv2007

On the non-Boltzmannian nature of quasi-stationary states in long-range interacting systems

We discuss the non-Boltzmannian nature of quasi-stationary states in the Hamiltonian Mean Field (HMF) model, a paradigmatic model for long-range interacting classical many-body systems. We present a theorem excluding the Boltzmann-Gibbs exponential weight in Gibbs $Γ$-space of microscopic configurations, and comment a paper recently published by Baldovin and Orlandini (2006). On the basis of the points here discussed, the ongoing debate on the possible application, within appropriate limits, of the generalized $q$-statistics to long-range Hamiltonian systems remains open.

cond-mat.stat-mech↗
arXiv2007

Deviation from Gaussianity in the cosmic microwave background temperature fluctuations

Recent measurements of the temperature fluctuations of the cosmic microwave background (CMB) radiation from the WMAP satellite provide indication of a non-Gaussian behavior. Although the observed feature is small, it is detectable and analyzable. Indeed, the temperature distribution P^{CMB}(Delta T) of these data can be quite well fitted by the anomalous probability distribution emerging within nonextensive statistical mechanics, based on the entropy S_q = k (1 - \int dx [P(x)]^q)/(q - 1) (where in the limit case q -> 1 we obtain the Boltzmann-Gibbs entropy S_1 = - k \int dx P(x) ln[P(x)]). For the CMB frequencies analysed, ν= 40.7, 60.8, and 93.5 GHz, P^{CMB}(Delta T) is well described by P_q(Delta T) \propto 1/[1 + (q-1) B(ν) (Delta T)^2]^{1/(q-1)}, with q = 1.04 \pm 0.01, the strongest non-Gaussian contribution coming from the South-East sector of the celestial sphere. Moreover, Monte Carlo simulations exclude, at the 99% confidence level, P_1(Delta T) \propto e^{- B(ν) (Delta T)^2} to fit the three-year WMAP data.

astro-ph↗
arXiv2007

Multivariate Generalizations of the q--Central Limit Theorem

We study multivariate generalizations of the $q$-central limit theorem, a generalization of the classical central limit theorem consistent with nonextensive statistical mechanics. Two types of generalizations are addressed, more precisely the {\it direct} and {\it sequential} $q$-central limit theorems are proved. Their relevance to the asymptotic scale invariance of some specially correlated systems is studied. A $q$-analog of the classic weak convergence is introduced and its equivalence to the $q$-convergence is proved for $q>1$.

cond-mat.stat-mech↗
arXiv2007

Roundoff-induced attractors and reversibility in conservative two-dimensional maps

We numerically study two conservative two-dimensional maps, namely the baker map (whose Lyapunov exponent is known to be positive), and a typical one (exhibiting a vanishing Lyapunov exponent) chosen from the generalized shift family of maps introduced by C. Moore [Phys Rev Lett {\bf 64}, 2354 (1990)] in the context of undecidability. We calculated the time evolution of the entropy $S_q \equiv \frac{1-\sum_{i=1}^Wp_i^q}{q-1}$ ($S_1=S_{BG}\equiv -\sum_{i=1}^Wp_i \ln p_i$), and exhibited the dramatic effect introduced by numerical precision. Indeed, in spite of being area-preserving maps, they present, {\it well after} the initially concentrated ensemble has spread virtually all over the phase space, unexpected {\it pseudo-attractors} (fixed-point like for the baker map, and more complex structures for the Moore map). These pseudo-attractors, and the apparent time (partial) reversibility they provoke, gradually disappear for increasingly large precision. In the case of the Moore map, they are related to zero Lebesgue-measure effects associated with the frontiers existing in the definition of the map. In addition to the above, and consistently with the results by V. Latora and M. Baranger [Phys. Rev. Lett. {\bf 82}, 520 (1999)], we find that the rate of the far-from-equilibrium entropy production of baker map, numerically coincides with the standard Kolmogorov-Sinai entropy of this strongly chaotic system.

cond-mat.stat-mech↗
arXiv2007

Unified model for network dynamics exhibiting nonextensive statistics

We introduce a dynamical network model which unifies a number of network families which are individually known to exhibit $q$-exponential degree distributions. The present model dynamics incorporates static (non-growing) self-organizing networks, preferentially growing networks, and (preferentially) rewiring networks. Further, it exhibits a natural random graph limit. The proposed model generalizes network dynamics to rewiring and growth modes which depend on internal topology as well as on a metric imposed by the space they are embedded in. In all of the networks emerging from the presented model we find q-exponential degree distributions over a large parameter space. We comment on the parameter dependence of the corresponding entropic index q for the degree distributions, and on the behavior of the clustering coefficients and neighboring connectivity distributions.

cond-mat.stat-mech↗
arXiv2006

Boltzmann-Gibbs thermal equilibrium distribution for classical systems and Newton law: A computational discussion

We implement a general numerical calculation that allows for a direct comparison between nonlinear Hamiltonian dynamics and the Boltzmann-Gibbs canonical distribution in Gibbs $Γ$-space. Using paradigmatic first-neighbor models, namely, the inertial XY ferromagnet and the Fermi-Pasta-Ulam $β$-model, we show that at intermediate energies the Boltzmann-Gibbs equilibrium distribution is a consequence of Newton second law (${\mathbf F}=m{\mathbf a}$). At higher energies we discuss partial agreement between time and ensemble averages.

cond-mat.stat-mech↗
arXiv2006

Long-range memory elementary 1D cellular automata: Dynamics and nonextensivity

We numerically study the dynamics of elementary 1D cellular automata (CA), where the binary state $σ_i(t) \in \{0,1\}$ of a cell $i$ does not only depend on the states in its local neighborhood at time $t-1$, but also on the memory of its own past states $σ_i(t-2), σ_i(t-3),...,σ_i(t-τ),...$. We assume that the weight of this memory decays proportionally to $τ^{-α}$, with $α\ge 0$ (the limit $α\to \infty$ corresponds to the usual CA). Since the memory function is summable for $α>1$ and nonsummable for $0\le α\le 1$, we expect pronounced %qualitative and quantitative changes of the dynamical behavior near $α=1$. This is precisely what our simulations exhibit, particularly for the time evolution of the Hamming distance $H$ of initially close trajectories. We typically expect the asymptotic behavior $H(t) \propto t^{1/(1-q)}$, where $q$ is the entropic index associated with nonextensive statistical mechanics. In all cases, the function $q(α)$ exhibits a sensible change at $α\simeq 1$. We focus on the class II rules 61, 99 and 111. For rule 61, $q = 0$ for $0 \le α\le α_c \simeq 1.3$, and $q<0$ for $α> α_c$, whereas the opposite behavior is found for rule 111. For rule 99, the effect of the long-range memory on the spread of damage is quite dramatic. These facts point at a rich dynamics intimately linked to the interplay of local lookup rules and the range of the memory. Finite size scaling studies varying system size $N$ indicate that the range of the power-law regime for $H(t)$ typically diverges $\propto N^z$ with $0 \le z \le 1$. Similar studies have been carried out for other rules, e.g., the famous "universal computer" rule 110.

cond-mat.stat-mech↗
arXiv2006

Influence of global correlations on central limit theorems and entropic extensivity

We consider probabilistic models of N identical distinguishable, binary random variables. If these variables are strictly or asymptotically independent, then, for N>>1, (i) the attractor in distribution space is, according to the standard central limit theorem, a Gaussian, and (ii) the Boltzmann-Gibbs-Shannon entropy is extensive, meaning that S_BGS(N) ~ N . If these variables have any nonvanishing global (i.e., not asymptotically independent) correlations, then the attractor deviates from the Gaussian. The entropy appears to be more robust, in the sense that, in some cases, S_BGS remains extensive even in the presence of strong global correlations. In other cases, however, even weak global correlations make the entropy deviate from the normal behavior. More precisely, in such cases the entropic form Sq can become extensive for some value of q different from unity . This scenario is illustrated with several new as well as previously described models. The discussion illuminates recent progress into q-describable nonextensive probabilistic systems, and the conjectured q-Central Limit Theorem (q-CLT) which posses a q-Gaussian attractor.

cond-mat.stat-mech↗
arXiv2006

Weak chaos and metastability in a symplectic system of many long-range-coupled standard maps

We introduce, and numerically study, a system of $N$ symplectically and globally coupled standard maps localized in a $d=1$ lattice array. The global coupling is modulated through a factor $r^{-α}$, being $r$ the distance between maps. Thus, interactions are {\it long-range} (nonintegrable) when $0\leqα\leq1$, and {\it short-range} (integrable) when $α>1$. We verify that the largest Lyapunov exponent $λ_M$ scales as $λ_{M} \propto N^{-κ(α)}$, where $κ(α)$ is positive when interactions are long-range, yielding {\it weak chaos} in the thermodynamic limit $N\to\infty$ (hence $λ_M\to 0$). In the short-range case, $κ(α)$ appears to vanish, and the behaviour corresponds to {\it strong chaos}. We show that, for certain values of the control parameters of the system, long-lasting metastable states can be present. Their duration $t_c$ scales as $t_c \propto N^{β(α)}$, where $β(α)$ appears to be numerically consistent with the following behavior: $β>0$ for $0 \le α< 1$, and zero for $α\ge 1$. All these results exhibit major conjectures formulated within nonextensive statistical mechanics (NSM). Moreover, they exhibit strong similarity between the present discrete-time system, and the $α$-XY Hamiltonian ferromagnetic model, also studied in the frame of NSM.

cond-mat.stat-mech↗
arXiv2006

A nonextensive approach to the dynamics of financial observables

We present results about financial market observables, specifically returns and traded volumes. They are obtained within the current nonextensive statistical mechanical framework based on the entropy $S_{q}=k\frac{1-\sum\limits_{i=1}^{W} p_{i} ^{q}}{1-q} (q\in \Re)$ ($S_{1} \equiv S_{BG}=-k\sum\limits_{i=1}^{W}p_{i} \ln p_{i}$). More precisely, we present stochastic dynamical mechanisms which mimic probability density functions empirically observed. These mechanisms provide possible interpretations for the emergence of the entropic indices $q$ in the time evolution of the corresponding observables. In addition to this, through multi-fractal analysis of return time series, we verify that the dual relation $q_{stat}+q_{sens}=2$ is numerically satisfied, $q_{stat}$ and $q_{sens}$ being associated to the probability density function and to the sensitivity to initial conditions respectively. This type of simple relation, whose understanding remains ellusive, has been empirically verified in various other systems.

physics.data-an↗
arXiv2005

Numerical indications of a q-generalised central limit theorem

We provide numerical indications of the $q$-generalised central limit theorem that has been conjectured (Tsallis 2004) in nonextensive statistical mechanics. We focus on $N$ 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 $q$-product with $q \le 1$. We show that, in the large $N$ limit (and after appropriate centering, rescaling, and symmetrisation), the emerging distributions are $q_e$-Gaussians, i.e., $p(x) \propto [1-(1-q_e) β(N) x^2]^{1/(1-q_e)}$, with $q_e=2-\frac{1}{q}$, and with coefficients $β(N)$ approaching finite values $β(\infty)$. The particular case $q=q_e=1$ recovers the celebrated de Moivre-Laplace theorem.

cond-mat.stat-mech↗
arXiv2005

Occupancy of phase space, extensivity of Sq, and q-generalized central limit theorem

Increasing the number $N$ of elements of a system typically makes the entropy to increase. The question arises on {\it what particular entropic form} we have in mind and {\it how it increases} with $N$. Thermodynamically speaking it makes sense to choose an entropy which increases {\it linearly} with $N$ for large $N$, i.e., which is {\it extensive}. If the $N$ elements are probabilistically {\it independent} (no interactions) or quasi-independent (e.g., {\it short}-range interacting), it is known that the entropy which is extensive is that of Boltzmann-Gibbs-Shannon, $S_{BG} \equiv -k \sum_{i=1}^W p_i \ln p_i$. If they are however {\it globally correlated} (e.g., through {\it long}-range interactions), the answer depends on the particular nature of the correlations. There is a large class of correlations (in one way or another related to scale-invariance) for which an appropriate entropy is that on which nonextensive statistical mechanics is based, i.e., $S_q \equiv k \frac{1-\sum_{i=1}^W p_i^q}{q-1}$ ($S_1=S_{BG}$), where $q$ is determined by the specific correlations. We briefly review and illustrate these ideas through simple examples of occupation of phase space. A very similar scenario emerges with regard to the central limit theorem. We present some numerical indications along these lines. The full clarification of such a possible connection would help qualifying the class of systems for which the nonextensive statistical concepts are applicable, and, concomitantly, it would enlighten the reason for which $q$-exponentials are ubiquitous in many natural and artificial systems.

cond-mat.stat-mech↗
arXiv2005

Temperature Fluctuations of the Cosmic Microwave Background Radiation: A Case of Nonextensivity?

Temperature maps of the Cosmic Microwave Background (CMB) radiation, as those obtained by the Wilkinson Microwave Anisotropy Probe (WMAP), provide one of the most precise data sets to test fundamental hypotheses of modern cosmology. One of these issues is related to the statistical properties of the CMB temperature fluctuations, which would have been produced by Gaussian random density fluctuations when matter and radiation were in thermal equilibrium in the early Universe. We analysed here the WMAP data and found that the distribution of the CMB temperature fluctuations P^{CMB}(Delta T) can be quite well fitted by the anomalous temperature distribution emerging within nonextensive statistical mechanics. This theory is based on the nonextensive entropy S_q = k (1 - \int dx [P_q(x)]^q) /(q-1), with the Boltzmann-Gibbs expression as the limit case q -> 1. For the frequencies investigated (ν= 40.7, 60.8, and 93.5 GHz), we found that P^{CMB}(Delta T) is well described by P_q(Delta T) \propto 1/[1 + (q-1) B(ν) (Delta T)^2]^{1/(q-1)}, with q = 1.055 \pm 0.002, which exclude, at the 99% confidence level, exact Gaussian temperature distributions P^{Gauss}(Delta T) \propto e^{- B(ν) Delta T^2}, corresponding to the q -> 1 limit, to properly represent the CMB temperature fluctuations measured by WMAP.

astro-ph↗
arXiv2005

Weak Chaos in large conservative system -- Infinite-range coupled standard maps

We study, through a new perspective, a globally coupled map system that essentially interpolates between simple discrete-time nonlinear dynamics and certain long-range many-body Hamiltonian models. In particular, we exhibit relevant similarities, namely (i) the existence of long-standing quasistationary states (QSS), and (ii) the emergence of weak chaos in the thermodynamic limit, between the present model and the Hamiltonian Mean Field model, a strong candidate for a nonxtensive statistical mechanical approach.

cond-mat.stat-mech↗