Nonextensive thermostatistics and the H theorem

The kinetic foundations of Tsallis' nonextensive thermostatistics are investigated through Boltzmann's transport equation approach. Our analysis follows from a nonextensive generalization of the "molecular chaos hypothesis." For q>0, the q-transport equation satisfies an H theorem based on Tsallis entropy. It is also proved that the collisional equilibrium is given by Tsallis' q-nonextensive velocity distribution.     

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.     

Long-term evolution of stellar self-gravitating systems away from thermal equilibrium: connection with nonextensive statistics

With particular attention to the recently postulated introduction of a nonextensive generalization of Boltzmann-Gibbs statistics, we study the long-term stellar dynamical evolution of self-gravitating systems on time scales much longer than the two-body relaxation time. In a self-gravitating N-body system confined in an adiabatic wall, we show that the quasiequilibrium sequence arising from the Tsallis entropy, so-called stellar polytropes, plays an important role in characterizing the transient states away from the Boltzmann-Gibbs equilibrium state.     

Nonextensive statistical mechanics of ionic solutions

Classical mean-field Poisson–Boltzmann theory of ionic solutions is revisited in the theoretical framework of nonextensive Tsallis statistics. The nonextensive equivalent of Poisson–Boltzmann equation is formulated revisiting the statistical mechanics of liquids and the Debye–Hückel framework is shown to be valid for highly diluted solutions even under circumstances where nonextensive thermostatistics must be applied. The lowest order corrections associated to nonadditive effects are identified for both symmetric and asymmetric electrolytes and the behavior of the average electrostatic potential in a homogeneous system is analytically and numerically analyzed for various values of the complexity measurement nonextensive parameter q.     

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.     

A generalized thermodynamics for power-law statistics

We show that there exists a natural way to define a condition of generalized thermal equilibrium between systems governed by Tsallis thermostatistics, under the hypotheses that (i) the coupling between the systems is weak, (ii) the structure functions of the systems have a power-law dependence on the energy. It is found that the q values of two such systems at equilibrium must satisfy a relationship involving the respective numbers of degrees of freedom. The physical properties of a Tsallis distribution can be conveniently characterized by a new parameter η which can vary between 0 and +∞, these limits corresponding, respectively, to the two opposite situations of a microcanonical distribution and of a distribution with a predominant power-tail at high energies. We prove that the statistical expression of the thermodynamic functions is univocally determined by the requirements that (a) systems at thermal equilibrium have the same temperature, (b) the definitions of temperature and entropy are consistent with the second law of thermodynamics. We find that, for systems satisfying the hypotheses (i) and (ii) specified above, the thermodynamic entropy is given by Rényi entropy.     

Generalized thermostatistics based on the Sharma-Mittal entropy and escort mean values

A generalized thermostatistics is developed for an entropy measure introduced by Sharma and Mittal. A maximum-entropy scheme involving the maximization of the Sharma and Mittal entropy under appropriate constraints expressed as escort mean values is advanced. Maximum-entropy distributions exhibiting a power law behavior in the asymptotic limit are obtained. Thus, results previously derived for the Renyi entropy and the Tsallis entropy are generalized. In addition, it is shown that for almost deterministic systems among all possible composable entropies with kernels that are described by power laws the Sharma-Mittal entropy is the only entropy measure that gives rise to a thermostatistics based on escort mean values and admitting of a partition function. Received 27 June 2002 Published online 31 December 2002     

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.     

The Minimal Length Uncertainty and the Nonextensive Thermodynamics

In this paper, we study the thermodynamics of quantum harmonic oscillator in the Tsallis framework and in the presence of a minimal length uncertainty. The existence of the minimal length is motivated by various theories such as string theory, loop quantum gravity, and black-hole physics. We analytically obtain the partition function, probability function, internal energy, and the specific heat capacity of the vibrational quantum system for \(1<q<\frac {3}{2}\) and compare the results with those of Tsallis and Boltzmann-Gibbs statistics without the minimal length scale.     

The Effect of Space Dimensions on a Generalized Hydrogen-Atom Specific Heat in the Generalized Boltzmann-Gibbs Statistics

In this article, the effect of the space dimensions on the generalized hydrogen-atom specific heat in the generalized Boltzmann-Gibbs statistics is studied. The temperature dependence of the specific heat for a few different values of q and for different low space dimensions using Tsallis statistics is numerically calculated. The results indicate that for a fixed value of q, as the space dimension increases the temperature range where the specific heat has a nonzero value, decreases, while the general behavior of the specific heat does not show any change. Also, there exits a q-independent quantity related to two specific temperatures of the system which is almost linearly dependent on the space dimensions.     

Generalized statistical mechanics for the N-body quantum problem — ideal gases

Within Tsallis generalized thermostatistics, the grand canonical ensemble is derived for quantum systems. In particular, the generalized Fermi-Dirac, Bose-Einstein and Maxwell-Boltzmann statistics are defined. The behavior of the chemical potential is depicted as a function of the temperature. Some thermodynamic quantities at high and low temperature are studied as well.     

Statistical mechanics based on Renyi entropy

In this work we show that it is possible to obtain a generalized statistical mechanics (thermostatistics) based on Renyi entropy, to be maximized with adequate constraints. The equilibrium probability distribution thus obtained has a very interesting property. Indeed, it reminds us the statistical distribution proposed by Tsallis, known to conveniently describe a variety of phenomena in nonextensive systems. Moreover, some examples are worked out in order to illustrate the main features of the herein introduced formalism.     

Weakly nonextensive thermostatistics and the Ising model with long-range interactions

We introduce a new nonextensive entropic measure that grows like , where N is the size of the system under consideration. This kind of nonextensivity arises in a natural way in some N-body systems endowed with long-range interactions described by interparticle potentials. The power law (weakly nonextensive) behavior exhibited by is intermediate between (1) the linear (extensive) regime characterizing the standard Boltzmann-Gibbs entropy and (2) the exponential law (strongly nonextensive) behavior associated with the Tsallis generalized q-entropies. The functional is parametrized by the real number in such a way that the standard logarithmic entropy is recovered when . We study the mathematical properties of the new entropy, showing that the basic requirements for a well behaved entropy functional are verified, i.e., possesses the usual properties of positivity, equiprobability, concavity and irreversibility and verifies Khinchin axioms except the one related to additivity since is nonextensive. For , the entropy becomes superadditive in the thermodynamic limit. The present formalism is illustrated by a numerical study of the thermodynamic scaling laws of a ferromagnetic Ising model with long-range interactions. Received 24 May 2000     

A family of nonextensive entropies

A generalized nonextensive two-parameter entropy is developed, along lines which unify current nonextensive frameworks. It recovers, as particular cases, the Tsallis and symmetric entropies, as well as the Boltzmann-Gibbs entropy. The properties of the new (q, q′)-entropy are analysed.     

Dynamics of normal and anomalous diffusion in nonlinear Fokker-Planck equations

Consequences of the connection between nonlinear Fokker-Planck equations and entropic forms are investigated. A particular emphasis is given to the feature that different nonlinear Fokker-Planck equations can be arranged into classes associated with the same entropic form and its corresponding stationary state. Through numerical integration, the time evolution of the solution of nonlinear Fokker-Planck equations related to the Boltzmann-Gibbs and Tsallis entropies are analyzed. The time behavior in both stages, in a time much smaller than the one required for reaching the stationary state, as well as towards the relaxation to the stationary state, are of particular interest. In the former case, by using the concept of classes of nonlinear Fokker-Planck equations, a rich variety of physical behavior may be found, with some curious situations, like an anomalous diffusion within the class related to the Boltzmann-Gibbs entropy, as well as a normal diffusion within the class of equations related to Tsallis’ entropy. In addition to that, the relaxation towards the stationary state may present a behavior different from most of the systems studied in the literature.