Maximum entropy with fluctuating constraints: The example of K-distributions

We indicate that in a maximum entropy setting, the thermodynamic β and the observation constraint are linked, so that fluctuations of the latter imposes fluctuations of the former. This gives an alternate viewpoint to ‘superstatistics’. While a Gamma model for fluctuations of the β parameter gives the so-called Tsallis distributions, we work out the case of a Gamma model for fluctuations of the observable, and show that this leads to K-distributions. We draw attention to the fact that these heavy-tailed distributions have high interest in physical applications, and we discuss them in some details.     

The imprints of superstatistics in multiparticle production processes

We provide an update of the overview of imprints of Tsallis nonextensive statistics seen in a multiparticle production processes. They reveal an ubiquitous presence of power law distributions of different variables characterized by the nonextensivity parameter q > 1. In nuclear collisions one additionally observes a q-dependence of the multiplicity fluctuations reflecting the finiteness of the hadronizing source. We present sum rules connecting parameters q obtained from an analysis of different observables, which allows us to combine different kinds of fluctuations seen in the data and analyze an ensemble in which the energy (E), temperature (T) and multiplicity (N) can all fluctuate. This results in a generalization of the so called Lindhard’s thermodynamic uncertainty relation. Finally, based on the example of nucleus-nucleus collisions (treated as a quasi-superposition of nucleon-nucleon collisions) we demonstrate that, for the standard Tsallis entropy with degree of nonextensivity q < 1, the corresponding standard Tsallis distribution is described by q′ = 2 − q > 1.     

Energy fluctuation and correlation in Tsallis statistics

We investigate the fluctuation of the energy in the framework of Tsallis statistics and find the correlation plays an important role in energy fluctuations. In Tsallis statistics, the correlation is induced by the nonextensivity of Tsallis entropy and exists between particles even if the particles are dynamically independent. By taking the generalized ideal gas as an example, we get that when the particle number N is large enough, the relative fluctuation of the energy is proportional to 1/N instead of in Boltzmann statistics. Thus, the relative energy fluctuation is much smaller in Tsallis statistics than that in Boltzmann statistics. Besides, we demonstrate that the introduction of correlation between particle energies leads to smaller energy fluctuations in Tsallis statistics.     

Test of h1(1830) made of $$K^* \bar K^*$$ with the $$\eta _c \to \varphi K^* \bar K^*$$ decay

The transverse momentum distributions of final-state particles produced in collisions at energies available at the Large Hadron Collider (LHC) are studied by using the two-Boltzmann distribution and Tsallis statistics. Experimental distributions described by the two-Boltzmann distribution can be described by the Tsallis statistics. The two-temperature emission described by the two-Boltzmann distribution reflects temperature fluctuation of interacting system. The Tsallis statistics can describe the temperature fluctuation and the degree of non-equilibrium. The results calculated by the two-Boltzmann distribution and the Tsallis statistics are in agreement with the experimental data available at the LHC energies. In some cases, the two-Boltzmann distribution degenerates to (single) Boltzmann distribution.

     

Inequalities for nonnegative numbers and information properties of qudit tomograms

We discuss some inequalities for N nonnegative numbers. We use these inequalities to obtain known inequalities for probability distributions and new entropic and information inequalities for quantum tomograms of qudit states. The inequalities characterize the degree of quantum correlations in addition to noncontextuality and quantum discord. We use the subadditivity and strong subadditivity conditions for qudit tomographic-probability distributions depending on the unitary-group parameters in order to derive new inequalities for Shannon, Rényi, and Tsallis entropies of spin states.     

Formal equivalence between Tsallis and extended Boltzmann-Gibbs statistics

A formal correspondence between the q-distribution obtained from the Tsallis entropy and non-Maxwellian distributions obtained from the Boltzmann-Gibbs (BG) entropy is obtained. This formal correspondence is obtained by imposing an infinite number of constraints when one maximizes the BG entropy. Different from the approach of Tsallis, Prato and Plastino [C. Tsallis, D. Prato, A.R. Plastino, Astrophys. Space Sci., 290 (2004) 259-274], we relate the constraints to the central moments, providing a natural meaning to the q-parameter.     

Nonextensive triangle equality and other properties of Tsallis relative-entropy minimization

Kullback–Leibler relative-entropy has unique properties in cases involving distributions resulting from relative-entropy minimization. Tsallis relative-entropy is a one-parameter generalization of Kullback–Leibler relative-entropy in the nonextensive thermostatistics. In this paper, we present the properties of Tsallis relative-entropy minimization and present some differences with the classical case. In the representation of such a minimum relative-entropy distribution, we highlight the use of the q-product, an operator that has been recently introduced to derive the mathematical structure behind the Tsallis statistics. One of our main results is the generalization of triangle equality of relative-entropy minimization to the nonextensive case.     

On the way towards a generalized entropy maximization procedure

We propose a generalized entropy maximization procedure, which takes into account the generalized averaging procedures and information gain definitions underlying the generalized entropies. This novel generalized procedure is then applied to Rényi and Tsallis entropies. The generalized entropy maximization procedure for Rényi entropies results in the exponential stationary distribution asymptotically for q∈(0,1] in contrast to the stationary distribution of the inverse power law obtained through the ordinary entropy maximization procedure. Another result of the generalized entropy maximization procedure is that one can naturally obtain all the possible stationary distributions associated with the Tsallis entropies by employing either ordinary or q-generalized Fourier transforms in the averaging procedure.     

q exponential is a Burr distribution

It is pointed out that the q-exponential distribution introduced in the ground-breaking paper by Tsallis [C. Tsallis, J. Stat. Phys. 52 (1988) 479] is contained by a family of distributions known since the 1940s. Some elementary statistical properties of this family are discussed. Six data sets on fracture roughness are used to demonstrate that this family provides much better models for fracture roughness than the q exponential distribution.     

Tsallis q-Stat and the Evidence of Long-Range Interactions in Soil Temperature Dynamics

The complexities in the variations of soil temperature and thermal diffusion poses a physical problem that requires more understanding. The quest for a better understanding of the complexities of soil temperature variation has prompted the study of the q-statistics in the soil temperature variation with the view of understanding the underlying dynamics of the temperature variation and thermal diffusivity of the soil. In this work, the values of Tsallis stationary state q index known as q-stat were computed from soil temperature measured at different stations in Nigeria. The intrinsic variations of the soil temperature were derived from the soil temperature time series by detrending method to extract the influences of other types of variations from the atmosphere. The detrended soil temperature data sets were further analysed to fit the q-Gaussian model. Our results show that our datasets fit into the Tsallis Gaussian distributions with lower values of q-stat during rainy season and around the wet soil regions of Nigeria and the values of q-stat obtained for monthly data sets were mostly in the range 1.2≤q≤2.9 for all stations, with very few values q closer to 1.2 for a few stations in the wet season. The distributions obtained from the detrended soil temperature data were mostly found to belong to the class of asymmetric q-Gaussians. The ability of the soil temperature data sets to fit into q-Gaussians might be due and the non-extensive statistical nature of the system and (or) consequently due to the presence of superstatistics. The possible mechanisms responsible this behaviour was further discussed.     

How fundamental is the character of thermal uncertainty relations?

We discuss here two different information measures of the Tsallis type, and their associated probability distributions, in order to repeat the Mandelbrot Cramer–Rao steps that lead to a thermal uncertainty relation for exponential distributions. We deal first with the original Tsallis measure and discuss afterwards a second entropic measure associated with the concept of escort distribution. In neither case it is possible to re-obtain a thermal uncertainty relationship. We conclude therefore that the thermal uncertainty, as derived from the Cramer–Rao inequality, cannot be as fundamental as the quantum one.     

Power laws in elementary and heavy-ion collisions A story of fluctuations and nonextensivity?

We review from the point of view of nonextensive statistics the ubiquitous presence in elementary and heavy-ion collisions of power law distributions. Special emphasis is placed on the conjecture that this is just a reflection of some intrinsic fluctuations existing in the hadronic systems considered. These systems are summarily described by a single parameter q playing the role of a nonextensivity measure in the nonextensive statistical models based on Tsallis entropy. This paper is part of the Topical Issue Statistical Power Law Tails in High-Energy Phenomena.     

Density operators that extremize Tsallis entropy and thermal stability effects

Quite general, analytical (both exact and approximate) forms for discrete probability distributions (PDs) that maximize Tsallis entropy for a fixed variance are here investigated. They apply, for instance, in a wide variety of scenarios in which the system is characterized by a series of discrete eigenstates of the Hamiltonian. Using these discrete PDs as “weights” leads to density operators of a rather general character. The present study allows one to vividly exhibit the effects of non-extensivity. Varying Tsallis’ non-extensivity index q one is seen to pass from unstable to stable systems and even to unphysical situations of infinite energy.     

Generalised Tsallis statistics in electron-positron collisions

The scaling of charged hadron fragmentation functions to the Tsallis distribution for the momentum fraction 0.01?x?0.2 is presented for various e⁺e⁻ collision energies. A possible microcanonical generalisation of the Tsallis distribution is proposed, which gives good agreement with measured data up to x≈1. The proposal is based on superstatistics and a Koba-Nielsen-Olesen (KNO) like scaling of multiplicity distributions in e⁺e⁻ experiments.     

Modified exponential function: the connection between nonextensivity and q-deformation

Tsallis thermostatistics has deep-rooted connection with quantum group formalism. Assuming that the modification of the standard exponential function considered in Tsallis thermostatistics has the same functional form as the one appearing in the q-calculus formalism and using the appropriate internal energy constraint, we derive the temperature dependent connection between the nonextensivity parameter and deformation parameter.     

X-ray binary systems and nonextensivity

We study the x-ray intensity variations obtained from the time series of 155 light curves of x-ray binary systems collected by the instrument All Sky Monitor on board the satellite Rossi X-Ray Timing Explorer. These intensity distributions are adequately fitted by q-Gaussian distributions which maximize the Tsallis entropy and in turn satisfy a nonlinear Fokker-Planck equation, indicating their nonextensive and nonequilibrium behavior. From the values of the entropic index q obtained, we give a physical interpretation of the dynamics in x-ray binary systems based on the kinetic foundation of generalized thermostatistics. The present findings indicate that the binary systems display a nonextensive and turbulent behavior.     

On the construction of complex networks with optimal Tsallis entropy

In this work, we first formulate the Tsallis entropy in the context of complex networks. We then propose a network construction whose topology maximizes the Tsallis entropy. The growing network model has two main ingredients: copy process and random attachment mechanism (C-R model). We show that the resulting degree distribution exactly agrees with the required degree distribution that maximizes the Tsallis entropy. We also provide another example of network model using a combination of preferential and random attachment mechanisms (P-R model) and compare it with the distribution of the Tsallis entropy. In this case, we show that by adequately identifying the exponent factor q, the degree distribution can also be written in the q-exponential form. Taken together, our findings suggest that both mechanisms, copy process and preferential attachment, play a key role for the realization of networks with maximum Tsallis entropy. Finally, we discuss the interpretation of q parameter of the Tsallis entropy in the context of complex networks.     

An interacting-agent model of financial markets from the viewpoint of nonextensive statistical mechanics

In this paper we present an interacting-agent model of stock markets. We describe a stock market through an Ising-like model in order to formulate the tendency of traders to be influenced by the other traders’ investment attitudes [Kaizoji, Physica A 287 (2000) 493], and formulate the traders’ decision-making regarding investment as the maximum entropy principle for nonextensive entropy [C. Tsallis, J. Stat. Phys. 52 (1988) 479]. We demonstrate that the equilibrium probability distribution function of the traders’ investment attitude is the q-exponential distribution. We also show that the power-law distribution of the volatility of price fluctuations, which is often demonstrated in empirical studies can be explained naturally by our model which originates in the collective crowd behavior of many interacting-agents.