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Constantino Tsallis

Publications and source records attributed to Constantino Tsallis.

At least 55 records · Page 3Linked to original sources
arXiv2021

Generalized Box-Muller method for generating q-Gaussian random deviates

Addendum: The generalized Box-Müller algorithm provides a methodology for generating q-Gaussian random variates. The parameter $-\infty<q\leq3$ is related to the shape of the tail decay; $q<1$ for compact-support including parabola $(q=0)$; $1<q\leq3$ for heavy-tail including Cauchy $(q=2)$. This addendum clarifies the transformation $q'=((3q-1)/(q+1))$ within the algorithm is due to a difference in the dimensions d of the generalized logarithm and the generalized distribution. The transformation is clarified by the decomposition of $q=1+2κ/(1+dκ)$, where the shape parameter $-1<κ\leq\infty$ quantifies the magnitude of the deformation from exponential. A simpler specification for the generalized Box- Müller algorithm is provided using the shape of the tail decay. Original: The q-Gaussian distribution is known to be an attractor of certain correlated systems, and is the distribution which, under appropriate constraints, maximizes the entropy Sq, basis of nonextensive statistical mechanics. This theory is postulated as a natural extension of the standard (Boltzmann-Gibbs) statistical mechanics, and may explain the ubiquitous appearance of heavy-tailed distributions in both natural and man-made systems. The q-Gaussian distribution is also used as a numerical tool, for example as a visiting distribution in Generalized Simulated Annealing. We develop and present a simple, easy to implement numerical method for generating random deviates from a q-Gaussian distribution based upon a generalization of the well known Box-Muller method. Our method is suitable for a larger range of q values, q<3, than has previously appeared in the literature, and can generate deviates from q-Gaussian distributions of arbitrary width and center. MATLAB code showing a straightforward implementation is also included.

cond-mat.stat-mech↗
arXiv2020

Connecting complex networks to nonadditive entropies

Boltzmann-Gibbs statistical mechanics applies satisfactorily to a plethora of systems. It fails however for complex systems generically involving strong space-time entanglement. Its generalization based on nonadditive $q$-entropies adequately handles a wide class of such systems. We show here that scale-invariant networks belong to this class. We numerically study a $d$-dimensional geographically located network with weighted links and exhibit its 'energy' distribution per site at its quasi-stationary state. Our results strongly suggest a correspondence between the random geometric problem and a class of thermal problems within the generalised thermostatistics. The Boltzmann-Gibbs exponential factor is generically substituted by its $q$-generalisation, and is recovered in the $q=1$ limit when the nonlocal effects fade away. The present connection should cross-fertilise experiments in both research areas.

cond-mat.stat-mech↗
arXiv2020

Dynamical robustness of discrete conservative systems: Harper and generalized standard maps

In recent years, statistical characterization of the discrete conservative dynamical systems (more precisely, paradigmatic examples of area-preserving maps such as the standard and the web maps) has been analyzed extensively and shown that, for larger parameter values for which the Lyapunov exponents are largely positive over the entire phase space, the probability distribution is a Gaussian, consistent with Boltzmann-Gibbs (BG) statistics. On the other hand, for smaller parameter values for which the Lyapunov exponents are virtually zero over the entire phase space, we verify this distribution appears to approach a $q$-Gaussian (with $q \simeq 1.935$), consistent with $q$-statistics. Interestingly, if the parameter values are in between these two extremes, then the probability distributions happen to exhibit a linear combination of these two behaviours. Here, we numerically show that the Harper map is also in the same universality class of the maps discussed so far. This constitutes one more evidence on the robustness of this behavior whenever the phase space consists of stable orbits. Then, we propose a generalization of the standard map for which the phase space includes many sticky regions, changing the previously observed simple linear combination behavior to a more complex combination.

cond-mat.stat-mech↗
arXiv2019

A generalised model for asymptotically-scale-free geographical networks

We consider a generalised d-dimensional model for asymptotically-scale-free geographical networks. Central to many networks of this kind, when considering their growth in time, is the attachment rule, i.e. the probability that a new node is attached to one (or more) preexistent nodes. In order to be more realistic, a fitness parameter $η_i \in [0,1]$ for each node $i$ of the network is also taken into account to reflect the ability of the nodes to attract new ones. Our d-dimensional model takes into account the geographical distances between nodes, with different probability distribution for $η$ which sensibly modifies the growth dynamics. The preferential attachment rule is assumed to be $Π_i\propto k_i η_i r_{ij}^{-α_A} $ where $k_i$ is the connectivity of the $i$th pre-existing site and $α_A$ characterizes the importance of the euclidean distance r for the network growth. For special values of the parameters, this model recovers respectively the Bianconi-Barabási and the Barabási-Albert ones. The present generalised model is asymptotically scale-free in all cases, and its degree distribution is very well fitted with q-exponential distributions, which optimise the nonadditive entropy $S_q$, given by $p(k) \propto e_q^{-k/κ} \equiv 1/[1+(q-1)k/κ]^{1/(q-1)}$, with $(q,κ)$ depending uniquely only on the ratio $α_A/d$ and the fitness distribution. Hence this model constitutes a realization of asymptotically-scale-free geographical networks within nonextensive statistical mechanics, where $k$ plays the role of energy and $κ$ plays the role of temperature. General scaling laws are also found for q as a function of the parameters of the model.

physics.soc-ph↗
arXiv2019

Möbius transforms, cycles and q-triplets in statistical mechanics

In the realm of Boltzmann-Gibbs (BG) statistical mechanics and its q-generalisation for complex systems, we analyse observed sequences of q-triplets, or q-doublets if one of them is the unity, in terms of cycles of successive Möbius transforms of the line preserving unity ( q=1 corresponds to the BG theory). Such transforms have the form q --> (aq + 1-a)/[(1+a)q -a], where a is a real number; the particular cases a=-1 and a=0 yield respectively q --> (2-q) and q --> 1/q, currently known as additive and multiplicative dualities. This approach seemingly enables the organisation of various complex phenomena into different classes, named N-complete or incomplete. The classification that we propose here hopefully constitutes a useful guideline in the search, for non-BG systems whenever well described through q-indices, of new possibly observable physical properties.

cond-mat.stat-mech↗
arXiv2019

Scaling properties of d-dimensional complex networks

The area of networks is very interdisciplinary and exhibits many applications in several fields of science. Nevertheless, there are few studies focusing on geographically located $d$-dimensional networks. In this paper, we study scaling properties of a wide class of $d$-dimensional geographically located networks which grow with preferential attachment involving Euclidean distances through $r_{ij}^ {-α_A} \;(α_A \geq 0)$. We have numerically analyzed the time evolution of the connectivity of sites, the average shortest path, the degree distribution entropy, and the average clustering coefficient, for $d=1,2,3,4$, and typical values of $α_A$. Remarkably enough, virtually all the curves can be made to collapse as functions of the scaled variable $α_A/d$. These observations confirm the existence of three regimes. The first one occurs in the interval $α_A/d \in [0,1]$; it is non-Boltzmannian with very-long-range interactions in the sense that the degree distribution is a $q$-exponential with $q$ constant and above unity. The critical value $α_A/d =1$ that emerges in many of these properties is replaced by $α_A/d =1/2$ for the $β$-exponent which characterizes the time evolution of the connectivity of sites. The second regime is still non-Boltzmannian, now with moderately long-range interactions, and reflects in an index $q$ monotonically decreasing with $α_A/d$ increasing from its critical value to a characteristic value $α_A/d \simeq 5$. Finally, the third regime is Boltzmannian (with $q=1$), and corresponds to short-range interactions.

cond-mat.stat-mech↗
arXiv2018

Validity and Failure of the Boltzmann Weight

The dynamics and thermostatistics of a classical inertial XY model, characterized by long-range interactions, are investigated on $d$-dimensional lattices ($d=1,2,$ and 3), through molecular dynamics. The interactions between rotators decay with the distance $r_{ij}$ like~$1/r_{ij}^α$ ($α\geq 0$), where $α\to\infty$ and $α=0$ respectively correspond to the nearest-neighbor and infinite-range interactions. We verify that the momenta probability distributions are Maxwellians in the short-range regime, whereas $q$-Gaussians emerge in the long-range regime. Moreover, in this latter regime, the individual energy probability distributions are characterized by long tails, corresponding to $q$-exponential functions. The present investigation strongly indicates that, in the long-range regime, central properties fall out of the scope of Boltzmann-Gibbs statistical mechanics, depending on $d$ and $α$ through the ratio $α/d$.

cond-mat.stat-mech↗
arXiv2018

Overdamped dynamics of particles with repulsive power-law interactions

We investigate the dynamics of overdamped $D$-dimensional systems of particles repulsively interacting through short-ranged power-law potentials, $V(r)\sim r^{-λ}\;(λ/D>1)$. 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 $q=1-λ/D$. 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.

cond-mat.stat-mech↗
arXiv2017

Fermi--Pasta--Ulam--Tsingou problems: Passage from Boltzmann to $q$-statistics

The 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 $P(ε) \sim e^{-βε}$ describes its equilibrium distribution of one-body energies, and its velocity distribution is Maxwellian, i.e., $P(v) \sim e^{- βv^2/2}$. We consider here a generalized system where the quartic coupling constant between sites decays as $1/d_{ij}^α$ $(α\ge 0; d_{ij} = 1,2,\dots)$. Through {\it first-principle} molecular dynamics we demonstrate that, for large $α$ (above $α\simeq 1$), 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 $α$ (below $α\simeq 1$), 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, $P(ε) \sim e_{q_ε}^{-β_ε ε} \equiv [1+(q_ε - 1) β_εε]^{-1/(q_ε-1)}$ with $q_ε > 1$, and its velocity distribution is given by $P(v) \sim e_{q_v}^{ - β_v v^2/2}$ with $q_v > 1$. Moreover, within small error bars, we verify $q_ε = q_v = q$, which decreases from an extrapolated value q $\simeq$ 5/3 to q=1 when $α$ increases from zero to $α\simeq 1$, and remains q = 1 thereafter.

cond-mat.stat-mech↗
arXiv2017

Statistical characterization of discrete conservative systems: The web map

We 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.

cond-mat.stat-mech↗
arXiv2017

Role of dimensionality in preferential attachment growth in the Bianconi-Barabási model

Scale-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 $Π_{i} \propto η_i k_i/r_{ij}^{α_A}$ $(1 \leq i < j; α_A \geq 0)$, where $η_i$ 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 $P(k) \propto e_q^{-k/κ}$, where $e_q^{-k/κ}$ is the $q$-exponential function naturally emerging within nonextensive statistical mechanics. We determine the index $q$ and $κ$ as functions of the quantities $α_A$ and $d$, and numerically verify that both present a universal behavior with respect to the scaled variable $α_A/d$. The same behavior also has been displayed by the dynamical $β$ exponent which characterizes the steadily growing number of links of a given site.

cond-mat.stat-mech↗
arXiv2017

Entanglement can preserve the compact nature of the phase-space occupancy

We 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.

cond-mat.stat-mech↗
arXiv2017

Thermal conductance of the coupled-rotator chain: Influence of temperature and size

Thermal 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 $T^{-α}$ with $α=2/(q-1) \simeq 3.64$.

cond-mat.stat-mech↗
arXiv2016

Nonlinear inhomogeneous Fokker-Planck equations: entropy and free-energy time evolution

We 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.

cond-mat.stat-mech↗
arXiv2016

Statistical characterization of the standard map

The 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 $ξ$ to the initial conditions, i.e., $ξ\sim e_{q_{\text{sen}}}^{λ_{q_{\text{sen}}} \,t}\;(λ_{q_{\text{sen}}}>0)$, 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.

cond-mat.stat-mech↗
arXiv2016

Dynamics and Statistics of the Fermi--Pasta--Ulam $β$--model with different ranges of particle interactions

In the present work we study the Fermi--Pasta--Ulam (FPU) $β$--model involving long--range interactions (LRI) in both the quadratic and quartic potentials, by introducing two independent exponents $α_1$ and $α_2$ 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 $0\leq α_2<1$, 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.

nlin.CD↗
arXiv2016

The limit distribution in the $q$-CLT for $q \ge 1$ is unique and can not have a compact support

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.

cond-mat.stat-mech↗