results for au:Tsallis_C in:cond-mat

- May 16 2018 cond-mat.stat-mech arXiv:1805.05892v1We investigate the dynamics of overdamped $D$-dimensional systems of particles repulsively interacting through short-ranged power-law potentials, $V(r)\sim r^{-\lambda}\;(\lambda/D>1)$. We show that such systems obey a non-linear diffusion equation, and that their stationary state extremizes a $q$-generalized nonadditive entropy. Here we focus on the dynamical evolution of these systems. Our first-principle $D=1,2$ many-body numerical simulations (based on Newton's law) confirm the predictions obtained from the time-dependent solution of the non-linear diffusion equation, and show that the one-particle space-distribution $P(x,t)$ appears to follow a compact-support $q$-Gaussian form, with $q=1-\lambda/D$. We also calculate the velocity distributions $P(v_x,t)$ and, interestingly enough, they follow the same $q$-Gaussian form (apparently precisely for $D=1$, and nearly so for $D=2$). The satisfactory match between the continuum description and the molecular dynamics simulations in a more general, time-dependent, framework neatly confirms the idea that the present dissipative systems indeed represent suitable applications of the $q$-generalized thermostatistical theory.
- Sep 27 2017 cond-mat.stat-mech arXiv:1709.08729v1The Fermi-Pasta-Ulam (FPU) one-dimensional Hamiltonian includes a quartic term which guarantees ergodicity of the system in the thermodynamic limit. Consistently, the Boltzmann factor $P(\epsilon) \sim e^{-\beta \epsilon}$ describes its equilibrium distribution of one-body energies, and its velocity distribution is Maxwellian, i.e., $P(v) \sim e^{- \beta v^2/2}$. We consider here a generalized system where the quartic coupling constant between sites decays as $1/d_{ij}^{\alpha}$ $(\alpha \ge 0; d_{ij} = 1,2,\dots)$. Through \it first-principle molecular dynamics we demonstrate that, for large $\alpha$ (above $\alpha \simeq 1$), i.e., short-range interactions, Boltzmann statistics (based on the \it additive entropic functional $S_B[P(z)]=-k \int dz P(z) \ln P(z)$) is verified. However, for small values of $\alpha$ (below $\alpha \simeq 1$), i.e., long-range interactions, Boltzmann statistics dramatically fails and is replaced by q-statistics (based on the \it nonadditive entropic functional $S_q[P(z)]=k (1-\int dz [P(z)]^q)/(q-1)$, with $S_1 = S_B$). Indeed, the one-body energy distribution is q-exponential, $P(\epsilon) \sim e_{q_{\epsilon}}^{-\beta_{\epsilon} \epsilon} \equiv [1+(q_{\epsilon} - 1) \beta_{\epsilon}{\epsilon}]^{-1/(q_{\epsilon}-1)}$ with $q_{\epsilon} > 1$, and its velocity distribution is given by $P(v) \sim e_{q_v}^{ - \beta_v v^2/2}$ with $q_v > 1$. Moreover, within small error bars, we verify $q_{\epsilon} = q_v = q$, which decreases from an extrapolated value q $\simeq$ 5/3 to q=1 when $\alpha$ increases from zero to $\alpha \simeq 1$, and remains q = 1 thereafter.
- Aug 15 2017 cond-mat.stat-mech nlin.CD arXiv:1708.03705v1We numerically study the two-dimensional, area preserving, web map. When the map is governed by ergodic behavior, it is, as expected, correctly described by Boltzmann-Gibbs statistics, based on the additive entropic functional $S_{BG}[p(x)] = -k\int dx\,p(x) \ln p(x)$. In contrast, possible ergodicity breakdown and transitory sticky dynamical behavior drag the map into the realm of generalized $q$-statistics, based on the nonadditive entropic functional $S_q[p(x)]=k\frac{1-\int dx\,[p(x)]^q}{q-1}$ ($q \in {\cal R}; S_1=S_{BG}$). We statistically describe the system (probability distribution of the sum of successive iterates, sensitivity to the initial condition, and entropy production per unit time) for typical values of the parameter that controls the ergodicity of the map. For small (large) values of the external parameter $K$, we observe $q$-Gaussian distributions with $q=1.935\dots$ (Gaussian distributions), like for the standard map. In contrast, for intermediate values of $K$, we observe a different scenario, due to the fractal structure of the trajectories embedded in the chaotic sea. Long-standing non-Gaussian distributions are characterized in terms of the kurtosis and the box-counting dimension of chaotic sea.
- Jul 27 2017 cond-mat.stat-mech arXiv:1707.08527v1We study the one-dimensional transverse-field spin-1/2 Ising ferromagnet at its critical point. We consider an $L$-sized subsystem of a $N$-sized ring, and trace over the states of $(N-L)$ spins, with $N\to\infty$. The full $N$-system is in a pure state, but the $L$-system is in a statistical mixture. As well known, for $L >>1$, the Boltzmann-Gibbs-von Neumann entropy violates thermodynamical extensivity, namely $S_{BG}(L) \propto \log L$, whereas the nonadditive entropy $S_q$ is extensive for $q=q_c=\sqrt{37}-6 $, namely $S_{q_c}(L) \propto L$. When this problem is expressed in terms of independent fermions, we show that the usual thermostatistical sums emerging within Fermi-Dirac statistics can, for $L>>1$, be indistinctively taken up to $L$ terms or up to $\log L$ terms. This is interpreted as a compact occupancy of phase-space of the $L$-system, hence standard BG quantities with an effective length $V \equiv \log L$ are appropriate and are explicitly calculated. In other words, the calculations are to be done in a phase-space whose effective dimension is $2^{\log L}$ instead of $2^L$. The whole scenario is strongly reminiscent of a usual phase transition of a spin-1/2 $d$-dimensional system, where the phase-space dimension is $2^{L^d}$ in the disordered phase, and effectively $2^{L^d/2}$ in the ordered one.
- We discuss a generalized representation of the Dirac delta function in $d$ dimensions in terms of $q$-exponential functions. We apply this new representation to the study of the so-called $q$-Fourier transform, proving its invertibility for any value of $d$. We finally illustrate the effect of the $q$-deformation on the Gibbs phenomenon.
- May 02 2017 cond-mat.stat-mech arXiv:1705.00014v2Scale-free networks are quite popular nowadays since many systems are well represented by such structures. In order to study these systems, several models were proposed. However, most of them do not take into account the node-to-node Euclidean distance, i.e., the geographical distance. In real networks, the distance between sites can be very relevant, e.g., those cases where it is intended to minimize costs. Within this scenario we studied the role of dimensionality $d$ in the Bianconi-Barabási model with a preferential attachment growth involving Euclidean distances. The preferential attachment in this model follows the rule $\Pi_{i} \propto \eta_i k_i/r_{ij}^{\alpha_A}$ $(1 \leq i < j; \alpha_A \geq 0)$, where $\eta_i$ characterizes the fitness of the $i$-th site and is randomly chosen within the $(0,1]$ interval. We verified that the degree distribution $P(k)$ for dimensions $d=1,2,3,4$ are well fitted by $P(k) \propto e_q^{-k/\kappa}$, where $e_q^{-k/\kappa}$ is the $q$-exponential function naturally emerging within nonextensive statistical mechanics. We determine the index $q$ and $\kappa$ as functions of the quantities $\alpha_A$ and $d$, and numerically verify that both present a universal behavior with respect to the scaled variable $\alpha_A/d$. The same behavior also has been displayed by the dynamical $\beta$ exponent which characterizes the steadily growing number of links of a given site.
- Mar 24 2017 cond-mat.stat-mech arXiv:1703.07813v1Thermal conductance of a homogeneous 1D nonlinear lattice system with neareast neighbor interactions has recently been computationally studied in detail by Li et al [Eur. Phys. J. B \bf 88, 182 (2015)], where its power-law dependence on temperature $T$ for high temperatures is shown. Here, we address its entire temperature dependence, in addition to its dependence on the size $N$ of the system. We obtain a neat data collapse for arbitrary temperatures and system sizes, and numerically show that the thermal conductance curve is quite satisfactorily described by a fat-tailed $q$-Gaussian dependence on $TN^{1/3}$ with $q \simeq 1.55$. Consequently, its $T \to\infty$ asymptotic behavior is given by $T^{-\alpha}$ with $\alpha=2/(q-1) \simeq 3.64$.
- Dec 13 2016 cond-mat.stat-mech arXiv:1612.03658v1The standard map, paradigmatic conservative system in the $(x,p)$ phase space, has been recently shown to exhibit interesting statistical behaviors directly related to the value of the standard map parameter $K$. A detailed numerical description is achieved in the present paper. More precisely, for large values of $K$, the Lyapunov exponents are neatly positive over virtually the entire phase space, and, consistently with Boltzmann-Gibbs (BG) statistics, we verify $q_{\text{ent}}=q_{\text{sen}}=q_{\text{stat}}=q_{\text{rel}}=1$, where $q_{\text{ent}}$ is the $q$-index for which the nonadditive entropy $S_q \equiv k \frac{1-\sum_{i=1}^W p_i^q}{q-1}$ (with $S_1=S_{BG} \equiv -k\sum_{i=1}^W p_i \ln p_i$) grows linearly with time before achieving its $W$-dependent saturation value; $q_{\text{sen}}$ characterizes the time increase of the sensitivity $\xi$ to the initial conditions, i.e., $\xi \sim e_{q_{\text{sen}}}^{\lambda_{q_{\text{sen}}} \,t}\;(\lambda_{q_{\text{sen}}}>0)$, where $e_q^z \equiv[1+(1-q)z]^{1/(1-q)}$; $q_{\text{stat}}$ is the index associated with the $q_{\text{stat}}$-Gaussian distribution of the time average of successive iterations of the $x$-coordinate; finally, $q_{\text{rel}}$ characterizes the $q_{\text{rel}}$-exponential relaxation with time of the entropy $S_{q_{\text{ent}}}$ towards its saturation value. In remarkable contrast, for small values of $K$, the Lyapunov exponents are virtually zero over the entire phase space, and, consistently with $q$-statistics, we verify $q_{\text{ent}}=q_{\text{sen}}=0$, $q_{\text{stat}} \simeq 1.935$, and $q_{\text{rel}} \simeq1.4$. The situation corresponding to intermediate values of $K$, where both stable orbits and a chaotic sea are present, is discussed as well. The present results transparently illustrate when BG or $q$-statistical behavior are observed.
- Sep 21 2016 cond-mat.stat-mech arXiv:1609.05980v2We extend a recently introduced free-energy formalism for homogeneous Fokker-Planck equations to a wide, and physically appealing, class of inhomogeneous nonlinear Fokker-Planck equations. In our approach, the free-energy functional is expressed in terms of an entropic functional and an auxiliary potential, both derived from the coefficients of the equation. With reference to the introduced entropic functional, we discuss the entropy production in a relaxation process towards equilibrium. The properties of the stationary solutions of the considered Fokker-Planck equations are also discussed.
- In a paper by Umarov, Tsallis and Steinberg (2008), a generalization of the Fourier transform, called the $q$-Fourier transform, was introduced and applied for the proof of a $q$-generalized central limit theorem ($q$-CLT). Subsequently, Hilhorst illustrated (2009 and 2010) that the $q$-Fourier transform for $q>1$ is not invertible in the space of density functions. Indeed, using an invariance principle, he constructed a family of densities with the same $q$-Fourier transform and noted that "as a consequence, the $q$-central limit theorem falls short of achieving its stated goal". The distributions constructed there have compact support. We prove now that the limit distribution in the $q$-CLT is unique and can not have a compact support. This result excludes all the possible counterexamples which can be constructed using the invariance principle and fills the gap mentioned by Hilhorst.
- Sep 06 2016 cond-mat.stat-mech arXiv:1609.00972v2Nonlinear 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.
- Aug 15 2016 cond-mat.stat-mech arXiv:1608.03599v1Boltzmann introduced in the 1870's a logarithmic measure for the connection between the thermodynamical entropy and the probabilities of the microscopic configurations of the system. His entropic functional for classical systems was extended by Gibbs to the entire phase space of a many-body system, and by von Neumann in order to cover quantum systems as well. Finally, it was used by Shannon within the theory of information. The simplest expression of this functional corresponds to a discrete set of $W$ microscopic possibilities, and is given by $S_{BG}= -k\sum_{i=1}^W p_i \ln p_i$ ($k$ is a positive universal constant; \it BG stands for \it Boltzmann-Gibbs). This relation enables the construction of BG statistical mechanics. The BG theory has provided uncountable important applications. Its application in physical systems is legitimate whenever the hypothesis of \it ergodicity is satisfied. However, \it what can we do when ergodicity and similar simple hypotheses are violated?, which indeed happens in very many natural, artificial and social complex systems. It was advanced in 1988 the possibility of generalizing BG statistical mechanics through a family of nonadditive entropies, namely $S_q=k\frac{1-\sum_{i=1}^W p_i^q}{q-1}$, which recovers the additive $S_{BG}$ entropy in the $q \to1$ limit. The index $q$ is to be determined from mechanical first principles. Along three decades, this idea intensively evolved world-wide (see Bibliography in \urlhttp://tsallis.cat.cbpf.br/biblio.htm), and led to a plethora of predictions, verifications, and applications in physical systems and elsewhere. As expected whenever a \it paradigm shift is explored, some controversy naturally emerged as well in the community. The present status of the general picture is here described, starting from its dynamical and thermodynamical foundations, and ending with its most recent physical applications.
- A plethora of natural, artificial and social complex systems exists which violate the basic hypothesis (e.g., ergodicity) of Boltzmann-Gibbs (BG) statistical mechanics. Many of such cases can be satisfactorily handled by introducing nonadditive entropic functionals, such as $S_q\equiv k\frac{1-\sum_{i=1}^W p_i^q}{q-1} \; \Bigl(q \in {\cal R}; \, \sum_{i=1}^W p_i=1 \Bigr)$, with $S_1=S_{BG}\equiv -k\sum_{i=1}^W p_i \ln p_i$. Each class of such systems can be characterized by a set of values $\{q\}$, directly corresponding to its various physical/dynamical/geometrical properties. A most important subset is usually referred to as the $q$-triplet, namely $(q_{sensitivity}, q_{relaxation}, q_{stationary\,state})$, defined in the body of this paper. In the BG limit we have $q_{sensitivity}=q_{relaxation}=q_{stationary\,state}=1$. For a given class of complex systems, the set $\{q\}$ contains only a few independent values of $q$, all the others being functions of those few. An illustration of this structure was given in 2005 [Tsallis, Gell-Mann and Sato, Proc. Natl. Acad. Sc. USA \bf 102, 15377; TGS]. This illustration enabled a satisfactory analysis of the Voyager 1 data on the solar wind. But the general form of these structures still is an open question. This is so, for instance, for the challenging $q$-triplet associated with the edge of chaos of the logistic map. We introduce here a transformation which sensibly generalizes the TGS one, and which might constitute an important step towards the general solution.
- May 30 2016 nlin.CD cond-mat.stat-mech arXiv:1605.08562v3In the present work we study the Fermi--Pasta--Ulam (FPU) $\beta $--model involving long--range interactions (LRI) in both the quadratic and quartic potentials, by introducing two independent exponents $\alpha_1$ and $\alpha_2$ respectively, which make the forces decay with distance $r$. Our results demonstrate that weak chaos, in the sense of decreasing Lyapunov exponents, and $q$--Gaussian probability density functions (pdfs) of sums of the momenta, occurs only when long--range interactions are included in the quartic part. More importantly, for $0\leq \alpha_2<1$, we obtain extrapolated values for $q \equiv q_\infty >1$, as $N\rightarrow \infty$, suggesting that these pdfs persist in that limit. On the other hand, when long--range interactions are imposed only on the quadratic part, strong chaos and purely Gaussian pdfs are always obtained for the momenta. We have also focused on similar pdfs for the particle energies and have obtained $q_E$-exponentials (with $q_E>1$) when the quartic-term interactions are long--ranged, otherwise we get the standard Boltzmann-Gibbs weight, with $q=1$. The values of $q_E$ coincide, within small discrepancies, with the values of $q$ obtained by the momentum distributions.
- The so called $q$-triplets were conjectured in 2004 and then found in nature in 2005. A relevant further step was achieved in 2005 when the possibility was advanced that they could reflect an entire infinite algebra based on combinations of the self-dual relations $q \to 2-q$ (\it additive duality) and $q \to 1/q$ (\it multiplicative duality). The entire algebra collapses into the single fixed point $q=1$, corresponding to the Boltzmann-Gibbs entropy and statistical mechanics. For $q \ne 1$, an infinite set of indices $q$ appears, corresponding in principle to an infinite number of physical properties of a given complex system describable in terms of the so called $q$-statistics. The basic idea that is put forward is that, for a given universality class of systems, a small number (typically one or two) of independent $q$ indices exist, the infinite others being obtained from these few ones by simply using the relations of the algebra. The $q$-triplets appear to constitute a few central elements of the algebra. During the last decade, an impressive amount of $q$-triplets have been exhibited in analytical, computational, experimental and observational results in natural, artificial and social systems. Some of them do satisfy the available algebra constructed solely with the additive and multiplicative dualities, but some others seem to violate it. In the present work we generalize those two dualities with the hope that a wider set of systems can be handled within. The basis of the generalization is given by the \it selfdual relation $q \to q_a(q) \equiv \frac{(a+2) -aq}{a-(a-2)q} \,\, (a \in {\cal R})$. We verify that $q_a(1)=1$, and that $q_2(q)=2-q$ and $q_0(q)=1/q$. To physically motivate this generalization, we briefly review illustrative applications of $q$-statistics, in order to exhibit possible candidates where the present generalized algebras could be useful.
- Feb 01 2016 cond-mat.stat-mech arXiv:1601.07951v2We analyze the distribution that extremizes a linear combination of the Boltzmann--Gibbs entropy and the nonadditive $q$-entropy. We show that this distribution can be expressed in terms of a Lambert function. Both the entropic functional and the extremizing distribution can be associated with a nonlinear Fokker--Planck equation obtained from a master equation with nonlinear transition rates. Also, we evaluate the entropy extremized by a linear combination of a Gaussian distribution (which extremizes the Boltzmann--Gibbs entropy) and a $q$-Gaussian distribution (which extremizes the $q$-entropy). We give its explicit expression for $q=0$, and discuss the other cases numerically. The entropy that we obtain can be expressed, for $q=0$, in terms of Lambert functions, and exhibits a discontinuity in the second derivative for all values of $q<1$. The entire discussion is closely related to recent results for type-II superconductors and for the statistics of the standard map.
- A plethora of natural, artificial and social systems exist which do not belong to the Boltzmann-Gibbs (BG) statistical-mechanical world, based on the standard additive entropy $S_{BG}$ and its associated exponential BG factor. Frequent behaviors in such complex systems have been shown to be closely related to $q$-statistics instead, based on the nonadditive entropy $S_q$ (with $S_1=S_{BG}$), and its associated $q$-exponential factor which generalizes the usual BG one. In fact, a wide range of phenomena of quite different nature exist which can be described and, in the simplest cases, understood through analytic (and explicit) functions and probability distributions which exhibit some universal features. Universality classes are concomitantly observed which can be characterized through indices such as $q$. We will exhibit here some such cases, namely concerning the distribution of inter-occurrence (or inter-event) times in the areas of finance, earthquakes and genomes.
- Oct 05 2015 cond-mat.stat-mech arXiv:1510.00415v1We 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 $p=\hbar k$, $E=\hbar \omega$ 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 $\omega$ and a $q$-Gaussian square modulus profile.
- Sep 25 2015 cond-mat.stat-mech arXiv:1509.07141v1Deep connections are known to exist between scale-free networks and non-Gibbsian statistics. For example, typical degree distributions at the thermodynamical limit are of the form $P(k) \propto e_q^{-k/\kappa}$, where the $q$-exponential form $e_q^z \equiv [1+(1-q)z]^{\frac{1}{1-q}}$ optimizes the nonadditive entropy $S_q$ (which, for $q\to 1$, recovers the Boltzmann-Gibbs entropy). We introduce and study here $d$-dimensional geographically-located networks which grow with preferential attachment involving Euclidean distances through $r_{ij}^{-\alpha_A} \; (\alpha_A \ge 0)$. Revealing the connection with $q$-statistics, we numerically verify (for $d$ =1, 2, 3 and 4) that the $q$-exponential degree distributions exhibit, for both $q$ and $\kappa$, universal dependences on the ratio $\alpha_A/d$. Moreover, the $q=1$ limit is rapidly achieved by increasing $\alpha_A/d$ to infinity.
- Sep 16 2015 cond-mat.stat-mech arXiv:1509.04697v2We introduce a generalized $d$-dimensional Fermi-Pasta-Ulam (FPU) model in presence of long-range interactions, and perform a first-principle study of its chaos for $d=1,2,3$ through large-scale numerical simulations. The nonlinear interaction is assumed to decay algebraically as $d_{ij}^{-\alpha}$ ($\alpha \ge 0$), $\{d_{ij}\}$ being the distances between $N$ oscillator sites. Starting from random initial conditions we compute the maximal Lyapunov exponent $\lambda_{max}$ as a function of $N$. Our $N>>1$ results strongly indicate that $\lambda_{max}$ remains constant and positive for $\alpha/d>1$ (implying strong chaos, mixing and ergodicity), and that it vanishes like $N^{-\kappa}$ for $0 \le \alpha/d < 1$ (thus approaching weak chaos and opening the possibility of breakdown of ergodicity). The suitably rescaled exponent $\kappa$ exhibits universal scaling, namely that $(d+2) \kappa$ depends only on $\alpha/d$ and, when $\alpha/d$ increases from zero to unity, it monotonically decreases from unity to zero, remaining so for all $\alpha/d >1$. The value $\alpha/d=1$ can therefore be seen as a critical point separating the ergodic regime from the anomalous one, $\kappa$ playing a role analogous to that of an order parameter. This scaling law is consistent with Boltzmann-Gibbs statistics for $\alpha/d > 1$, and possibly with $q$-statistics for $0 \le \alpha/d < 1$.
- Jul 20 2015 cond-mat.stat-mech arXiv:1507.05058v1A multi-parametric version of the nonadditive entropy $S_{q}$ is introduced. This new entropic form, denoted by $S_{a,b,r}$, possesses many interesting statistical properties, and it reduces to the entropy $S_q$ for $b=0$, $a=r:=1-q$ (hence Boltzmann-Gibbs entropy $S_{BG}$ for $b=0$, $a=r \to 0$). The construction of the entropy $S_{a,b,r}$ is based on a general group-theoretical approach recently proposed by one of us \citeTempesta2. Indeed, essentially all the properties of this new entropy are obtained as a consequence of the existence of a rational group law, which expresses the structure of $S_{a,b,r}$ with respect to the composition of statistically independent subsystems. Depending on the choice of the parameters, the entropy $S_{a,b,r}$ can be used to cover a wide range of physical situations, in which the measure of the accessible phase space increases say exponentially with the number of particles $N$ of the system, or even stabilizes, by increasing $N$, to a limiting value. This paves the way to the use of this entropy in contexts where a system "freezes" some or many of its degrees of freedom by increasing the number of its constituting particles or subsystems.
- We introduce three deformations, called $\alpha$-, $\beta$- and $\gamma$-deformation respectively, of a $N$-body probabilistic model, first proposed by Rodríguez et al. (2008), having $q$-Gaussians as $N\to\infty$ limiting probability distributions. The proposed $\alpha$- and $\beta$-deformations are asymptotically scale-invariant, whereas the $\gamma$-deformation is not. We prove that, for both $\alpha$- and $\beta$-deformations, the resulting deformed triangles still have $q$-Gaussians as limiting distributions, with a value of $q$ independent (dependent) on the deformation parameter in the $\alpha$-case ($\beta$-case). In contrast, the $\gamma$-case, where we have used the celebrated $Q$-numbers and the Gauss binomial coefficients, yields other limiting probability distribution functions, outside the $q$-Gaussian family. These results suggest that scale-invariance might play an important role regarding the robustness of the $q$-Gaussian family.
- Mar 31 2015 cond-mat.stat-mech nlin.CD arXiv:1503.08685v1We focus on a linear chain of $N$ first-neighbor-coupled logistic maps at their edge of chaos in the presence of a common noise. This model, characterised by the coupling strength $\epsilon$ and the noise width $\sigma_{max}$, was recently introduced by Pluchino et al [Phys. Rev. E \bf 87, 022910 (2013)]. They detected, for the time averaged returns with characteristic return time $\tau$, possible connections with $q$-Gaussians, the distributions which optimise, under appropriate constraints, the nonadditive entropy $S_q$, basis of nonextensive statistics mechanics. We have here a closer look on this model, and numerically obtain probability distributions which exhibit a slight asymmetry for some parameter values, in variance with simple $q$-Gaussians. Nevertheless, along many decades, the fitting with $q$-Gaussians turns out to be numerically very satisfactory for wide regions of the parameter values, and we illustrate how the index $q$ evolves with $(N, \tau, \epsilon, \sigma_{max})$. It is nevertheless instructive on how careful one must be in such numerical analysis. The overall work shows that physical and/or biological systems that are correctly mimicked by the Pluchino et al model are thermostatistically related to nonextensive statistical mechanics when time-averaged relevant quantities are studied.
- Using a simple probabilistic model, we illustrate that a small part of a strongly correlated many-body classical system can show a paradoxical behavior, namely asymptotic stochastic independence. We consider a triangular array such that each row is a list of $n$ strongly correlated random variables. The correlations are preserved even when $n\to\infty$, since the standard central limit theorem does not hold for this array. We show that, if we choose a fixed number $m<n$ of random variables of the $n$th row and trace over the other $n-m$ variables, and then consider $n\to\infty$, the $m$ chosen ones can, paradoxically, turn out to be independent. However, the scenario can be different if $m$ increases with $n$. Finally, we suggest a possible experimental verification of our results near criticality of a second-order phase transition.
- Dec 02 2014 cond-mat.stat-mech arXiv:1412.0006v1We study a symmetric generalization $\mathfrak{p}^{(N)}_k(\eta, \alpha)$ of the binomial distribution recently introduced by Bergeron et al, where $\eta \in [0,1]$ denotes the win probability, and $\alpha$ is a positive parameter. This generalization is based on $q$-exponential generating functions ($e_{q^{gen}}^z \equiv [1+(1-q^{gen})z]^{1/(1-q^{gen})};\,e_{1}^z=e^z)$ where $q^{gen}=1+1/\alpha$. The numerical calculation of the probability distribution function of the number of wins $k$, related to the number of realizations $N$, strongly approaches a discrete $q^{disc}$-Gaussian distribution, for win-loss equiprobability (i.e., $\eta=1/2$) and all values of $\alpha$. Asymptotic $N\to \infty$ distribution is in fact a $q^{att}$-Gaussian $e_{q^{att}}^{-\beta z^2}$, where $q^{att}=1-2/(\alpha-2)$ and $\beta=(2\alpha-4)$. The behavior of the scaled quantity $k/N^\gamma$ is discussed as well. For $\gamma<1$, a large-deviation-like property showing a $q^{ldl}$-exponential decay is found, where $q^{ldl}=1+1/(\eta\alpha)$. For $\eta=1/2$, $q^{ldl}$ and $q^{att}$ are related through $1/(q^{ldl}-1)+1/(q^{att}-1)=1$, $\forall \alpha$. For $\gamma=1$, the law of large numbers is violated, and we consistently study the large-deviations with respect to the probability of the $N\to\infty$ limit distribution, yielding a power law, although not exactly a $q^{LD}$-exponential decay. All $q$-statistical parameters which emerge are univocally defined by $(\eta, \alpha)$. Finally we discuss the analytical connection with the Pólya urn problem.
- Dec 02 2014 hep-ph cond-mat.stat-mech arXiv:1412.0474v1Multiparticle production processes in $pp$ collisions at the central rapidity region are usually considered to be divided into independent "soft" and "hard" components. The first is described by exponential (thermal-like) transverse momentum spectra in the low-$p_T$ region with a scale parameter $T$ associated with the temperature of the hadronizing system. The second is governed by a power-like distributions of transverse momenta with power index $n$ at high-$p_T$ associated with the hard scattering between partons. We show that the hard-scattering integral can be approximated as a nonextensive distribution of a quasi-power-law containing a scale parameter $T$ and a power index $n=1/(q -1)$, where $q$ is the nonextensivity parameter. We demonstrate that the whole region of transverse momenta presently measurable at LHC experiments at central rapidity (in which the observed cross sections varies by $14$ orders of magnitude down to the low $p_T$ region) can be adequately described by a single nonextensive distribution. These results suggest the dominance of the hard-scattering hadron-production process and the approximate validity of a "no-hair" statistical-mechanical description of the $p_T$ spectra for the whole $p_T$ region at central rapidity for $pp$ collisions at high-energies.
- Nov 03 2014 cond-mat.stat-mech arXiv:1410.8591v1We briefly review the connection between statistical mechanics and thermodynamics. We show that, in order to satisfy thermodynamics and its Legendre transformation mathematical frame, the celebrated Boltzmann-Gibbs~(BG) statistical mechanics is sufficient but not necessary. Indeed, the $N\to\infty$ limit of statistical mechanics is expected to be consistent with thermodynamics. For systems whose elements are generically independent or quasi-independent in the sense of the theory of probabilities, it is well known that the BG theory (based on the additive BG entropy) does satisfy this expectation. However, in complete analogy, other thermostatistical theories (\emphe.g., $q$-statistics), based on nonadditive entropic functionals, also satisfy the very same expectation. We illustrate this standpoint with systems whose elements are strongly correlated in a specific manner, such that they escape the BG realm.
- Sep 12 2014 hep-ph cond-mat.stat-mech arXiv:1409.3278v1We analyze LHC available data measuring the distribution probability of transverse momenta~$p_T$ in proton-proton collisions at $\sqrt{s}=0.9\,\textrm{TeV}$ (CMS, ALICE, ATLAS) and $\sqrt{s}=7\,\textrm{TeV}$ (CMS, ATLAS). A remarkably good fitting can be obtained, along fourteen decades in magnitude, by phenomenologically using $q$-statistics for a \it single particle of a two-dimensional relativistic ideal gas. The parameters that have been obtained by assuming $\textrm{d}N/p_T\textrm{d}p_T\textrm{d}y \propto e_q^{-E_T/T}$ at mid-rapidity are, in all cases, $q \simeq 1.1$ and $T\simeq 0.13\,\textrm{GeV}$ (which satisfactorily compares with the pion mass). This fact suggests the approximate validity of a "no-hair" statistical-mechanical description of the hard-scattering hadron-production process in which the detailed mechanisms of parton scattering, parton cascades, parton fragmentation, running coupling and other information can be subsumed under the stochastic dynamics in the lowest-order description. In addition to that basic structure, a finer analysis of the data suggests a small oscillatory structure on top of the leading $q$-exponential. The physical origin of such intriguing oscillatory behavior remains elusive, though it could be related to some sort of fractality or scale-invariance within the system.
- Jul 24 2014 cond-mat.stat-mech arXiv:1407.6052v1In 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.
- We consider correlated random variables $X_1,\dots,X_n$ taking values in $\{0,1\}$ such that, for any permutation $\pi$ of $\{1,\dots,n\}$, the random vectors $(X_1,\dots,X_n)$ and $(X_{\pi(1)},\dots,X_{\pi(n)})$ have the same distribution. This distribution, which was introduced by Rodríguez et al (2008) and then generalized by Hanel et al (2009), is scale-invariant and depends on a real parameter $\nu>0$ ($\nu\to\infty$ implies independence). Putting $S_n=X_1+\cdots+X_n$, the distribution of $S_n-n/2$ approaches a $Q$-Gaussian distribution with compact support ($Q=1-1/(\nu-1)<1$) as $n$ increases, after appropriate scaling. In the present article, we show that the distribution of $S_n/n$ converges, as $n\to\infty$, to a beta distribution with both parameters equal to $\nu$. In particular, the law of large numbers does not hold since, if $0\le x<1/2$, then $\mathbb{P}(S_n/n\le x)$, which is the probability of the event $\{S_n/n\le x\}$ (large deviation), does not converges to zero as $n\to\infty$. For $x=0$ and every real $\nu>0$, we show that $\mathbb{P}(S_n=0)$ decays to zero like a power law of the form $1/n^\nu$ with a subdominant term of the form $1/n^{\nu+1}$. If $0<x\le 1$ and $\nu>0$ is an integer, we show that we can analytically find upper and lower bounds for the difference between $\mathbb{P}(S_n/n\le x)$ and its ($n\to\infty$) limit. We also show that these bounds vanish like a power law of the form $1/n$ with a subdominant term of the form $1/n^2$.
- Jun 12 2014 cond-mat.stat-mech arXiv:1406.2733v1We deduce a nonlinear and inhomogeneous Fokker-Planck equation within a generalized Stratonovich, or stochastic $\alpha$-, prescription ($\alpha=0$, $1/2$ 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, $p_{st}(x) = N_q[1 - (1-q)\beta V(x)]^{1/(1-q)}$, with an index $q$ which does not depend on $\alpha$ in the presence of any nonvanishing nonlinearity. This is in contrast with the linear case, for which the index $q$ is $\alpha$-dependent.
- May 15 2014 nlin.CD cond-mat.stat-mech arXiv:1405.3528v1We introduce and numerically study a long-range-interaction generalization of the one-dimensional Fermi-Pasta-Ulam (FPU) $\beta-$ model. The standard quartic interaction is generalized through a coupling constant that decays as $1/r^\alpha$ ($\alpha \ge 0$)(with strength characterized by $b>0$). In the $\alpha \to\infty$ limit we recover the original FPU model. Through classical molecular dynamics computations we show that (i) For $\alpha \geq 1$ the maximal Lyapunov exponent remains finite and positive for increasing number of oscillators $N$ (thus yielding ergodicity), whereas, for $0 \le \alpha <1$, it asymptotically decreases as $N^{- \kappa(\alpha)}$ (consistent with violation of ergodicity); (ii) The distribution of time-averaged velocities is Maxwellian for $\alpha$ large enough, whereas it is well approached by a $q$-Gaussian, with the index $q(\alpha)$ monotonically decreasing from about 1.5 to 1 (Gaussian) when $\alpha$ increases from zero to close to one. For $\alpha$ 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 $1/N \propto b^\delta /t_c^\gamma$ with $\gamma >0$ and $\delta >0$, 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).
- Apr 07 2014 cond-mat.stat-mech arXiv:1404.1257v2It is by now well known that the Boltzmann-Gibbs-von Neumann-Shannon logarithmic entropic functional ($S_{BG}$) is inadequate for wide classes of strongly correlated systems: see for instance the 2001 Brukner and Zeilinger's \it Conceptual inadequacy of the Shannon information in quantum measurements, among many other systems exhibiting various forms of complexity. On the other hand, the Shannon and Khinchin axioms uniquely mandate the BG form $S_{BG}=-k\sum_i p_i \ln p_i$; the Shore and Johnson axioms follow the same path. Many natural, artificial and social systems have been satisfactorily approached with nonadditive entropies such as the $S_q=k \frac{1-\sum_i p_i^q}{q-1}$ one ($q \in {\cal R}; \,S_1=S_{BG}$), basis of nonextensive statistical mechanics. Consistently, the Shannon 1948 and Khinchine 1953 uniqueness theorems have already been generalized in the literature, by Santos 1997 and Abe 2000 respectively, in order to uniquely mandate $S_q$. We argue here that the same remains to be done with the Shore and Johnson 1980 axioms. We arrive to this conclusion by analyzing specific classes of strongly correlated complex systems that await such generalization.
- Mar 24 2014 cond-mat.stat-mech arXiv:1403.5425v1The 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.
- Nov 12 2012 cond-mat.stat-mech arXiv:1211.2124v1The 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.
- Jun 28 2012 cond-mat.stat-mech arXiv:1206.6133v2We numerically study a one-dimensional system of $N$ classical localized planar rotators coupled through interactions which decay with distance as $1/r^\alpha$ ($\alpha \ge 0$). The approach is a first principle one (\textiti.e., based on Newton's law), and yields the probability distribution of momenta. For $\alpha$ large enough and $N\gg1$ we observe, for longstanding states, the Maxwellian distribution, landmark of Boltzmann-Gibbs thermostatistics. But, for $\alpha$ 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.
- Jun 12 2012 cond-mat.stat-mech nlin.CD arXiv:1206.2152v4We 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.
- May 29 2012 cond-mat.stat-mech gr-qc arXiv:1205.6084v1We 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 $p=\hbar k$, $E=\hbar \omega$ 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 $i\hbar \frac{\partial}{\partial t}(\frac{\Phi}{\Phi_0}) = - \frac{1}{2-q}\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} [(\frac{\Phi}{\Phi_0})^{2-q}] + V(x)(\frac{\Phi}{\Phi_0})^{q}$ with $V(x)=max$. Remarkably enough, the potential $V$ couples to $\Phi^q$, instead of coupling to $\Phi$, as happens in the familiar linear case ($q=1$).
- We briefly review a perspective along which the Boltzmann-Gibbs statistical mechanics, the strongly chaotic dynamical systems, and the Schroedinger, Klein-Gordon and Dirac partial differential equations are seen as linear physics, and are characterized by an index q = 1. We exhibit in what sense q ≠ 1 yields nonlinear physics, which turn out to be quite rich and directly related to what is nowadays referred to as complexity, or complex systems. We first discuss a few central points like the distinction between additivity and extensivity, and the Central Limit Theorem as well as the large-deviation theory. Then we comment the case of gravitation (which within the present context corresponds to q ≠ 1, and to similar nonlinear approaches), with special focus onto the entropy of black holes. Finally we briefly focus on recent nonlinear generalizations of the Schroedinger, Klein-Gordon and Dirac equations, and mention various illustrative predictions, verifications and applications within physics (in both low- and high-energy regimes) as well as out of it.
- Feb 11 2012 cond-mat.stat-mech arXiv:1202.2154v2As 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 \empharea 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 \emphnot 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.
- 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 $e^{-\beta E_i}$. 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 $e_q^{-\beta_q E_i}$ ($e_q^z \equiv [1+(1-q)z]^{1/(1-q)};\,e_1^z=e^z)$. We numerically study large deviations for a strongly correlated model which depends on the indices $Q \in [1,2)$ and $\gamma \in (0,1)$. This model provides, in the $N\to\infty$ limit ($\forall \gamma$), $Q$-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 $q= \frac{Q-1}{\gamma (3-Q)}+1 \ge 1$. This $q$-generalized illustration opens wide the door towards a desirable large-deviation foundation of nonextensive statistical mechanics.
- 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.
- 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 $\nu>0$, whose value determines the degree of correlation between $N$ $(d+1)$-valued random variables. There are $(d+1)^N$ different events, and the limit $\nu\to\infty$ 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 $q=(\nu-2)/(\nu-1)$ for $\nu \in [1,\infty)$) as $N\to\infty$ limit probability distributions for the sum of the $N$ binary variables [A. Rodrı́guez \em et al, J. Stat. Mech. (2008) P09006; R. Hanel \em et al, Eur. Phys. J. B \bf 72, 263 (2009)]. In the $\nu\to\infty$ limit the $d-$dimensional multinomial distribution is recovered for the sums, which approach a $d-$dimensional Gaussian distribution for $N\to\infty$. For any $\nu$, 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 $\nu$. 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.
- Jul 01 2011 nlin.CD cond-mat.stat-mech arXiv:1106.6226v1We 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.
- Jun 21 2011 cond-mat.stat-mech arXiv:1106.3761v1We 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.
- Jun 21 2011 cond-mat.stat-mech arXiv:1106.3781v1Clausius 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.
- Jun 16 2011 cond-mat.stat-mech arXiv:1106.3100v1We 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 $\frac{\partial p(x,t)}{\partial t}=-\frac{\partial}{\partial x}[F(x) p(x,t)] + 1/2D \frac{\partial^2}{\partial x^2} [\phi(x,p)p(x,t)]$, 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 $\phi(x,p)$ 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 $\phi(x,p)=[A+BV(x)]^\theta [p(x,t)]^{\eta}$, where $A$, $B$, $\theta$ and $\eta$ are (real) constants. Our main result can be sumarized as follows: For $\theta \neq 1$ and arbitrary confining potential $V(x)$, $p(x,\infty) \propto \lbrace 1-\beta(1-q)V(x)\rbrace ^{1/(1-q)} \equiv e_q^{-\beta V(x)}$, where $q= 1+ \eta/(\theta-1)$. The present approach unifies into a single mechanism, essentially \it long memory, results currently discussed and applied in the literature.
- Jun 01 2011 cond-mat.stat-mech arXiv:1105.6184v2We 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).
- Apr 29 2011 cond-mat.other quant-ph arXiv:1104.5461v1Generalizations 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 $q$.
- Feb 14 2011 cond-mat.stat-mech arXiv:1102.2408v1We 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.
- In a recent paper Hilhorst \citeHilhorst2010 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.
- 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.
- 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.
- Apr 07 2010 cond-mat.stat-mech arXiv:1004.0722v1We consider a class of single-particle one-dimensional stochastic equations which include external field, additive and multiplicative noises. We use a parameter $\theta \in [0,1]$ which enables the unification of the traditional Itô and Stratonovich approaches, now recovered respectively as the $\theta=0$ and $\theta=1/2$ 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 $q = \frac{\tau + 2M (2 - \theta)}{\tau + 2M (1 - \theta)}<3$, where $\tau \ge 0$ 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 $\kappa_1$ and the $q$-generalized kurtosis $\kappa_q$ (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.
- We present a generalization of the representation in plane waves of Dirac delta, $\delta(x)=(1/2\pi)\int_{-\infty}^\infty e^{-ikx}\,dk$, namely $\delta(x)=(2-q)/(2\pi)\int_{-\infty}^\infty e_q^{-ikx}\,dk$, 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 $\pi$. 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.
- Mar 12 2010 cond-mat.stat-mech arXiv:1003.2232v2As 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.
- Nov 12 2009 cond-mat.stat-mech arXiv:0911.2009v2The $\alpha$-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 $(q,\alpha)$-stable distributions. These sequences are generalizations of i.i.d. $\alpha$-stable distributions, and have not been previously studied. Long-range dependent $(q,\alpha)$-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 $\alpha$-stable Lévy distributions. In the present paper we establish basic properties of $(q,\alpha)$-stable distributions, and generalize the result of Umarov, Tsallis and Steinberg (2008), where the particular case $\alpha=2, q\in [1,3),$ was considered, to the whole range of stability and nonextensivity parameters $\alpha \in (0,2]$ and $q \in [1,3),$ respectively. We also discuss possible further extensions of the results that we obtain, and formulate some conjectures.
- Nov 09 2009 cond-mat.stat-mech arXiv:0911.1263v1We 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.
- Sep 01 2009 cond-mat.stat-mech arXiv:0908.4438v1Extremization 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.
- Jun 09 2009 cond-mat.stat-mech arXiv:0906.1262v1In 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.
- Jan 28 2009 cond-mat.stat-mech arXiv:0901.4292v1We 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.
- Dec 22 2008 cond-mat.stat-mech arXiv:0812.3855v1We 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.
- May 26 2008 cond-mat.stat-mech arXiv:0805.3652v2In 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.
- Apr 22 2008 cond-mat.stat-mech arXiv:0804.3362v1The 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 \emphporous-medium equation, is investigated through both numerical and analytical approaches. It is shown that an \emphinitial $q$-Gaussian, characterized by an index $q_i$, approaches the \emphfinal, 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 \emphrelaxation 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.
- Apr 10 2008 cond-mat.stat-mech arXiv:0804.1488v2In 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 $\nu=1,2,3,...$, unifying the Leibnitz triangle ($\nu=1$) and the case of independent variables ($\nu\to\infty$); (ii) two slightly different discretizations of $q$--Gaussians; (iii) a special family, characterized by the parameter $\chi$, which generalizes the usual case of independent variables (recovered for $\chi=1/2$). 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 $q=(\nu -2)/(\nu-1)$. 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.
- Feb 13 2008 cond-mat.stat-mech arXiv:0802.1698v2Escort 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.
- Feb 11 2008 cond-mat.stat-mech arXiv:0802.1138v3The 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.
- 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.
- Jan 09 2008 cond-mat.stat-mech arXiv:0801.1311v1A 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 $\mathcal{G}_{q}=\{f = a e_{q}^{-\beta x2}, \, a>0, \, \beta>0 \}$. This constitutes a first step towards a general representation of the inverse $q$-Fourier operation, which would enable interesting physical and other applications.
- Dec 28 2007 cond-mat.stat-mech astro-ph.CO arXiv:0712.4165v2A 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.