arXiv subjects

Constantino Tsallis

Publications and source records attributed to Constantino Tsallis.

At least 91 records · Page 5Linked to original sources

arXiv2014

Boltzmann-Gibbs entropy is sufficient but not necessary for the likelihood factorization required by Einstein

In 1910 Einstein published a crucial aspect of his understanding of Boltzmann entropy. He essentially argued that the likelihood function of any system composed by two probabilistically independent subsystems {\it ought} to be factorizable into the likelihood functions of each of the subsystems. Consistently he was satisfied by the fact that Boltzmann (additive) entropy fulfills this epistemologically fundamental requirement. We show here that entropies (e.g., the $q$-entropy on which nonextensive statistical mechanics is based) which generalize the BG one through violation of its well known additivity can {\it also} fulfill the same requirement. This fact sheds light on the very foundations of the connection between the micro- and macro-scopic worlds.

cond-mat.stat-mech↗

arXiv2014

Nonlinear inhomogeneous Fokker-Planck equation within a generalized Stratonovich prescription

We deduce a nonlinear and inhomogeneous Fokker-Planck equation within a generalized Stratonovich, or stochastic $α$-, prescription ($α=0$, $1/2$ and $1$ respectively correspond to the Itô, Stratonovich and anti-Itô prescriptions). We obtain its stationary state $p_{st}(x)$ for a class of constitutive relations between drift and diffusion and show that it has a $q$-exponential form, $p_{st}(x) = N_q[1 - (1-q)βV(x)]^{1/(1-q)}$, with an index $q$ which does not depend on $α$ in the presence of any nonvanishing nonlinearity. This is in contrast with the linear case, for which the index $q$ is $α$-dependent.

cond-mat.stat-mech↗

arXiv2014

Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics

We introduce and numerically study a long-range-interaction generalization of the one-dimensional Fermi-Pasta-Ulam (FPU) $β-$ model. The standard quartic interaction is generalized through a coupling constant that decays as $1/r^α$ ($α\ge 0$)(with strength characterized by $b>0$). In the $α\to\infty$ limit we recover the original FPU model. Through classical molecular dynamics computations we show that (i) For $α\geq 1$ the maximal Lyapunov exponent remains finite and positive for increasing number of oscillators $N$ (thus yielding ergodicity), whereas, for $0 \le α<1$, it asymptotically decreases as $N^{- κ(α)}$ (consistent with violation of ergodicity); (ii) The distribution of time-averaged velocities is Maxwellian for $α$ large enough, whereas it is well approached by a $q$-Gaussian, with the index $q(α)$ monotonically decreasing from about 1.5 to 1 (Gaussian) when $α$ increases from zero to close to one. For $α$ small enough, the whole picture is consistent with a crossover at time $t_c$ from $q$-statistics to Boltzmann-Gibbs (BG) thermostatistics. More precisely, we construct a "phase diagram" for the system in which this crossover occurs through a frontier of the form $1/N \propto b^δ/t_c^γ$ with $γ>0$ and $δ>0$, in such a way that the $q=1$ ($q>1$) behavior dominates in the $\lim_{N \to\infty} \lim_{t \to\infty}$ ordering ($\lim_{t \to\infty} \lim_{N \to\infty}$ ordering).

nlin.CD↗

arXiv2014

An introduction to nonadditive entropies and a thermostatistical approach of inanimate and living matter

The possible distinction between inanimate and living matter has been of interest to humanity since thousands of years. Clearly, such a rich question can not be answered in a single manner, and a plethora of approaches naturally do exist. However, during the last two decades, a new standpoint, of thermostatistical nature, has emerged. It is related to the proposal of nonadditive entropies in 1988, in order to generalise the celebrated Boltzmann-Gibbs additive functional, basis of standard statistical mechanics. Such entropies have found deep fundamental interest and uncountable applications in natural, artificial and social systems. In some sense, this perspective represents an epistemological paradigm shift. These entropies crucially concern complex systems, in particular those whose microscopic dynamics violate ergodicity. Among those, living matter and other living-like systems play a central role. We briefly review here this approach, and present some of its predictions, verifications and applications.

cond-mat.stat-mech↗

arXiv2013

Influence of the interaction range on the thermostatistics of a classical many-body system

We numerically study a one-dimensional system of $N$ classical localized planar rotators coupled through interactions which decay with distance as $1/r^α$ ($α\ge 0$). The approach is a first principle one (\textit{i.e.}, based on Newton's law), and yields the probability distribution of momenta. For $α$ large enough and $N\gg1$ we observe, for longstanding states, the Maxwellian distribution, landmark of Boltzmann-Gibbs thermostatistics. But, for $α$ small or comparable to unity, we observe instead robust fat-tailed distributions that are quite well fitted with $q$-Gaussians. These distributions extremize, under appropriate simple constraints, the nonadditive entropy $S_q$ upon which nonextensive statistical mechanics is based. The whole scenario appears to be consistent with nonergodicity and with the thesis of the $q$-generalized Central Limit Theorem.

cond-mat.stat-mech↗

arXiv2013

Black hole thermodynamical entropy

As early as 1902, Gibbs pointed out that systems whose partition function diverges, e.g. gravitation, lie outside the validity of the Boltzmann-Gibbs (BG) theory. Consistently, since the pioneering Bekenstein-Hawking results, physically meaningful evidence (e.g., the holographic principle) has accumulated that the BG entropy $S_{BG}$ of a $(3+1)$ black hole is proportional to its area $L^2$ ($L$ being a characteristic linear length), and not to its volume $L^3$. Similarly it exists the \emph{area law}, so named because, for a wide class of strongly quantum-entangled $d$-dimensional systems, $S_{BG}$ is proportional to $\ln L$ if $d=1$, and to $L^{d-1}$ if $d>1$, instead of being proportional to $L^d$ ($d \ge 1$). These results violate the extensivity of the thermodynamical entropy of a $d$-dimensional system. This thermodynamical inconsistency disappears if we realize that the thermodynamical entropy of such nonstandard systems is \emph{not} to be identified with the BG {\it additive} entropy but with appropriately generalized {\it nonadditive} entropies. Indeed, the celebrated usefulness of the BG entropy is founded on hypothesis such as relatively weak probabilistic correlations (and their connections to ergodicity, which by no means can be assumed as a general rule of nature). Here we introduce a generalized entropy which, for the Schwarzschild black hole and the area law, can solve the thermodynamic puzzle.

cond-mat.stat-mech↗

arXiv2013

Noise, Synchrony and Correlations at the Edge of Chaos

We study the effect of a weak random additive noise in a linear chain of N locally-coupled logistic maps at the edge of chaos. Maps tend to synchronize for a strong enough coupling, but if a weak noise is added, very intermittent fluctuations in the returns time series are observed. This intermittency tends to disappear when noise is increased. Considering the pdfs of the returns, we observe the emergence of fat tails which can be satisfactorily reproduced by $q$-Gaussians curves typical of nonextensive statistical mechanics. Interoccurrence times of these extreme events are also studied in detail. Similarities with recent analysis of financial data are also discussed.

cond-mat.stat-mech↗

arXiv2012

Reply to Comment on "Towards a large deviation theory for strongly correlated systems"

The paper that is commented by Touchette contains a computational study which opens the door to a desirable generalization of the standard large deviation theory (applicable to a set of $N$ nearly independent random variables) to systems belonging to a special, though ubiquitous, class of strong correlations. It focuses on three inter-related aspects, namely (i) we exhibit strong numerical indications which suggest that the standard exponential probability law is asymptotically replaced by a power-law as its dominant term for large $N$; (ii) the subdominant term appears to be consistent with the $q$-exponential behavior typical of systems following $q$-statistics, thus reinforcing the thermodynamically extensive entropic nature of the exponent of the $q$-exponential, basically $N$ times the $q$-generalized rate function; (iii) the class of strong correlations that we have focused on corresponds to attractors in the sense of the Central Limit Theorem which are $Q$-Gaussian distributions (in principle $1 < Q < 3$), which relevantly differ from (symmetric) Lévy distributions, with the unique exception of Cauchy-Lorentz distributions (which correspond to $Q = 2$), where they coincide, as well known. In his Comment, Touchette has agreeably discussed point (i), but, unfortunately, points (ii) and (iii) have, as we detail here, visibly escaped to his analysis. Consequently, his conclusion claiming the absence of special connection with $q$-exponentials is unjustified.

cond-mat.stat-mech↗

arXiv2012

Nonlinear Schroedinger Equation in the Presence of Uniform Acceleration

We consider a recently proposed nonlinear Schroedinger equation exhibiting soliton-like solutions of the power-law form $e_q^{i(kx-wt)}$, involving the $q$-exponential function which naturally emerges within nonextensive thermostatistics [$e_q^z \equiv [1+(1-q)z]^{1/(1-q)}$, with $e_1^z=e^z$]. Since these basic solutions behave like free particles, obeying $p=\hbar k$, $E=\hbar ω$ and $E=p^2/2m$ ($1 \le q<2$), it is relevant to investigate how they change under the effect of uniform acceleration, thus providing the first steps towards the application of the aforementioned nonlinear equation to the study of physical scenarios beyond free particle dynamics. We investigate first the behaviour of the power-law solutions under Galilean transformation and discuss the ensuing Doppler-like effects. We consider then constant acceleration, obtaining new solutions that can be equivalently regarded as describing a free particle viewed from an uniformly accelerated reference frame (with acceleration $a$) or a particle moving under a constant force $-ma$. The latter interpretation naturally leads to the evolution equation $i\hbar \frac{\partial}{\partial t}(\fracΦ{Φ_0}) = - \frac{1}{2-q}\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} [(\fracΦ{Φ_0})^{2-q}] + V(x)(\fracΦ{Φ_0})^{q}$ with $V(x)=max$. Remarkably enough, the potential $V$ couples to $Φ^q$, instead of coupling to $Φ$, as happens in the familiar linear case ($q=1$).

cond-mat.stat-mech↗

arXiv2011

Towards a large deviation theory for statistical-mechanical complex systems

The theory of large deviations constitutes a mathematical cornerstone in the foundations of Boltzmann-Gibbs statistical mechanics, based on the additive entropy $S_{BG}=- k_B\sum_{i=1}^W p_i \ln p_i$. Its optimization under appropriate constraints yields the celebrated BG weight $e^{-βE_i}$. An elementary large-deviation connection is provided by $N$ independent binary variables, which, in the $N\to\infty$ limit yields a Gaussian distribution. The probability of having $n \ne N/2$ out of $N$ throws is governed by the exponential decay $e^{-N r}$, where the rate function $r$ is directly related to the relative BG entropy. To deal with a wide class of complex systems, nonextensive statistical mechanics has been proposed, based on the nonadditive entropy $S_q=k_B\frac{1- \sum_{i=1}^W p_i^q}{q-1}$ ($q \in {\cal R}; \,S_1=S_{BG}$). Its optimization yields the generalized weight $e_q^{-β_q E_i}$ ($e_q^z \equiv [1+(1-q)z]^{1/(1-q)};\,e_1^z=e^z)$. We numerically study large deviations for a strongly correlated model which depends on the indices $Q \in [1,2)$ and $γ\in (0,1)$. This model provides, in the $N\to\infty$ limit ($\forall γ$), $Q$-Gaussian distributions, ubiquitously observed in nature ($Q\to 1$ recovers the independent binary model). We show that its corresponding large deviations are governed by $e_q^{-N r_q}$ ($\propto 1/N^{1/(q-1)}$ if $q>1$) where $q= \frac{Q-1}{γ(3-Q)}+1 \ge 1$. This $q$-generalized illustration opens wide the door towards a desirable large-deviation foundation of nonextensive statistical mechanics.

cond-mat.stat-mech↗

arXiv2011

Restricted random walk model as a new testing ground for the applicability of q-statistics

We present exact results obtained from Master Equations for the probability function P(y,T) of sums $y=\sum_{t=1}^T x_t$ of the positions x_t of a discrete random walker restricted to the set of integers between -L and L. We study the asymptotic properties for large values of L and T. For a set of position dependent transition probabilities the functional form of P(y,T) is with very high precision represented by q-Gaussians when T assumes a certain value $T^*\propto L^2$. The domain of y values for which the q-Gaussian apply diverges with L. The fit to a q-Gaussian remains of very high quality even when the exponent $a$ of the transition probability g(x)=|x/L|^a+p with 0<p<<1 is different from 1, all though weak, but essential, deviation from the q-Gaussian does occur for $a\neq1$. To assess the role of correlations we compare the T dependence of P(y,T) for the restricted random walker case with the equivalent dependence for a sum y of uncorrelated variables x each distributed according to 1/g(x).

cond-mat.stat-mech↗

arXiv2011

A scale-invariant probabilistic model based on Leibniz-like pyramids

We introduce a family of probabilistic {\it scale-invariant} Leibniz-like pyramids and $(d+1)$-dimensional hyperpyramids ($d=1,2,3,...$), characterized by a parameter $ν>0$, whose value determines the degree of correlation between $N$ $(d+1)$-valued random variables. There are $(d+1)^N$ different events, and the limit $ν\to\infty$ corresponds to independent random variables, in which case each event has a probability $1/(d+1)^N$ to occur. The sums of these $N$ $\,(d+1)$-valued random variables correspond to a $d-$dimensional probabilistic model, and generalizes a recently proposed one-dimensional ($d=1$) model having $q-$Gaussians (with $q=(ν-2)/(ν-1)$ for $ν\in [1,\infty)$) as $N\to\infty$ limit probability distributions for the sum of the $N$ binary variables [A. Rodr\'ıguez {\em et al}, J. Stat. Mech. (2008) P09006; R. Hanel {\em et al}, Eur. Phys. J. B {\bf 72}, 263 (2009)]. In the $ν\to\infty$ limit the $d-$dimensional multinomial distribution is recovered for the sums, which approach a $d-$dimensional Gaussian distribution for $N\to\infty$. For any $ν$, the conditional distributions of the $d-$dimensional model are shown to yield the corresponding joint distribution of the $(d-1)$-dimensional model with the same $ν$. For the $d=2$ case, we study the joint probability distribution, and identify two classes of marginal distributions, one of them being asymmetric and scale-invariant, while the other one is symmetric and only asymptotically scale-invariant. The present probabilistic model is proposed as a testing ground for a deeper understanding of the necessary and sufficient conditions for having $q$-Gaussian attractors in the $N\to\infty$ limit, the ultimate goal being a neat mathematical view of the causes clarifying the ubiquitous emergence of $q$-statistics verified in many natural, artificial and social systems.

cond-mat.stat-mech↗

arXiv2011

The nonadditive entropy $S_q$: A door open to the nonuniversality of the mathematical expression of the Clausius thermodynamic entropy in terms of the probabilities of the microscopic configurations

Clausius introduced, in the 1860s, a thermodynamical quantity which he named {\it entropy} $S$. This thermodynamically crucial quantity was proposed to be {\it extensive}, i.e., in contemporary terms, $S(N) \propto N$ in the thermodynamic limit $N \to\infty$. A decade later, Boltzmann proposed a functional form for this quantity which connects $S$ with the occurrence probabilities of the microscopic configurations (referred to as {\it complexions} at that time) of the system. This functional is, if written in modern words referring to a system with $W$ possible discrete states, $S_{BG}=-k_B \sum_{i=1}^W p_i \ln p_i$. The BG entropy is {\it additive}, meaning that, if A and B are two probabilistically independent systems, then $S_{BG}(A+B)=S_{BG}(A)+S_{BG}(B)$. The words, {\it extensive} and {\it additive}, were practically treated, for over more than one century, as almost synonyms, and $S_{BG}$ was considered to be the unique form that $S$ could take. In other words, the functional $S_{BG}$ was considered to be universal. It has become increasingly clear today that it is {\it not} so, and that those two words are {\it not} synonyms, but happen to coincide whenever we are dealing with paradigmatic Hamiltonians involving {\it short-range} interactions between their elements, presenting no strong frustration and other "pathologies". These facts constitute the basis of a generalization of the BG entropy and statistical mechanics, introduced in 1988, and frequently referred to as nonadditive entropy $S_q$ and nonextensive statistical mechanics, respectively. We briefly review herein these points, and exhibit recent as well as typical applications of these concepts in natural, artificial, and social systems, as shown through theoretical, experimental, observational and computational predictions and verifications.

cond-mat.stat-mech↗

arXiv2011

Sensitivity, entropy, and escape rates at the onset of chaos

We analytically link three properties of nonlinear dynamical systems, namely sensitivity to initial conditions, entropy production, and escape rate, in $z$-logistic maps for both positive and zero Lyapunov exponents. We unify these relations at chaos, where the Lyapunov exponent is positive, and at its onset, where it vanishes. Our result unifies, in particular, two already known cases, namely (i) the standard entropy rate in the presence of escape, valid for exponential functionality rates with strong chaos, and (ii) the Pesin-like identity with no escape, valid for the power-law behavior present at points such as the Feigenbaum one.

cond-mat.stat-mech↗

arXiv2011

Long memory constitutes a unified mesoscopic mechanism consistent with nonextensive statistical mechanics

We unify two paradigmatic mesoscopic mechanisms for the emergence of nonextensive statistics, namely the multiplicative noise mechanism leading to a {\it linear} Fokker-Planck (FP) equation with {\it inhomogenous} diffusion coefficient, and the non-Markovian process leading to the {\it nonlinear} FP equation with {\it homogeneous} diffusion coefficient. More precisely, we consider the equation $\frac{\partial p(x,t)}{\partial t}=-\frac{\partial}{\partial x}[F(x) p(x,t)] + 1/2D \frac{\partial^2}{\partial x^2} [ϕ(x,p)p(x,t)]$, where $D \in {\cal R}$ and $F(x)=-\partial V(x) /\partial x$, $V(x)$ being the potential under which diffusion occurs. Our aim is to find whether $ϕ(x,p)$ exists such that the inhomogeneous linear and the homogeneous nonlinear FP equations become unified in such a way that the (ubiquitously observed) $q$-exponentials remain as stationary solutions. It turns out that such solutions indeed exist for a wide class of systems, namely when $ϕ(x,p)=[A+BV(x)]^θ[p(x,t)]^η$, where $A$, $B$, $θ$ and $η$ are (real) constants. Our main result can be sumarized as follows: For $θ\neq 1$ and arbitrary confining potential $V(x)$, $p(x,\infty) \propto \lbrace 1-β(1-q)V(x)\rbrace ^{1/(1-q)} \equiv e_q^{-βV(x)}$, where $q= 1+ η/(θ-1)$. The present approach unifies into a single mechanism, essentially {\it long memory}, results currently discussed and applied in the literature.

cond-mat.stat-mech↗

arXiv2011

Nonlinear Relativistic and Quantum Equations with a Common Type of Solution

Generalizations of the three main equations of quantum physics, namely, the Schrödinger, Klein-Gordon, and Dirac equations, are proposed. Nonlinear terms, characterized by exponents depending on an index $q$, are considered in such a way that the standard linear equations are recovered in the limit $q \rightarrow 1$. Interestingly, these equations present a common, soliton-like, travelling solution, which is written in terms of the $q$-exponential function that naturally emerges within nonextensive statistical mechanics. In all cases, the well-known Einstein energy-momentum relation is preserved for arbitrary values of $q$.

cond-mat.other↗

arXiv2011

Some Open Points in Nonextensive Statistical Mechanics

We present and discuss a list of some interesting points that are currently open in nonextensive statistical mechanics. Their analytical, numerical, experimental or observational advancement would naturally be very welcome.

cond-mat.stat-mech↗

arXiv2010

Limit distribution in the $q$-CLT for $q \ge 1$ can not have a compact support

In a recent paper Hilhorst \cite{Hilhorst2010} illustrated that the $q$-Fourier transform for $q>1$ is not invertible in the space of density functions. Using an invariance principle he constructed a family of densities with the same $q$-Fourier transform and claimed that $q$-Gaussians are not mathematically proved to be attractors. We show here that none of the distributions constructed in Hilhorst's counterexamples can be a limit distribution in the $q$-CLT, except the one whose support covers the whole real axis, which is precisely the $q$-Gaussian distribution.

cond-mat.stat-mech↗