Resumo em Inglês:

Statistical mechanics constitutes one of the pillars of contemporary physics. Recognized as such - together with mechanics (classical, quantum, relativistic), electromagnetism and thermodynamics -, it is one of the mandatory theories studied at virtually all the intermediate-and advanced-level courses of physics around the world. As it normally happens with such basic scientific paradigms, it is placed at a crossroads of various other branches of knowledge. In the case of statistical mechanics, the standard theory - hereafter referred to as the Boltzmann-Gibbs (BG) statistical mechanics - exhibits highly relevant connections at a variety of microscopic, mesoscopic and macroscopic physical levels, as well as with the theory of probabilities (in particular, with the Central Limit Theorem, CLT ). In many circumstances, the ubiquitous efects of the CLT , with its Gaussian attractors (in the space of the distributions of probabilities), are present. Within this complex ongoing frame, a possible generalization of the BG theory was advanced in 1988 (C.T., J. Stat. Phys. 52, 479). The extension of the standard concepts is intended to be useful in those "pathological", and nevertheless very frequent, cases where the basic assumptions (molecular chaos hypothesis, ergodicity) for applicability of the BG theory would be violated. Such appears to be, for instance, the case in classical long-range-interacting many-body Hamiltonian systems (at the so-called quasi-stationary state). Indeed, in such systems, the maximal Lyapunov exponent vanishes in the thermodynamic limit N → ∞. This fact creates a quite novel situation with regard to typical BG systems, which generically have a positive value for this central nonlinear dynamical quantity. This peculiarity has sensible effects at all physical micro-, meso-and macroscopic levels. It even poses deep challenges at the level of the CLT . In the present occasion, after 20 years of the 1988 proposal, we undertake here an overview of some selected successes of the approach, and of some interesting points that still remain as open questions. Various theoretical, experimental, observational and computational aspects will be addressed.

Resumo em Inglês:

We provide an overview on superstatistical techniques applied to complex systems with time scale separation. Three examples of recent applications are dealt with in somewhat more detail: the statistics of small-scale velocity differences in Lagrangian turbulence experiments, train delay statistics on the British rail network, and survival statistics of cancer patients once diagnosed with cancer. These examples correspond to three different universality classes: Lognormal superstatistics, χ2-superstatistics and inverse χ2 superstatistics.

Resumo em Inglês:

As an essential component in the demonstration of an atypical, q-deformed, statistical mechanical structure in the dynamics of the Feigenbaum attractor we expose, at a previously unknown level of detail, the features of the dynamics of trajectories that either evolve towards the Feigenbaum attractor or are captured by its matching repellor. Amongst these features are the following: i) The set of preimages of the attractor and of the repellor are embedded (dense) into each other. ii) The preimage layout is obtained as the limiting form of the rank structure of the fractal boundaries between attractor and repellor positions for the family of supercycle attractors. iii) The joint set of preimages for each case form an infinite number of families of well-defined phase-space gaps in the attractor or in the repellor. iv) The gaps in each of these families can be ordered with decreasing width in accord to power laws and are seen to appear sequentially in the dynamics generated by uniform distributions of initial conditions. v) The power law with log-periodic modulation associated to the rate of approach of trajectories towards the attractor (and to the repellor) is explained in terms of the progression of gap formation. vi) The relationship between the law of rate of convergence to the attractor and the inexhaustible hierarchy feature of the preimage structure is elucidated. We discuss the function of these properties in the atypical thermodynamic framework existing at the period-doubling transition to chaos.

Resumo em Inglês:

In this talk I first review at an elementary level a selection of central limit theorems, including some lesser known cases, for sums and maxima of uncorrelated and correlated random variables. I recall why several of them appear in physics. Next, I show that there is room for new versions of central limit theorems applicable to specific classes of problems. Finally, I argue that we have insufficient evidence that, as a consequence of such a theorem, q-Gaussians occupy a special place in statistical physics.

Resumo em Inglês:

Experimental measurements in terrestrial laboratory, space and astrophysical observations of variation and fluctuation of nuclear decay constants, measurements of large enhancements in fusion reaction rate of deuterons implanted in metals and electron capture by nuclei in solar core indicate that these processes depend on the environment where they take place and possibly also on the fluctuation of some extensive parameters and eventually on stellar energy production. Electron screening is the first important environment effect. We need to develop a treatment beyond the Debye-Hückel screening approach, commonly adopted within global thermodynamic equilibrium. Advances in the description of these processes can be obtained by means of q-thermostatistics and/or superstatistics for metastable states. This implies to handle, without ambiguities, the case q < 1.

Resumo em Inglês:

Starting from the idea of Tsallis on non-extensive statistical mechanics and the q-entropy notion, we recall the Theil index Th and transform it into the Th q index. Both indices can be used to map onto themselves any time series in a non linear way. We develop an application of the Th q to the GDP evolution of 20 rich countries in the time interval [1950 -2003] and search for a proof of globalization of their economies. First we calculate the distances between the "new" time series and to their mean, from which such data simple networks are constructed. We emphasize that it is useful to, and we do, take into account different time "parameters": (i) the moving average time window for the raw time series to calculate the Th q index; (ii) the moving average time window for calculating the time series distances; (iii) a correlation time lag. This allows us to deduce optimal conditions to measure the features of the network, i.e. the appearance in 1970 of a globalization process in the economy of such countries and the present beginning of deviations. The q value hereby used is that which measures the overall data distribution and is equal to 1.8125.

Resumo em Inglês:

A family of entropy indices constructed in the framework of Tsallis entropy formalism is used to investigate ecological diversity. It represents a new perspective in ecology because a simple equation can incorporate all aspects of α-diversity, from richness to dominance and can be also related to a measure of species rarity. In addition, a generalized Kullback-Leibler distance, constructed in the framework of a nonextensive formalism, is recalled and used as a measure of β-diversity between two systems. These tools are applied to data relative to the macrophytes collected from two not far apart arms of Itaipu Reservoir, in Paraná River basin.

Resumo em Inglês:

Nonextensive statistical mechanics has been a source of investigation in mathematical structures such as deformed algebraic structures. In this work, we present some consequences of q-operations on the construction of q-numbers for all numerical sets. Based on such a construction, we present a new product that distributes over the q-sum. Finally, we present different patterns of q-Pascal's triangles, based on q-sum, whose elements are q-numbers.

Resumo em Inglês:

We investigate entropic aspects of the quantum entanglement dynamics of two-qubits systems interacting with an environment. In particular we consider the detection, based on the violation of classical entropic inequalities involving q-entropies, of the phenomenon of entanglement disappearance and subsequent entanglement revival during the alluded two-qubits' evolution.

Resumo em Inglês:

Limit distributions are not limited to uncorrelated variables but can be constructively derived for a large class of correlated random variables, as was shown e.g. in the context of large deviation theory [1], and recently in a very general setting by Hilhorst and Schehr [2]. At the same time it has been conjectured, based on numerical evidence, that several limit distributions originating from specific correlated random processes follow q-Gaussians. It could be shown that this is not the case for some of these situations, and more complicated limit distributions are necessary. In this work we show the derivation of the analytical form of entropy which -under the maximum entropy principle, imposing ordinary constraints- provides exactly these limit distributions. This is a concrete example for the necessity of more general entropy functionals beyond q statistics.

Resumo em Inglês:

A numerically efficient transfer matrix approach is used to investigate the validity of the Tsallis scaling hypothesis in the long-range Ising spin chain with competitive interactions. In this model, the interaction between two spins i and j placed r lattice steps apart is Ji, j = (-1)ζ(i,j)J0/rα, where ζ(i, j) is either 0 or 1. This procedure has succeeded to show the validity of the scaling hypothesis for the well investigated ferromagnetic version of the model, i.e., ζ(i, j)= 0,∀i, j, ∀α > 0. Results are reported for some models of a set, which is defined by requiring ζ(i, j) to be a periodic sequence of 0's and 1's. As expected from symmetry arguments, we find that the hypothesis is not valid when ζ(i, j)= 1,∀i, j and α < 1. however, it is verified, with high degree of numerical accuracy, when α < 1, for sequences in which the occurrence of ζ(i, j)= 0 is more frequent than that of ζ(i, j)= 1.

Resumo em Inglês:

In this work we analyze the implications of using a power law distribution of vertice's quality in the growth dynamics of a network studied by Bianconi and Barabási. Using this suggested distribution we show the degree distribution interpolates the Barabási et al. model and Bianconi et al. model. This modified model (with power law distribution) can help us understand the evolution of complex systems. Additionally, we determine the exponent gamma related to the degree distribution, the time evolution of the average number of links,< ki >∝ (t/i)β (i coincindes with the input-time of the i th node), the average path length and the clustering coefficient.

Resumo em Inglês:

Dynamical evolution of earthquake network is studied. Through the analysis of the real data taken from California and Japan, it is found that the values of the clustering coefficient exhibit a specific behavior around the moment of a main shock: the coefficient remains stationary before a main shock, suddenly jumps up at the main shock, and then slowly decreases to become stationary again. Thus, the network approach to seismicity dynamically characterizes main shocks in a peculiar manner.

Resumo em Inglês:

In this paper we analyze a model for the dynamics of the immune system interacting with a target population. The model consists in a set of two-dimensional delayed differential equations. The model is effectively infinite dimensional due to the presence of the delay and chaotic regimes can be supported. We show that a delayed response induces sustained oscillations and larger delay times implies in a series of bifurcations leading to chaos. The characteristic exponent of the critical power law relaxation towards the stationary state is obtained as well as the critical exponent governing the vanishing of the order parameter in the vicinity of the chaotic transition.

Resumo em Inglês:

A reduced (stereo-chemical) model is employed to study kinetic aspects of globular protein folding process, by Monte Carlo simulation. Nonextensive statistical approach is used: transition probability p i j between configurations i → j is given by p i j =[1 +(1 - q)ΔGi j/kB T ]1/(1-q), where q is the nonextensive (Tsallis) parameter. The system model consists of a chain of 27 beads immerse in its solvent; the beads represent the sequence of amino acids along the chain by means of a 10-letter stereo-chemical alphabet; a syntax (rule) to design the amino acid sequence for any given 3D structure is embedded in the model. The study focuses mainly kinetic aspects of the folding problem related with the protein folding time, represented in this work by the concept of first passage time (FPT). Many distinct proteins, whose native structures are represented here by compact self avoiding (CSA) configurations, were employed in our analysis, although our results are presented exclusively for one representative protein, for which a rich statistics was achieved. Our results reveal that there is a specific combinations of value for the nonextensive parameter q and temperature T, which gives the smallest estimated folding characteristic time (t). Additionally, for q = 1.1, (t) stays almost invariable in the range 0.9 < T < 1.3, slightly oscillating about its average value <img border=0 width=32 height=32 src="../../../../../../../img/revistas/bjp/v39n2a/a16txt01.gif" align=absmiddle > or = 27 ±σ, where σ = 2 is the standard deviation. This behavior is explained by comparing the distribution of the folding times for the Boltzmann statistics (q → 1), with respect to the nonextensive statistics for q = 1.1, which shows that the effect of the nonextensive parameter q is to cut off the larger folding times present in the original (q → 1) distribution. The distribution of natural logarithm of the folding times for Boltzmann statistics is a triple peaked Gaussian, while, for q = 1.1 (Tsallis), it is a double peaked Gaussian, suggesting that a log-normal process with two characteristic times replaced the original process with three characteristic times. Finally we comment on the physical meaning of the present results, as well its significance in the near future works.

Resumo em Inglês:

In a series of molecular dynamics simulations we analyzed structural and dynamics properties of water at different temperatures (213 K to 360 K), using the Simple Point Charge-Extended (SPC/E) water. We detected a q-exponential behavior in the history-dependent bond correlation function of hydrogen bonds. We found that q increases with T -1 below approximately 300 K and is correlated to the increase of the tetrahedral structure of water and the subdiffusive motion of the molecules.

Resumo em Inglês:

We address this work to investigate symbolic sequences with long-range correlations by using computational simulation. We analyze sequences with two, three and four symbols that could be repeated l times, with the probability distribution p(l) ∝ 1/l µ. For these sequences, we verified that the usual entropy increases more slowly when the symbols are correlated and the Tsallis entropy exhibits, for a suitable choice of q, a linear behavior. We also study the chain as a random walk-like process and observe a nonusual diffusive behavior depending on the values of the parameter µ.

Resumo em Inglês:

In this manuscript we give thought to the aftermath on the stable probability density function when standard multiplicative cascades are generalised cascades based on the q-product of Borges that emerged in the context of non-extensive statistical mechanics.

Resumo em Inglês:

It is shown that the Poissonian-like process with slowly diffusing-like time-dependent average interevent time may be represented as the superstatistical one and exhibits 1/ f noise. The distribution of the Poissonian-like interevent time may be expressed as q-exponential distribution of the Nonextensive Statistical Mechanics.

Resumo em Inglês:

Following the basic prescriptions of the Tsallis' nonextensive relativistic thermodynamics, we investigate the relevance of nonextensive statistical effects on the relativistic nuclear and subnuclear equation of state. In this framework, we study the first order phase transition from hadronic to quark-gluon plasma phase by requiring the Gibbs conditions on the global conservation of the electric and the baryon charges. The relevance of small deviations from the standard extensive statistics is investigated in the context of intermediate and high energy heavy-ion collisions.

Resumo em Inglês:

We investigate the quantum entanglement-related features of the many-body model of Plastino and Moszkowski [N. Cimento 47 (1978) 470]. This is an exactly solvable N-body, SU2 two-level model exhibiting several quantum phase transitions. We show that these transitions happen to be also entanglement-transitions associated with different many-body Dicke states. The main properties of the model considered here make it particularly well suited to study, by recourse to exact analytical computations, some connections between quantum phase transitions and quantum entanglement-theory.

Resumo em Inglês:

The nonextensive statistical mechanics proposed by Tsallis is today an intense and growing research field. Probability distributions which emerges from the nonextensive formalism(q-distributions) have been applied to an impressive variety of problems. In particular, the role of q-distributions in the interdisciplinary field of complex systems has been expanding. Here, we make a brief review of q-exponential, q-Gaussian and q-Weibull distributions focusing some of their basic properties and recent applications. The richness of systems analyzed may indicate future directions in this field.

Resumo em Inglês:

In the framework of the statistical mechanics based on the Sharma-Taneja-Mittal entropy we derive a family of nonlinear Fokker-Planck equations characterized by the associated non-increasing Lyapunov functional. This class of equations describes kinetic processes in anomalous mediums where both super-diffusive and subdiffusive mechanisms arise contemporarily and competitively. We classify the Lie symmetries and derive the corresponding group-invariant solutions for the physically meaningful Uhlenbeck-Ornstein process. For the purely diffusive process we show that any localized state asymptotically approaches a bell shape well fitted by a generalized Gaussian which is, in general, a quasi-self-similar solution for this class of purely diffusive equations.

Resumo em Inglês:

We investigate solutions, by using the Green function approach, for a system governed by a non-Markovian Fokker-Planck equation and subjected to a Comb structure. This structure consists of the axis of structure as the backbone and fingers which are attached perpendicular to the axis. For this system, we consider an arbitrary initial condition, in the presence of time dependent diffusion coefficients and spatial fractional derivative, and analyze the connection to the anomalous diffusion.

Resumo em Inglês:

In this paper we apply an evolving stochastic method to construct simple and effective Artificial Neural Networks, based on the theory of Tsallis statistical mechanics. Our aim is to establish an automatic process for building a smaller network with high classification performance. We aim to assess the utility of the method based on statistical mechanics for the estimation of transparent coating material on security papers and cholesterol levels in blood samples. Our experimental study verifies that there are indeed improvements in the overall performance in terms of classification success and at the size of network compared to other efficient backpropagation learning methods.

Resumo em Inglês:

We have previously described neurotic psychopathology and psychoanalytic working-through by an associative memory mechanism, based on a neural network. Memory was initially modelled by a Boltzmann machine (BM). We simulated known microscopic mechanisms that control synaptic properties and showed that the network self-organizes to a hierarchical, clustered structure. Some properties of the complex networks which result from this self-organization indicate that a generalization of the BM may be necessary to model memory. We have thus generalized the memory model, using Generalized Simulated Annealing, derived from the nonextensive formalism, and show some properties of the resulting memory access mechanism.

Resumo em Inglês:

In the present contribution, we propose a possible way to discuss the distributions of words in a given text. We have devoted our study to discuss some relevant properties observed in Spanish texts of Latin-American writers. We start analyzing the appearance of distributions of the frequency of occurrence in the Zipf perspective. We identify two regions of behavior separated by a special point. In order to correctly define such a point, we work beyond the Zipf law, defining other probability distribution that takes the frequency of repetition of a particular word among other different words into account. At this point, we take the linguistic problem to a statistical level. We make an effort to characterize the point of separation between two regions, via the Binder cumulant of fourth order, as it is made in the characterization of critical points in phase transitions of physical systems.

Resumo em Inglês:

An important manifestation of the Uncertainty Principle, one of the cornerstones of our present understanding of Nature, is that related to semiclassical localization in phase-space. We wish here to add some notes on the subject with reference to the canonical harmonic oscillator problem, with emphasis in the concepts of semiclassical Husimi distributions, the associated Wehrl entropy, escort distributions, and Fisher's information measure.

Resumo em Inglês:

We argue that nonlocal effects represented by fractionary terms in the kinetic energy can be relevant to achieve a satisfactory phenomenological description of the thermal behavior of the specific heat of non-crystalline solids at very low temperature. We propose a simple model formed by the direct sum of two Hamiltonians, one of which is obtained by incorporating fractional derivatives in the kinetic energy of a conventional Hamiltonian, and the other one accounts for the presence of phonons in the system. Some experimental data are used to support the proposed description.

Resumo em Inglês:

In this paper we studied the research activity of Brazilian Institutions for all sciences and also their performance in the area of physics between 1945 and December 2008. All the data come from the Web of Science database for this period. The analysis of the experimental data shows that, within a nonextensive thermostatistical formalism, the Tsallis q-exponential distribution N(c) can constitute a new characterization of the research impact for Brazilian Institutions. The data examined in the present survey can successfully be fitted by applying a universal curve namely, <img border=0 width=32 height=32 src="../../../../../../../img/revistas/bjp/v39n2a/a31tex01.gif" align=absmiddle>with q <img border=0 width=32 height=32 src="../../../../../../../img/revistas/bjp/v39n2a/a04tex01.gif" align=absmiddle>4/3 for all the available citations c, T being an "effective temperature". The present analysis ultimately suggests that via the "effective temperature" T, we can provide a new performance metric for the impact level of the research activity in Brazil, taking into account the number of the publications and their citations. This new performance metric takes the "quantity" (number of publications) and the "quality" (number of citations) for different Brazilian Institutions into account. In addition we analyzed the research performance in Brazil to show how the scientific research activity changes with time, for instance between 1945 to 1985, then during the period 1986-1990, 1991-1995, and so on until the present. Finally, this work intends to show a new methodology that can be used to analyze and compare institutions within a given country.