Tsallis Q-statistics Research Articles

Tsallis q-Statistics Fingerprints in Precipitation Data across Sicily.

Precipitation patterns are critical for understanding the hydrological and climatological dynamics of any region. Sicily, the largest island in the Mediterranean sea, with its diverse topography and climatic conditions, serves as an ideal case study for analyzing precipitation data, to gain insights into regional water resources, agricultural productivity, and climate change impacts. This paper employs advanced statistical physics methods, particularly Tsallis q-statistics, to analyze sub-hourly precipitation data from 2002 to 2023, provided by the Sicilian Agrometeorological Informative System (SIAS). We investigate several critical variables related to rainfall events, including duration, depth, maximum record, and inter-event time. The study spans two decades (2002-2012 and 2013-2023), analyzing the distributions of relevant variables. Additionally, we examine the simple returns of these variables to identify significant temporal changes, fitting these returns with q-Gaussian distributions. Our findings reveal the scale-invariant nature of precipitation events, the presence of long-range interactions, and memory effects, characteristic of complex environmental processes.

Tsallis q-Statistics in Seismology.

Non-extensive statistical mechanics (or q-statistics) is based on the so-called non-additive Tsallis entropy. Since its introduction by Tsallis, in 1988, as a generalization of the Boltzmann-Gibbs equilibrium statistical mechanics, it has steadily gained ground as a suitable theory for the description of the statistical properties of non-equilibrium complex systems. Therefore, it has been applied to numerous phenomena, including real seismicity. In particular, Tsallis entropy is expected to provide a guiding principle to reveal novel aspects of complex dynamical systems with catastrophes, such as seismic events. The exploration of the existing connections between Tsallis formalism and real seismicity has been the focus of extensive research activity in the last two decades. In particular, Tsallis q-statistics has provided a unified framework for the description of the collective properties of earthquakes and faults. Despite this progress, our present knowledge of the physical processes leading to the initiation of a rupture, and its subsequent growth through a fault system, remains quite limited. The aim of this paper was to provide an overview of the non-extensive interpretation of seismicity, along with the contributions of the Tsallis formalism to the statistical description of seismic events.

Tsallis meets Boltzmann: q-index for a finite ideal gas and its thermodynamic limit.

Nonadditive Tsallis q-statistics has successfully been applied for a plethora of systems in natural sciences and other branches of knowledge. Nevertheless, its foundations have been severely criticized by some authors based on the standard additive Boltzmann-Gibbs approach, thereby remaining a quite controversial subject. In order to clarify some polemical concepts, the distribution function for an ideal gas with a finite number of point particles and its q-index are analytically determined. The two-particle correlation function is also derived. The degree of correlation diminishes continuously with the growth of the number of particles. The ideal finite gas system is usually correlated, becomes less correlated when the number of particles grows, and is finally fully uncorrelated when the molecular chaos regime is reached. It is also advocated that both approaches can be confronted through a careful kinetic spectroscopic experiment. The analytical results derived here suggest that Tsallis q-statistics may play a physical role more fundamental than usually discussed in the literature.

Modified Tsallis and Weibull distributions for multiplicities in e+e− collisions

Multiplicity distributions of charged particles produced in the e^{+}e^{-} collisions at energies ranging from 14 to 91 GeV are studied using Tsallis q-statistics and the recently proposed Weibull distribution functions, in both restricted rapidity windows as well as in full phase space. It is shown that Tsallis $q$-statistics explains the data excellently in all rapidity ranges while the Weibull distribution fails to reproduce the data in full phase space. Modifications to the distributions are proposed to establish manifold improvements in the fitting of the data.

Statistical hadronization and multiplicities in high-energy hadron–nucleus collisions

Concepts from statistical mechanics and ensemble theory are applied to study the characteristic properties of charged particle production in hadron–nucleus collisions at high energy. In the present study we utilize the predictions from different approaches using statistical mechanics. The Tsallis q-statistics is used to study the regularity in multiplicity distributions in hadron–nucleus collisions at high energies as one of the interesting options. Gamma distributions and a possible microcanonical generalization of Tsallis distribution have also been exploited to describe the data.

Boundedness vs unboundedness of a noise linked to Tsallis q-statistics: The role of the overdamped approximation

An apparently ideal way to generate continuous bounded stochastic processes is to consider the stochastically perturbed motion of a point of small mass in an infinite potential well, under overdamped approximation. Here, however, we show that the aforementioned procedure can be fallacious and lead to incorrect results. We indeed provide a counter-example concerning one of the most employed bounded noises, hereafter called Tsallis-Stariolo-Borland (TSB) noise, which admits the well known Tsallis q-statistics as stationary density. In fact, we show that for negative values of the Tsallis parameter q (corresponding to sufficiently large diffusion coefficient of the stochastic force), the motion resulting from the overdamped approximation is unbounded. We then investigate the cause of the failure of Kramers first type approximation, and we formally show that the solutions of the full Newtonian non-approximated model are bounded, following the physical intuition. Finally, we provide a new family of bounded noises extending the TSB noise, the boundedness of whose solutions we formally show.

Universality of non-extensive Tsallis statistics and time series analysis: Theory and applications

We investigate Tsallis q-statistics for various complex physical systems. In particular, the Tsallis q-triplet is estimated for space plasmas, atmospheric dynamics, seismicity, and brain and cardiac activity. Remarkable agreement between theoretical Tsallis predictions and experimental estimations was observed in all cases. We also present theoretical extensions of Tsallis q-statistics for intermittent turbulence theory, non-equilibrium statistics, and fractal extension of dynamics.

Generalized Statistical Mechanics and Scaling Behavior for Non-equilibrium Polymer Chains: II. Monomers Connected by Springs

Using molecular dynamics simulation, we find that the velocity distributions of monomers in the system of non-equilibrium polymer chains and Lennard-Jones molecules in a wide range of simulation time can be well described by Tsallis q-statistics and a single scaling function; the value of q is related to the conformation constraining potential, the interactions with background fluid, or the destruction of chain homogeneity. This approach can be applied to other non-equilibrium systems.

Rapidity spectra analysis in terms of non-extensive statistic approach

We provide the description of rapidity spectra of particles produced in pp̄ collisions using the anomalous diffusion approach to account for their non-equilibrium character. In particular, we exhibit connections between multiproduction processes and anomalous diffusion described through the nonlinear Focker-Planck equation with a nonlinearity given by the nonextensivity parameter q describing the underlying Tsallis q-statistics. We demonstrate how this leads to the Feynman scaling violation in these collisions. The q parameter obtained in this way turns out to be closely connected to the parameter 1 k converting the original poissonian multiplicity distribution to its observed Negative Binomial form. The inelasticity of the reaction has also been calculated and found to slightly decrease with increasing reaction energy.

Lévy flights in external force fields: Langevin and fractional Fokker-Planck equations and their solutions

We consider Levy flights subject to external force fields. This anomalous transport process is described by two approaches, a Langevin equation with Levy noise and the corresponding generalized Fokker-Planck equation containing a fractional derivative in space. The cases of free flights, constant force and linear Hookean force are analyzed in detail, and we corroborate our findings with results from numerical simulations. We discuss the non-Gibbsian character of the stationary solution for the case of the Hookean force, i.e. the deviation from Boltzmann equilibrium for long times. The possible connection to Tsallis's q-statistics is studied.