Tsallis Statistical Mechanics Research Articles

Thermal leptogenesis in nonextensive cosmology

Thermal leptogenesis is a mechanism that explains the observed asymmetry between matter and antimatter in the early universe. In this study, we review the impact of nonextensive Tsallis statistical mechanics on the early universe and study its effect on thermal leptogenesis. The study has found that the use of nonextensive statistical mechanics can affect the production of baryon asymmetry in thermal leptogenesis by modifying the equilibrium abundance of particles, decay, and washout parameters. Also, we show that nonextensive statistical mechanics potentially reduce the required right-handed neutrino mass scale.

How, Why and When Tsallis Statistical Mechanics Provides Precise Descriptions of Natural Phenomena.

The limit of validity of ordinary statistical mechanics and the pertinence of Tsallis statistics beyond it is explained considering the most probable evolution of complex systems processes. To this purpose we employ a dissipative Landau-Ginzburg kinetic equation that becomes a generic one-dimensional nonlinear iteration map for discrete time. We focus on the Renormalization Group (RG) fixed-point maps for the three routes to chaos. We show that all fixed-point maps and their trajectories have analytic closed-form expressions, not only (as known) for the intermittency route to chaos but also for the period-doubling and the quasiperiodic routes. These expressions have the form of q-exponentials, while the kinetic equation's Lyapunov function becomes the Tsallis entropy. That is, all processes described by the evolution of the fixed-point trajectories are accompanied by the monotonic progress of the Tsallis entropy. In all cases the action of the fixed-point map attractor imposes a severe impediment to access the system's built-in configurations, leaving only a subset of vanishing measure available. Only those attractors that remain chaotic have ineffective configuration set reduction and display ordinary statistical mechanics. Finally, we provide a brief description of complex system research subjects that illustrates the applicability of our approach.

Sparse Regularized Optimal Transport with Deformed q-Entropy.

Optimal transport is a mathematical tool that has been a widely used to measure the distance between two probability distributions. To mitigate the cubic computational complexity of the vanilla formulation of the optimal transport problem, regularized optimal transport has received attention in recent years, which is a convex program to minimize the linear transport cost with an added convex regularizer. Sinkhorn optimal transport is the most prominent one regularized with negative Shannon entropy, leading to densely supported solutions, which are often undesirable in light of the interpretability of transport plans. In this paper, we report that a deformed entropy designed by q-algebra, a popular generalization of the standard algebra studied in Tsallis statistical mechanics, makes optimal transport solutions supported sparsely. This entropy with a deformation parameter q interpolates the negative Shannon entropy (q=1) and the squared 2-norm (q=0), and the solution becomes more sparse as q tends to zero. Our theoretical analysis reveals that a larger q leads to a faster convergence when optimized with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. In summary, the deformation induces a trade-off between the sparsity and convergence speed.

The spectrum of Hawking radiation in Tsallis statistical mechanics

Hawking radiation is one of the cores in modern gravitational theory. Several articles have calculated the spectrum of Hawking radiation in Boltzmann–Gibbs statistical mechanics. However, based on recent researches, gravitational systems cannot be studied by the standard statistical mechanics. In this article, we calculate the modification to the spectrum of Hawking radiation in Tsallis statistical mechanics. We obtain the modified Stefan–Boltzmann’s law and modified power of Hawking radiation. We confirm the conclusion proposed by Giddings, namely, the radiation should originate from the effective radius, which extends well outside the horizon of black-hole. The lifetime of black hole and the effect of large q are discussed as well.

Finite Tsallis gravitational partition function for a system of galaxies

In this paper, we will study the large scale structure formation using the gravitational partition function. We will assertively argue that the system of gravitating galaxies can be analyzed using the Tsallis statistical mechanics. The divergences in the Tsallis gravitational partition function can be removed using the mathematical riches of the generalization of the dimensional regularization (GDR). The finite gravitational partition function thus obtained will be used to evaluate the thermodynamics of the system of galaxies and thus, to understand the clustering of galaxies in the universe. The correlation function which is beleived to contain the information of clustering of galaxies will also be discussed in this formalism.

Work relations with measurement and feedback control on nonuniform temperature systems

The relation between the work performed to a system and the change of its free energy during a certain process is important in nonequilibrium statistical mechanics. In particular, the work relation with measurement and feedback control has attracted much attention, because it resolved the paradox concerning Maxwell's demon. Most studies, however, assume that their target systems are isolated or isothermal. In this paper, by considering a nonisothermal system, we generalize the Sagawa-Ueda-Jarzynski relation, which involves measurement and feedback control, and apply it to a realistic model. Furthermore, when the temperature profile is quadratic, we see that the system is governed by Tsallis statistical mechanics. In addition, we show that our formulation provides the generalized version of the second law of information thermodynamics and a set of work relations for isothermal systems.

Radial distribution function within the framework of the Tsallis statistical mechanics

This study is conducted to obtain the radial distribution function (RDF) within the Tsallis statistical mechanics. To this end, probability distribution functions are applied in the first and fourth versions of the Tsallis statistics. Moreover, a closed formula is proposed for RDF. The power nature of the probability distribution in the Tsallis statistics makes it difficult to separate kinetic energy and configurational potential parts. By using the Taylor expansion around q=1 of the power distribution, it is possible to show the independency of momenta and coordinates through integrating over the phase space variables. In addition, at low densities, numerical calculations have been performed for the RDF. Our results show that the correlation increases as q values increase.

Can we use the [formula omitted]-Gaussian of ambient noise fluctuations as a vulnerability index? A case study in Cultural Heritage buildings

In this work we study in terms of Tsallis statistical mechanics the properties of microtremors’ fluctuations in two church bell towers, which are monuments of cultural heritage, in the city of Chania (Crete, Greece). We have shown that fluctuations of ambient vibrations recordings in the Church tower bells follow a q-Gaussian distribution. The behavior of Tsallis q parameter with the level (high) of the measuring point within the tower and the amplification factors at that points as extracted from horizontal-to-vertical (HVSR) spectral ratios are presented and discussed. Since q decreases as the amplification factor increases, we could suggest q as a vulnerability index, where, as q decreases approaching unity, then the structural system is getting more vulnerable. The latter approach suggests that introducing ideas of Tsallis statistics could be useful in characterizing extremely complex processes as that governed the estimation of seismic vulnerability in which a multidisciplinary approach is required.

Generalized polarization force acting on dust grains in a dusty plasma

Abstract The polarization force acting on dust particles in a dusty plasma is revisited within the theoretical framework of the Tsallis statistical mechanics. The generalized nonextensive polarization force expression is derived. As application, the modifications arising in the propagation of dust-acoustic solitary waves, and dust sheath formation are analyzed. Our results should be of wide relevance to explain and interpret the sheath formation and its structure in nonequilibrium plasmas related process such as surface treatments and ion implantation.

Biophysical approach to investigate temperature effects on protein dynamics

The main purpose of this work is to gain more understanding of the temperature dependence of Min protein dynamics employing spot tracking technique (STT) and biophysical characterization. We observed and confirmed the variation of protein cluster dynamics at evaluated temperatures. We found that the time MinD was retained at the cell poles played an important role in this variation. From 25 °C to 37 °C, the MinD protein oscillation period decreased 2.3 times and the anomalous dynamic exponent increased 2.4 times. The time-varied anomalous diffusion coefficient was found to be temperature dependent, which was qualitatively consistent with the prediction by Tsallis statistical mechanics. Furthermore, the average apparent effective potential depth of membrane-bound MinD protein decreased from 10.01 to 3.54 k B T . These results showed that the diffusive mode and the MinD protein cluster distribution at the cell poles were altered with temperature and this then affected the whole Min protein dynamics.

How the Nonextensivity Parameter Affects Energy Fluctuations

In this article, the effect of the nonextensivity parameter on the energy fluctuations of nonextensive systems is studied in two different versions of the Tsallis statistical mechanics. Once a general expression has been reported for the energy fluctuations in the second version (Tsallis work in 1988), the energy fluctuations of an ideal gas and a harmonic oscillator are studied in the second and fourth(OLM choice for the mean energy constrain) versions of the Tsallis statistical mechanics. The results for the fourth version indicate that relative energy fluctuations are strongly affected by the nonextensivity parameter via the number of accessible states. In fact, in the case of subextensive systems, the nonextensivity parameter leads to the fewer accessible states compared to that of the extensive systems and, therefore, smaller relative energy fluctuations are expected. For super-extensive systems, however, relative energy fluctuations are found to be larger than those in extensive systems because of the greater accessible states available. Our studies show that very large relative energy fluctuations are caused as the result of un-normalized nature of the second version which, in some cases, limits its application.

A fractional Fokker-Planck model for anomalous diffusion

In this paper, we present a study of anomalous diffusion using a Fokker-Planck description with fractional velocity derivatives. The distribution functions are found using numerical means for varying degree of fractionality of the stable Lévy distribution. The statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy in terms of Tsallis statistical mechanics. We find that the ratio of the generalized entropy and expectation is increasing with decreasing fractionality towards the well known so-called sub-diffusive domain, indicating a self-organising behavior.

Study of the temporal correlations in the magnitude time series before major earthquakes in Japan

AbstractA characteristic change of seismicity has been recently uncovered when the precursory Seismic Electric Signals activities initiate before an earthquake occurrence. In particular, the fluctuations of the order parameter of seismicity exhibit a simultaneous distinct minimum upon analyzing the seismic catalog in a new time domain termed natural time and employing a sliding natural time window comprising a number of events that would occur in a few months. Here we focus on the minima preceding all earthquakes of magnitude 8 (and 9) class that occurred in Japanese area from 1 January 1984 to 11 March 2011 (the day of the M9 Tohoku earthquake). By applying Detrended Fluctuation Analysis to the earthquake magnitude time series, we find that each of these minima is preceded as well as followed by characteristic changes of temporal correlations between earthquake magnitudes. In particular, we identify the following three main features. The minima are observed during periods when long‐range correlations have been developed, but they are preceded by a stage in which an evident anticorrelated behavior appears. After the minima, the long‐range correlations breakdown to an almost random behavior possibly turning to anticorrelation. The minima that precede M≥7.8 earthquakes are distinguished from other minima which are either nonprecursory or followed by smaller earthquakes. Furthermore, it is discussed whether Tsallis statistical mechanics, in the frame of which it has been suggested that kappa distributions arise, can capture the effects of temporal correlations between earthquake magnitudes.

Effect of q-non-extensive distribution of electrons on the plasma sheath floating potential

In this paper, a collisionless unmagnetized plasma sheath consisting of electrons following non-extensive q-distribution, and cold mobile inertial ions is studied in the stationary state. In this type of plasma with non-Maxwellian electron distribution (Tsallis statistical mechanics), the effective electron temperature (Te, eff) and electron screening temperature (Te,*) are evaluated. The other plasma sheath phenomena such as the Bohm sheath criterion, Debye shielding, floating potential, and sheath length are investigated in the presence of q-non-extensive velocity-distributed electrons. It is observed that above-mentioned phenomena depend significantly on the non-extensive parameter q.

New bounds for Tsallis parameter in a noncommutative phase–space entropic gravity and nonextensive Friedmann equations

In this paper, we have analyzed the nonextensive Tsallis statistical mechanics in the light of Verlinde’s formalism. We have obtained, with the aid of a noncommutative phase–space entropic gravity, a new bound for Tsallis nonextensive (NE) parameter (TNP) that is clearly different from the ones present in the current literature. We derived the Friedmann equations in a NE scenario. We also obtained here a relation between the gravitational constant and the TNP.

Propagation of two-solitons in an electron acoustic waves in a plasma with electrons featuring Tsallis distribution

A first theoretical work is presented to study the propagation of two-solitons in an electron acoustic waves (EAWs) within the theoretical framework of the Tsallis statistical mechanics. For this purpose, cylindrical and spherical Korteweg-de Vries (KdV) equations are derived for electron acoustic solitary waves (EASWs) in an unmagnetized three species plasma system comprised of cold electrons, immobile ions and hot electrons featuring Tsallis statistics by employing the standard reductive perturbation method. The effects of electron nonextensivity and the fractional number density of the hot electrons relative to that of the cold ones number density (α) on the profiles of two-soliton structures are investigated numerically. Results would be helpful for understanding the localized structures that may occur in space plasmas.

Dust ion-acoustic solitary waves in a dusty plasma with nonextensive electrons

The dust-modified ion-acoustic waves of Shukla and Silin are revisited within the theoretical framework of the Tsallis statistical mechanics. Nonextensivity may originate from correlation or long-range plasma interactions. Interestingly, we find that owing to electron nonextensivity, dust ion-acoustic (DIA) solitary waves may exhibit either compression or rarefaction. Our analysis is then extended to include self-consistent dust charge fluctuation. In this connection, the correct nonextensive electron charging current is rederived. The Korteweg-de Vries equation, as well as the Korteweg-de Vries-Burgers equation, is obtained, making use of the reductive perturbation method. The DIA waves are then analyzed for parameters corresponding to space dusty plasma situations.

Solitary ion-acoustic wave propagation in the presence of electron trapping and background nonextensivity

Solitary ion-acoustic wave propagation in the presence of electron trapping is investigated within the theoretical framework of the Tsallis statistical mechanics. A physically meaningful Schamel-like distribution is outlined. In the small amplitude limit, the nonlinear dispersion relation is derived to analyze the global dependency of the main solitary wave quantities. It is found that for a given amplitude and trapping state, the solitary potential structure speeds up and broadens as the electron nonextensivity strengthens. Our results may be of basic interest for experiments that involve particle trapping. The flexibility provided by the nonextensive q-parameter enables one to obtain a good agreement between theory and experiment.

Comment on “Debye shielding in a nonextensive plasma” [Phys. Plasmas 18, 062102 (2011)

Gougam and Tribeche [Phys. Plasmas 18, 062102 (2011)] revisit the phenomenon of Debye shielding within the theoretical framework of Tsallis statistical mechanics. The effect of electron nonextensivity on the Bohm sheath criterion is also investigated, and a modified critical Mach number is derived. However, their results are questionable, since they are based on a somewhat unjustified assumption about the parameter that specifies the width of the distribution function.

The physics aspect of price index changes: Super- diffusive approach

Attempts were made to quantify the buying and selling interaction of worldwide financial markets into quantitative findings. We introduce a probability density derived from non-extensive Tsallis statistical mechanics that can be applied to the interpretation of percent price index changes for important indices such as NYSE Composite, DJIA, S&P 500, NASDAQ Composite, FTSE 100, NIKKEI 225, Hang Seng, Straits Times and SET index. Results of applying Tsallis’ probability density through markets’ observation illustrated the behavior of all indices indicating super diffusive dynamics. Furthermore, an Ito-Langevin equation with a time-dependent diffusion coefficient and the nonlinear Fokker–Planck equation can exhibit investment risk of each price index. Finally, we not only explained the complex behavior of financial indices in Physics aspect, but simplified it into quantitative meanings able to be virtually used further as well. Key words: Econophysics, Tsallis’ probability density, Fokker-Planck equation, financial market index.