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In this work, we study the effectiveness of employing archetypal aperiodic sequencing -- namely Fibonacci, Thue-Morse, and Rudin-Saphiro -- on the Parrondian effect. From a capital gain perspective, our results show that these series do yield a Parrondo's Paradox with the Thue-Morse based strategy outperforming not only the other two aperiodic strategies but benchmark Parrondian games with random and periodical ($AABBAABB\ldots$) switching as well. The least performing of the three aperiodic strategies is the Rudin-Shapiro. To elucidate the underlying causes of these results, we analyze the cross-correlation between the capital generated by the switching protocols and that of the isolated losing games. This analysis reveals that a pronounced anti-correlation (below -0.95) with both isolated games is typically required to achieve a robust manifestation of Parrondo's effect. We also study the influence of the sequencing on the capital using the lacunarity and persistence measures. In general, we observe that the switching protocols tend to become less performing in terms of the capital as one increases the persistence and thus approaches the features of an isolated losing game. For the (log-)lacunarity, a property related to heterogeneity, we notice that for small persistence (less than 0.5) the performance increases with the lacunarity with a maximum around 0.4. In respect of this, our work shows that the optimisation of a switching protocol is strongly dependent on a fine tune between persistence and heterogeneity., Comment: 9 pages, 8 figures and 2 appendices

We study the Galam's majority-rule model in the presence of an independent behavior that can be driven intrinsically or can be mediated by information regarding the collective opinion of the whole population. We first apply the mean-field approach where we obtained an explicit time-dependent solution for the order parameter of the model. We complement our results with Monte Carlo simulations where our findings indicate that independent opinion leads to order-disorder continuous nonequilibrium phase transitions. Finite-size scaling analysis show that the model belongs to the mean-field Ising model universality class. Moreover, results from an approach with the Kramers-Moyal coefficients provide insights about the social volatility., Comment: Invited paper for the special issue "In honor of professor Serge Galam for his 70th birthday and forty years of sociophysics" (special issue editors: A. Martins, T. Cheon, X.Tang, B. Chopard, S. Biswas), to appear in Physics

We investigate the transport and entanglement properties exhibited by quantum walks with coin operators concatenated in a space-time fractal structure. Inspired by recent developments in photonics, we choose the paradigmatic Sierpinski gasket. The 0-1 pattern of the fractal is mapped into an alternation of the generalized Hadamard-Fourier operators. In fulfilling the blank space on the analysis of the impact of disorder in quantum walk properties -- specifically, fractal deterministic disorder --, our results show a robust effect of entanglement enhancement as well as an interesting novel road to superdiffusive spreading with a tunable scaling exponent attaining effective ballistic diffusion. Namely, with this fractal approach it is possible to obtain an increase in quantum entanglement without jeopardizing spreading. Alongside those features, we analyze further properties such as the degree of interference and visibility. The present model corresponds to a new application of fractals in an experimentally feasible setting, namely the building block for the construction of photonic patterned structures., Comment: 16 pages, 9 figures

We study a 2-D disordered time-discrete quantum walk based on 1-D `generalized elephant quantum walk' where an entangling coin operator is assumed and which paves the way to a new set of properties. We show that considering a given disorder in one direction, it is possible to control the degree of spreading and entanglement in the other direction. This observation helps assert that the random quantum walks of this ilk serve as a controllable decoherence channel with the degree of randomness being the tunable parameter and highlight the role of dimensionality in quantum systems regarding information and transport., Comment: 16 pages, 19 figures

Recently, it was introduced a generalization of a nonstandard step operator named the elephant quantum walk (EQW). With proper statistical distribution for the steps, that generalized EQW (gEQW) can be tuned to exhibit a myriad of dynamical scaling behavior ranging from standard diffusion to %and superdiffusion to ballistic and hyperballistic spreading. In this work, we study the influence of the statistics of the step size and the delocalization of the initial states on the entanglement entropy of the coin. Our results show that the gEQW generates maximally entangled states for almost all initial coin states and coin operators considering initially localized walkers and for the delocalized ones, taking the proper limit, the same condition is guaranteed. Differently from all the previous protocols that produce highly entangled states via QWs, this model is not upper-bounded by ballistic spreading and hence opens novel prospects for applications of dynamically disordered QWs as a robust maximal entanglement generator in programmable setups that ranges from slower-than-ballistic to faster-than ballistic., Comment: 14 pages, 14 figures, typos corrected, figure font size increased, new results subsection included for clarification

By considering an explicit nonequilibrium model, we analyze the statistics of the irreversible work, $w_{\rm irr}$, and irreversible entropy production, $\Delta_i s$, within the stochastic energetics framework. Restating the second law of thermodynamics as a function of $w_{\rm irr}$, we introduce the explicit probability of violating the canonical form of that second law for a different set of parameters and initial conditions of the model. Moreover, we study the irreversible entropy production along the same lines, since it can be cast as a generalization of the irreversible work. From an informational perspective, our result allows quantifying the probability of deleting information without performing work, contrarily to the Landauer's Principle, which we classify as an informational free lunch. We chose for initial conditions cases of low information content (equilibrium) and high information content (delta distributed)., Comment: 9 pages, 4 figures

Three different definitions of effective temperature -- $\mathcal{T}_{\rm k}$, $\mathcal{T}_{\rm i}$ and $\mathcal{T}_{\rm r}$ related to kinetic theory, system entropy and response theory, respectively -- are applied in the description of a non-equilibrium generalised massive Langevin model in contact with dichotomous noise. The differences between the definitions of $\mathcal{T}$ naturally wade out as the reservoir reaches its white-noise limit, approaching Gaussian features. The same framework is employed in its overdamped version as well, showing the loss of inertial contributions to the dynamics of the system also makes the three mentioned approaches for effective temperature equivalent.

In this work, we address a multicoupled dynamics on complex networks with tunable structural segregation. Specifically, we work on a networked epidemic spreading under a vaccination campaign with agents in favor and against the vaccine. Our results show that such coupled dynamics exhibits a myriad of phenomena such as nonequilibrium transitions accompanied by bistability. Besides we observe the emergence of an intermediate optimal segregation level where the community structure enhances negative opinions over vaccination but counterintuitively hinders - rather than favoring - the global disease spreading. Thus, our results hint vaccination campaigns should avoid policies that end up segregating excessively anti-vaccine groups so that they effectively work as echo chambers in which individuals look to confirmation without jeopardising the safety of the whole population., Comment: 11 pages, 8 figures, to appear in PRE

We investigate the phenomenology emerging from a 2-species dynamics under the scenario of a quasi-neutral competition within a metapopulation framework. We employ stochastic and deterministic approaches, namely spatially-constrained individual-based Monte Carlo simulations and coupled mean-field ODEs. Our results show the multifold interplay between competition, birth-death dynamics and spatial constraints induces a nonmonotonic relation between the ecological majority-minority switching and the diffusion between patches. This means that diffusion can set off birth-death ratios and enhance the preservation of a species., Comment: 6 figures, 10 pages, to appear in Nonlinear Dynamics

We investigate the emerging properties of quantum walks with temporal disorder engineered from a binary Markov chain with tailored correlation, $C$, and disorder strength, $r$. We show that when the disorder is weak -- $r \ll 1$ -- the introduction of negative correlation leads to a counter-intuitive higher production of spin-lattice entanglement entropy, $S_e$, than the setting with positive correlation, that is $S_e(-|C|)>S_e(|C|)$. These results show that negatively correlated disorder plays a more important role in quantum entanglement than it has been assumed in the literature., Comment: 9 pages and 5 figures

We show that a genuine Parrondo paradox can emerge in two-state quantum walks without resorting to experimentally intricate high-dimensional coins. To achieve such goal we employ a time-dependent coin operator without breaking the translation spatial invariance of the system.

We study the potential scenarios from a Susceptible-Infected-Recovered-Asymptomatic-Symptomatic-Dead (SIRASD) model. As a novelty, we consider populations that differ in their degree of compliance with social distancing policies following socioeconomic attributes that are observed in emerging and developing countries. Considering epidemiological parameters estimated from data of the propagation of SARS-CoV-2 in Brazil -- where there is a significant stake of the population making their living in the informal economy and thus prone to not follow self-isolation -- we assert that if the confinement measures are lifted too soon, namely as much as one week of consecutive declining numbers of new cases, it is very likely the appearance of a second peak. Our approach should be valid for any country where the number of people involved in the informal economy is a large proportion of the total labor force. In summary, our results point out the crucial relevance of target policies for supporting people in the informal economy to properly comply with preventive measures during the pandemic., Comment: 27 pages, 15 figures, 3 appendices, to appear in IJMPC

We analyze a set of discrete-time quantum walks for which the displacements on a chain follow binary aperiodic jumps according to three paradigmatic sequences: Fibonacci, Thue-Morse and Rudin-Shapiro. We use a generalized Hadamard coin $\widehat C_{H}$ as well as a generalized Fourier coin $\widehat C_{K}$. We verify the QW experiences a slowdown of the wavepacket spreading --- $\sigma ^2 (t) \sim t^\alpha $ --- by the aperiodic jumps whose exponent, $\alpha$, depends on the type of aperiodicity. Additional aperiodicity-induced effects also emerge, namely: (i) while the superdiffusive regime ($1<\alpha<2$) is predominant, $\alpha$ displays an unusual sensibility with the type of coin operator where the more pronounced differences emerge for the Rudin-Shapiro and random protocol; (ii) even though the angle $\theta$ of the coin operator is homogeneous in space and time, there is a nonmonotonic dependence of $\alpha$ with $\theta$. Fingerprints of the aperiodicity in the hoppings are also found when additional distributional measures such as Shannon entropy, IPR, Jensen-Shannon dissimilarity, and kurtosis are computed. Finally, we argue the spin-lattice entanglement is enhanced by aperiodic jumps., Comment: Updated version, close to the published one

We extend to the gamut of functional forms of the probability distribution of the time-dependent step-length a previous model dubbed Elephant Quantum Walk, which considers a uniform distribution and yields hyperballistic dynamics where the variance grows cubicly with time, $\sigma ^2 \propto t^3$, and a Gaussian for the position of the walker. We investigate this proposal both locally and globally with the results showing that the time-dependent interplay between interference, memory and long-range hopping leads to multiple transitions between dynamical regimes, namely ballistic $\rightarrow$ diffusive $\rightarrow$ superdiffusive $\rightarrow$ ballistic $\rightarrow$ hyperballistic for non-hermitian coin whereas the first diffusive regime is quelled for implementations using the Hadamard coin. In addition, we observe a robust asymptotic approach to maximal coin-space entanglement., Comment: 11 pages, 11 figures

How diffusion impacts on ecological dynamics under the Allee effect and spatial constraints? That is the question we address. Employing a microscopic minimal model in a metapopulation (without imposing nonlinear birth and death rates) we evince --- both numerically and analitically --- the emergence of an optimal diffusion that maximises the survival probability. Even though, at first such result seems counter-intuitive, it has empirical support from recent experiments with engineered bacteria. Moreover, we show that this optimal diffusion disappears for loose spatial constraints., Comment: 16 pages; 6 figures

We explore the impact of long-range memory on the properties of a family of quantum walks in a one-dimensional lattice and discrete time, which can be understood as the quantum version of the classical "Elephant Random Walk" non-Markovian process. This Elephant Quantum Walk is robustly superballistic with the standard deviation showing a constant exponent, $\sigma \propto t^{3/ 2}$ , whatever the quantum coin operator, on which the diffusion coefficient is dependent. On the one hand, this result indicates that contrarily to the classical case, the degree of superdiffusivity in quantum non- Markovian processes of this kind is mainly ruled by the extension of memory rather than other microscopic parameters that explicitly define the process. On the other hand, these parameters reflect on the diffusion coefficient., Comment: 4 figures, any comments is welcome. Accepted in PRA

In 1967, Z. Rieder, J. L. Lebowitz and E. Lieb (RLL) introduced a model of heat conduction on a crystal that became a milestone problem of non-equilibrium statistical mechanics. Along with its inability to reproduce Fourier's Law - which subsequent generalizations have been trying to amend - the RLL model is also characterized by awkward cusps at the ends of the non-equilibrium chain, an effect that has endured all these years without a satisfactory answer. In this paper, we first show that such trait stems from the insufficiency of pinning interactions between the chain and the substrate. Assuming the possibility of pinning the chain, the analysis of the temperature profile in the space of parameters reveals that for a proper combination of the border and bulk pinning values, the temperature profile may shift twice between the RLL cuspidal behavior and the expected monotonic local temperature evolution along the system, as a function of the pinning. At those inversions, the temperature profile along the chain is characterized by perfect plateaux: at the first threshold, the cumulants of the heat flux reach their maxima and the vanishing of the two-point velocity correlation function for all sites of the chain so that the system behaves similarly to a "phonon box". On the other hand, at the second change of the temperature profile, we still have the vanishing of the two-point correlation function but only for the bulk, which explains the emergence of the temperature plateau and thwarts the reaching of the maximal values of the cumulants of the heat flux.

Employing a recent technique which allows the representation of nonstationary data by means of a juxtaposition of locally stationary patches of different length, we introduce a comprehensive analysis of the key observables in a financial market: the trading volume and the price fluctuations. From the segmentation procedure we are able to introduce a quantitative description of a group of statistical features (stylizes facts) of the trading volume and price fluctuations, namely the tails of each distribution, the U-shaped profile of the volume in a trading session and the evolution of the trading volume autocorrelation function. The segmentation of the trading volume series provides evidence of slow evolution of the fluctuating parameters of each patch, pointing to the mixing scenario. Assuming that long-term features are the outcome of a statistical mixture of simple local forms, we test and compare different probability density functions to provide the long-term distribution of the trading volume, concluding that the log-normal gives the best agreement with the empirical distribution. Moreover, the segmentation of the magnitude price fluctuations are quite different from the results for the trading volume, indicating that changes in the statistics of price fluctuations occur at a faster scale than in the case of trading volume., Comment: 13 pages, 12 figures

We study the multifractal nature of the Central England Temperature (CET) anomaly, a time series that spans more than 200 years. The series is analyzed as a complete data set and considering a sliding window of 11 years. In both cases, we quantify the broadness of the multifractal spectrum as well as its components defined by the deviations from the Gaussian distribution and the influence of the dependence between measurements. The results show that the chief contribution to the multifractal structure comes from the dynamical dependencies, mainly the weak ones, followed by a residual contribution of the deviations from Gaussianity. However, using the sliding window, we verify that the spikes in the non-Gaussian contribution occur at very close dates associated with climate changes determined in previous works by component analysis methods. Moreover, the strong non-Gaussian contribution found in the multifractal measures from the 1960s onwards is in agreement with global results very recently proposed in the literature., Comment: 21 pages, 10 figures

Focussing on a paradigmatic small system consisting of two coupled damped oscillators, we survey the role of the L\'evy-It\^o nature of the noise in the thermal conductance. For white noises, we prove that the L\'evy-It\^o composition (Lebesgue measure) of the noise is irrelevant for the thermal conductance of a non-equilibrium linearly coupled chain, which signals the independence between mechanical and thermodynamical properties. On the other hand, for the non-linearly coupled case, the two types of properties mix and the explicit definition of the noise plays a central role., Comment: 9 pages, 2 figures. To be published in Physical Review E

Flocking is a paradigmatic example of collective animal behaviour, where decentralized interaction rules give rise to a globally ordered state. In the emergence of order out of self-organization we find similarities between biological systems, as bird flocks, and some physical systems, as ferromagnets. In both cases, the tendency of individuals to align to their neighbours gives rise to a polarized state. There is, however, one crucial difference: the interaction network within an animal group is not necessarily fixed in time, as each individual moves and may change its neighbours. Therefore, the dynamical interaction mechanism in biological and physical system can be quite different, not only due to the gross disparity in the complexity of the individual entities, but also because of the potential role of inter-individual motion. To assess the relevance of this mechanism it is necessary to gain quantitative experimental information about how much individuals move with respect to each other within the group. Here, by using data from field observations on starlings, we study the diffusion properties of individual birds within a flock and investigate the effect of diffusion on the dynamics of the interaction network. We find that birds diffuse faster than Brownian particles (superdiffusion) and in a strongly anisotropic way. We also find that neighbours change in time exclusively as a consequence of diffusion, so that no specific mechanism to keep one's neighbours seems to be enforced. Finally, we study the diffusion properties of birds at the border of the flock. We find that these individuals remain on the border significantly longer than what would be expected on the basis of a purely diffusional model, suggesting that there is a sort barrier a bird must cross to make the transition from border to interior of the flock., Comment: 22 pages, 10 figures

In this work, we inspect the reliability of controlling and quelling an epidemic disease mimicked by a Susceptible-Infected-Susceptible (SIS) model defined on a complex network by means of current and implementable quarantine and isolation policies. Specifically, we consider that each individual in the network is originally linked to two types of individuals: members of the same household and acquaintances. The topology of this network evolves taking into account a probability $q$ that aims at representing the quarantine or isolation process in which the connection with acquaintances is disrupted according to standard policies of control of epidemics. Within current policies of self-isolation and standard infection rates, our results show that the propagation is either only controllable for hypothetical rates of compliance or uncontrollable at all., Comment: 15 pages, 6 figures, to appear in JSTAT

In this work we study, under the Stratonovich definition, the problem of the damped oscillatory massive particle subject to a heterogeneous Poisson noise characterised by a rate of events, \lambda (t), and a magnitude, \Phi, following an exponential distribution. We tackle the problem by performing exact time-averages over the noise in a similar way to previous works analysing the problem of the Brownian particle. From this procedure we obtain the long-term equilibrium distributions of position and velocity as well as analytical asymptotic expressions for the injection and dissipation of energy terms. Considerations on the emergence of stochastic resonance in this type of system are also set forth., Comment: 21 pages, 5 figures. To be published in Journal of Statistical Mechanics: Theory and Experiment

We introduce a generalisation of the well-known ARCH process, widely used for generating uncorrelated stochastic time series with long-term non-Gaussian distributions and long-lasting correlations in the (instantaneous) standard deviation exhibiting a clustering profile. Specifically, inspired by the fact that in a variety of systems impacting events are hardly forgot, we split the process into two different regimes: a first one for regular periods where the average volatility of the fluctuations within a certain period of time is below a certain threshold and another one when the local standard deviation outnumbers it. In the former situation we use standard rules for heteroscedastic processes whereas in the latter case the system starts recalling past values that surpassed the threshold. Our results show that for appropriate parameter values the model is able to provide fat tailed probability density functions and strong persistence of the instantaneous variance characterised by large values of the Hurst exponent is greater than 0.8, which are ubiquitous features in complex systems., Comment: 18 pages, 5 figures, 1 table. To published in PLoS one

Using a single functional form which is able to represent several different classes of statistical distributions, we introduce a preliminary study of the ferromagnetic Ising model on the cubic lattices under the influence of non-Gaussian local external magnetic field. Specifically, depending on the value of the tail parameter, $\tau $ ($\tau < 3$), we assign a quenched random field that can be platykurtic (sub-Gaussian) or leptokurtic (fat-tailed) form. For $\tau< 5/3$, such distributions have finite standard deviation and they are either the Student-$t$ ($1< \tau< 5/3$) or the $r$-distribution ($\tau< 1$) extended to all plausible real degrees of freedom with the Gaussian being retrieved in the limit $\tau \rightarrow 1$. Otherwise, the distribution has got the same asymptotic power-law behaviour as the $\alpha$-stable L\'{e}vy distribution with $\alpha = (3 - \tau)/(\tau - 1)$. The uniform distribution is achieved in the limit $\tau \rightarrow \infty$. Our results purport the existence of ferromagnetic order at finite temperatures for all the studied values of $\tau$ with some mean-field predictions surviving in the three-dimensional case., Comment: Report of the first results of an ongoing project. 12 pages and 8 figures. Comments welcome

The return distributions of the coherent noise model are studied for the system size independent case. It is shown that, in this case, these distributions are in the shape of q-Gaussians, which are the standard distributions obtained in nonextensive statistical mechanics. Moreover, an exact relation connecting the exponent $\tau$ of avalanche size distribution and the q value of appropriate q-Gaussian has been obtained as q=(tau+2)/tau. Making use of this relation one can easily determine the q parameter values of the appropriate q-Gaussians a priori from one of the well-known exponents of the system. Since the coherent noise model has the advantage of producing different tau values by varying a model parameter \sigma, clear numerical evidences on the validity of the proposed relation have been achieved for different cases. Finally, the effect of the system size has also been analysed and an analytical expression has been proposed, which is corroborated by the numerical results., Comment: 14 pages, 3 figs

We consider Feller mean-reverting square-root diffusion, which has been applied to model a wide variety of processes with linearly state-dependent diffusion, such as stochastic volatility and interest rates in finance, and neuronal and populations dynamics in natural sciences. We focus on the statistical mixing (or superstatistical) process in which the parameter related to the mean value can fluctuate - a plausible mechanism for the emergence of heavy-tailed distributions. We obtain analytical results for the associated probability density function (both stationary and time dependent), its correlation structure and aggregation properties. Our results are applied to explain the statistics of stock traded volume at different aggregation scales., Comment: 16 pages, 3 figures. To be published in Journal of Statistical Mechanics: Theory and Experiment

In this manuscript we introduce a generalisation of the log-Normal distribution that is inspired by a modification of the Kaypten multiplicative process using the $q$-product of Borges [Physica A \textbf{340}, 95 (2004)]. Depending on the value of q the distribution increases the tail for small (when $q<1$) or large (when $q>1$) values of the variable upon analysis. The usual log-Normal distribution is retrieved when $q=1$. The main statistical features of this distribution are presented as well as a related random number generators and tables of quantiles of the Kolmogorov-Smirnov. Lastly, we illustrate the application of this distribution studying the adjustment of a set of variables of biological and financial origin., Comment: 25 pages, 7 figures

The influence of the tail features of the local magnetic field probability density function (PDF) on the ferromagnetic Ising model is studied in the limit of infinite range interactions. Specifically, we assign a quenched random field whose value is in accordance with a generic distribution that bears platykurtic and leptokurtic distributions depending on a single parameter $\tau < 3$ to each site. For $\tau< 5/3$, such distributions, which are basically Student-$t$ and $r$-distribution extended for all plausible real degrees of freedom, present a finite standard deviation, if not the distribution has got the same asymptotic power-law behavior as a $\alpha $-stable L\'evy distribution with $\alpha = (3 - \tau)/(\tau - 1)$. For every value of $\tau $, at specific temperature and width of the distribution, the system undergoes a continuous phase transition. Strikingly, we impart the emergence of an inflexion point in the temperature-PDF width phase diagrams for distributions broader than the Cauchy-Lorentz ($\tau = 2$) which is accompanied with a divergent free energy per spin (at zero temperature)., Comment: 12 pages, 4 figures To be published in Physical Review E

In this manuscript we discuss the effectiveness of the Kozachenko-Leonenko entropy estimator when generalised to cope with entropic forms customarily applied to study systems evincing asymptotic scale invariance and dependence (either linear or non-linear type). We show that when the variables are independently and identically distributed the estimator is only valuable along the whole domain if the data follow the uniform distribution, whereas for other distributions the estimator is only effectual in the limit of the Boltzmann-Gibbs-Shanon entropic form. We also analyse the influence of the dependence (linear and non-linear) between variables on the accuracy of the estimator between variables. As expected in the last case the estimator looses efficiency for the Boltzmann-Gibbs-Shanon entropic form as well., Comment: 10 pages. 5 Figures. 2 Appendices with 2 figures

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., Comment: To appear in the special issue of the Brazilian Journal of Physics "Proceedings of the 'International Conference on Nonextensive Statistical Mechanics and Applications'"

A metabolic model can be represented as bipartite graph comprising linked reaction and metabolite nodes. Here it is shown how a network of conserved fluxes can be assigned to the edges of such a graph by combining the reaction fluxes with a conserved metabolite property such as molecular weight. A similar flux network can be constructed by combining the primal and dual solutions to the linear programming problem that typically arises in constraint-based modelling. Such constructions may help with the visualisation of flux distributions in complex metabolic networks. The analysis also explains the strong correlation observed between metabolite shadow prices (the dual linear programming variables) and conserved metabolite properties. The methods were applied to recent metabolic models for Escherichia coli, Saccharomyces cerevisiae, and Methanosarcina barkeri. Detailed results are reported for E. coli; similar results were found for the other organisms., Comment: 9 pages, 4 figures, RevTeX 4.0, supplementary data available (excel)

In this manuscript, we analytically and numerically study statistical properties of an heteroskedastic process based on the celebrated ARCH generator of random variables whose variance is defined by a memory of $q_{m}$-exponencial, form ($e_{q_{m}=1}^{x}=e^{x}$). Specifically, we inspect the self-correlation function of squared random variables as well as the kurtosis. In addition, by numerical procedures, we infer the stationary probability density function of both of the heteroskedastic random variables and the variance, the multiscaling properties, the first-passage times distribution, and the dependence degree. Finally, we introduce an asymmetric variance version of the model that enables us to reproduce the so-called leverage effect in financial markets., Comment: 24 pages

In this manuscript we present a comparative study about the determination of the relaxation (\textit{i.e.}, independence) time scales obtained from the correlation function, the mutual information, and a criterion based on the evaluation of a nonextensive generalisation of mutual entropy. Our results show that, for systems with a small degree of complexity, standard mutual information and the criterion based on its nonextensive generalisation provide the same scale, whereas for systems with a higher complex dynamics the standard mutual information presents a time scale consistently smaller., Comment: 14 pages. To appear in Physica D

In this manuscript we analyse the behaviour of the probability density function of the sum of $N$ deterministic variables generated from the triangle map of Casati-Prosen. For the case in which the map is both ergodic and mixing the resulting probability density function quickly concurs with the Normal distribution. When these properties are modified the resulting probability density functions are described by power-laws. Moreover, contrarily to what it would be expected, as the number of added variables $N$ increases the distance to Gaussian distribution increases. This behaviour goes against standard central limit theorem. By extrapolation of our finite size results we preview that in the limit of $N$ going to infinity the distribution has the same asymptotic decay as a Lorenztian (or a $q=2$-Gaussian)., Comment: 14 pages. To appear in Physics Letters A

In the paper the question - Is the q-Fourier transform of a q-Gaussian a q'-Gaussian (with some q') up to a constant factor? - is studied for the whole range of $q\in (-\infty, 3)$. This question is connected with applicability of the q-Fourier transform in the study of limit processes in nonextensive statistical mechanics. We prove that the answer is affirmative if and only if q > 1, excluding two particular cases of q<1, namely, q = 1/2 and q = 2/3, which are also out of the theory valid for q \ge 1. We also discuss some applications of the q-Fourier transform to nonlinear partial differential equations such as the porous medium equation., Comment: 14 pages A new section on a related solution of the porous medium equation in comparison with the previous version has been introduce

In this manuscript we present a comprehensive study on the multifractal properties of high-frequency price fluctuations and instantaneous volatility of the equities that compose Dow Jones Industrial Average. The analysis consists about quantification of dependence and non-Gaussianity on the multifractal character of financial quantities. Our results point out an equivalent influence of dependence and non-Gaussianity on the multifractality of time series. Moreover, we analyse l-diagrams of price fluctuations. In the latter case, we show that the fractal dimension of these maps is basically independent of the lag between price fluctuations that we assume., Comment: 20 pages

In this article we review recent generalisations of the central limit theorem for the sum of specially correlated (or q-independent) variables, focusing on q greater or equal than 1. Specifically, this kind of correlation turns the probability density function known as q-Gaussian, which emerges upon maximisation of the entropy Sq, into an attractor in probability space. Moreover, we also discuss a q-generalisation of a-stable Levy distributions which can as well be stable for this special kind of correlation.Within this context, we verify the emergence of a triplet of entropic indices which relate the form of the attractor, the correlation, and the scaling rate, similar to the q-triplet that connects the entropic indices characterising the sensitivity to initial conditions, the stationary state, and relaxation to the stationary state in anomalous systems., Comment: 14 pages, 4 figures, and 1 table. To appear in the Proceedings of the conference CTNEXT07, Complexity, Metastability and Nonextensivity, Catania, Italy, 1-5 July 2007, Eds. S. Abe, H.J. Herrmann, P. Quarati, A. Rapisarda and C. Tsallis (American Institute of Physics, 2008) in press

In this article we review the standard versions of the Central and of the Levy-Gnedenko Limit Theorems, and illustrate their application to the convolution of independent random variables associated with the distribution known as q-Gaussian. This distribution emerges upon extremisation of the nonadditive entropy, basis of nonextensive statistical mechanics. It has a finite variance for q < 5/3, and an infinite one for q > 5/3. We exhibit that, in the case of (standard) independence, the q-Gaussian has either the Gaussian (if q < 5/3) or the a-stable Levy distributions (if q > 5/3) as its attractor in probability space. Moreover, we review a generalisation of the product, the q-product, which plays a central role in the approach of the specially correlated variables emerging within the nonextensive theory., Comment: 13 pages, 4 figures. To appear in the Proceedings of the conference CTNEXT07, Complexity, Metastability and Nonextensivity, Catania, Italy, 1-5 July 2007, Eds. S. Abe, H.J. Herrmann, P. Quarati, A. Rapisarda and C. Tsallis (American Institute of Physics, 2008) in press

In this manuscript we analyse the long-term probability density function of non-stationary dynamical processes which are enclosed inward the Feller class of processes with time varying exponents for multiplicative noise. The update in the value of the exponent occurs in the same conditions presented by Beck and Cohen for superstatistics. Moreover, we are able to provide a dynamical scenario for the emergence of a generalisation of the Weibull distribution previously introduced., Comment: 7 pages, 8 figures. A note about the application on turbulence models has been added to this final published version

In this manuscript we analyse the leading statistical properties of fluctuations of (log) 3-month US Treasury bill quotation in the secondary market, namely: probability density function, autocorrelation, absolute values autocorrelation, and absolute values persistency. We verify that this financial instrument, in spite of its high liquidity, shows very peculiar properties. Particularly, we verify that log-fluctuations belong to the Levy class of stochastic variables., Comment: To be published in EPJB

The ARCH process (R. F. Engle, 1982) constitutes a paradigmatic generator of stochastic time series with time-dependent variance like it appears on a wide broad of systems besides economics in which ARCH was born. Although the ARCH process captures the so-called "volatility clustering" and the asymptotic power-law probability density distribution of the random variable, it is not capable to reproduce further statistical properties of many of these time series such as: the strong persistence of the instantaneous variance characterised by large values of the Hurst exponent (H > 0.8), and asymptotic power-law decay of the absolute values self-correlation function. By means of considering an effective return obtained from a correlation of past returns that has a q-exponential form we are able to fix the limitations of the original model. Moreover, this improvement can be obtained through the correct choice of a sole additional parameter, $q_{m}$. The assessment of its validity and usefulness is made by mimicking daily fluctuations of SP500 financial index., Comment: 6 pages, 4 figures

In this article we analyse linear correlation and non-linear dependence of traded volume, $v$, of the 30 constituents of Dow Jones Industrial Average at different value scales. Specifically, we have raised $v$ to some real value $\alpha $ or $\beta $, which introduces a bias for small ($ \alpha, \beta <0$) or large ($\alpha, \beta >1$) values. Our results show that small values of $v$ are regularly \emph{anti-correlated} with values at other scales of traded volume. This is consistent with the high liquidity of the 30 equities analysed and the asymmetric form of the multi-fractal spectrum for traded volume which has supported the dynamical scenario presented by us., Comment: 6 pages, 2 figures

We present results about financial market observables, specifically returns and traded volumes. They are obtained within the current nonextensive statistical mechanical framework based on the entropy $S_{q}=k\frac{1-\sum\limits_{i=1}^{W} p_{i} ^{q}}{1-q} (q\in \Re)$ ($S_{1} \equiv S_{BG}=-k\sum\limits_{i=1}^{W}p_{i} \ln p_{i}$). More precisely, we present stochastic dynamical mechanisms which mimic probability density functions empirically observed. These mechanisms provide possible interpretations for the emergence of the entropic indices $q$ in the time evolution of the corresponding observables. In addition to this, through multi-fractal analysis of return time series, we verify that the dual relation $q_{stat}+q_{sens}=2$ is numerically satisfied, $q_{stat}$ and $q_{sens}$ being associated to the probability density function and to the sensitivity to initial conditions respectively. This type of simple relation, whose understanding remains ellusive, has been empirically verified in various other systems., Comment: Invited paper to appear in special issue of Eur. Phys. J. B dedicated to econophysics, edited by T. Di Matteo and T. Aste. 7 pages

In this article we study the dependence degree of the traded volume of the Dow Jones 30 constituent equities by using a nonextensive generalised form of the Kullback-Leibler information measure. Our results show a slow decay of the dependence degree as a function of the lag. This feature is compatible with the existence of non-linearities in this type time series. In addition, we introduce a dynamical mechanism whose associated stationary probability density function (PDF) presents a good agreement with the empirical results., Comment: 6 pages, 4 figures, 1 table. Based on the talk presented at "News, Expectations and Trends in Statistical Physics, NEXT-SigmaPhi 3rd International Conference. 13-18 August 2005, Kolymbari CRETE" Multi-fractal analysis section removed

The cornerstone of Boltzmann-Gibbs ($BG$) statistical mechanics is the Boltzmann-Gibbs-Jaynes-Shannon entropy $S_{BG} \equiv -k\int dx f(x)\ln f(x)$, where $k$ is a positive constant and $f(x)$ a probability density function. This theory has exibited, along more than one century, great success in the treatment of systems where short spatio/temporal correlations dominate. There are, however, anomalous natural and artificial systems that violate the basic requirements for its applicability. Different physical entropies, other than the standard one, appear to be necessary in order to satisfactorily deal with such anomalies. One of such entropies is $S_q \equiv k (1-\int dx [f(x)]^q)/(1-q)$ (with $S_1=S_{BG}$), where the entropic index $q$ is a real parameter. It has been proposed as the basis for a generalization, referred to as {\it nonextensive statistical mechanics}, of the $BG$ theory. $S_q$ shares with $S_{BG}$ four remarkable properties, namely {\it concavity} ($\forall q>0$), {\it Lesche-stability} ($\forall q>0$), {\it finiteness of the entropy production per unit time} ($q \in \Re$), and {\it additivity} (for at least a compact support of $q$ including $q=1$). The simultaneous validity of these properties suggests that $S_q$ is appropriate for bridging, at a macroscopic level, with classical thermodynamics itself. In the same natural way that exponential probability functions arise in the standard context,power-law tailed distributions, even with exponents {\it out} of the L\'evy range, arise in the nonextensive framework. In this review, we intend to show that many processes of interest in economy, for which fat-tailed probability functions are empirically observed, can be described in terms of the statistical mechanisms that underly the nonextensive theory., Comment: Invited review to appear in the "Third SPIE International Symposium 'Fluctuations and Noise'" proceedings

This manuscript reports a stochastic dynamical scenario whose associated stationary probability density function is exactly a previously proposed one to adjust high-frequency traded volume distributions. This dynamical conjecture, physically connected to superstatiscs, which is intimately related with the current nonextensive statistical mechanics framework, is based on the idea of local fluctuations in the mean traded volume associated to financial markets agents herding behaviour. The corroboration of this mesoscopic model is done by modelising NASDAQ 1 and 2 minute stock market traded volume.

The $GARCH$ algorithm is the most renowned generalisation of Engle's original proposal for modelising {\it returns}, the $ARCH$ process. Both cases are characterised by presenting a time dependent and correlated variance or {\it volatility}. Besides a memory parameter, $b$, (present in $ARCH$) and an independent and identically distributed noise, $\omega $, $GARCH$ involves another parameter, $c$, such that, for $c=0$, the standard $ARCH$ process is reproduced. In this manuscript we use a generalised noise following a distribution characterised by an index $q_{n}$, such that $q_{n}=1$ recovers the Gaussian distribution. Matching low statistical moments of $GARCH$ distribution for returns with a $q$-Gaussian distribution obtained through maximising the entropy $S_{q}=\frac{1-\sum_{i}p_{i}^{q}}{q-1}$, basis of nonextensive statistical mechanics, we obtain a sole analytical connection between $q$ and $(b,c,q_{n}) $ which turns out to be remarkably good when compared with computational simulations. With this result we also derive an analytical approximation for the stationary distribution for the (squared) volatility. Using a generalised Kullback-Leibler relative entropy form based on $S_{q}$, we also analyse the degree of dependence between successive returns, $z_{t}$ and $z_{t+1}$, of GARCH(1,1) processes. This degree of dependence is quantified by an entropic index, $q^{op}$. Our analysis points the existence of a unique relation between the three entropic indexes $q^{op}$, $q$ and $q_{n}$ of the problem, independent of the value of $(b,c)$.

We focus on the frontier between the chaotic and regular regions for the classical version of the quantum kicked top. We show that the sensitivity to the initial conditions is numerically well characterised by $\xi=e_q^{\lambda_q t}$, where $e_{q}^{x}\equiv [ 1+(1-q) x]^{\frac{1}{1-q}} (e_1^x=e^x)$, and $\lambda_q$ is the $q$-generalization of the Lyapunov coefficient, a result that is consistent with nonextensive statistical mechanics, based on the entropy $S_q=(1- \sum_ip_i^q)/(q-1) (S_1 =-\sum_i p_i \ln p_i$). Our analysis shows that $q$ monotonically increases from zero to unity when the kicked-top perturbation parameter $\alpha $ increases from zero (unperturbed top) to $\alpha_c$, where $\alpha_c \simeq 3.2$. The entropic index $q$ remains equal to unity for $\alpha \ge \alpha_c$, parameter values for which the phase space is fully chaotic., Comment: To appear in "Complexity, Metastability and Nonextensivity" (World Scientific, Singapore, 2005), Eds. C. Beck, A. Rapisarda and C. Tsallis

In this report I present a possible scenario which can lead to the emergence of a generalised Gamma distribution first presented by R. Osorio et al. as the distribution of traded volumes of stocks in financial markets. This propose is related with superstatics and the notion of moving average commonly used in econometrics., Comment: Brief Report containing the mainstream result. Final Article form to be presented soon. 2 Figures