Intensive variables in the framework of the non-extensive thermostatistics

By assuming an appropriate energy composition law between two systems governed by the same non-extensive entropy, we revisit the definitions of temperature and pressure, arising from the zeroth principle of thermodynamics, in a manner consistent with the thermostatistics structure of the theory. We show that the definitions of these quantities are sensitive to the composition law of entropy and internal energy governing the system. In this way, we can clarify some questions raised about the possible introduction of intensive variables in the context of non-extensive statistical mechanics.Comment: 14 pages, elsart style, version accepted on Physics Letters

Generalized Kinetic Equations for a System of Interacting Atoms and Photons: Theory and Simulations

In the present paper we introduce generalized kinetic equations describing the dynamics of a system of interacting gas and photons obeying to a very general statistics. In the space homogeneous case we study the equilibrium state of the system and investigate its stability by means of Lyapounov's theory. Two physically relevant situations are discussed in details: photons in a background gas and atoms in a background radiation. After having dropped the statistics generalization for atoms but keeping the statistics generalization for photons, in the zero order Chapmann-Enskog approximation, we present two numerical simulations where the system, initially at equilibrium, is perturbed by an external isotropic Dirac's delta and by a constant source of photons.Comment: 24 pages, 4 figures, IOP macro style, accepted on J. Phys. A: Math. Ge

A mechanism to derive multi-power law functions: an application in the econophysics framework

It is generally recognized that economical systems, and more in general complex systems, are characterized by power law distributions. Sometime, these distributions show a changing of the slope in the tail so that, more appropriately, they show a multi-power law behavior. We present a method to derive analytically a two-power law distribution starting from a single power law function recently obtained, in the frameworks of the generalized statistical mechanics based on the Sharma-Taneja-Mittal information measure. In order to test the method, we fit the cumulative distribution of personal income and gross domestic production of several countries, obtaining a good agreement for a wide range of data.Comment: 10pages, 3 figures. Presented at Int. Conf. on Application of Physics in Financial Analisys (APFA5), June 29 - July 1, 2006 Torino, Ital

Canonical partition function for anomalous systems described by the κ\kappa-entropy

Starting from the $\kappa$-distribution function, obtained by applying the maximal entropy principle to the $\kappa$-entropy [G. Kaniadakis, Phys. Rev. E 66 (2002), 056125], we derive the expression of the canonical $\kappa$-partition function and discuss its main properties. It is shown that all important macroscopical quantities of the system can be expressed employing only the $\kappa$-partition function. The relationship between the associated $\kappa$-free energy and the $\kappa$-entropy is also discussed.Comment: 8 pages, no figures. Work presented at the International conference Complexity and Nonextensivity: New Trends in Statistical Mechanics. - Yukawa Institute for Theoretical Physics - (14-18 March 2005) Kyoto, Japa

Lesche Stability of κ\kappa-Entropy

The Lesche stability condition for the Shannon entropy [B. Lesche, J. Stat. Phys. 27, 419 (1982)], represents a fundamental test, for its experimental robustness, for systems obeying the Maxwell-Boltzmann statistical mechanics. Of course, this stability condition must be satisfied by any entropic functional candidate to generate non-conventional statistical mechanics. In the present effort we show that the $\kappa$-entropy, recently introduced in literature [G. Kaniadakis, Phys. Rev. E 66, 056125 (2002)], satisfies the Lesche stability condition.Comment: Presented at next2003, Second Sardinian International Conference on News and Expectations in Thermostatistics, Villasimius (Cagliari) Italy, 21st-28th September 2003. In press Physica A (2004). Elsevier LaTeX macros, 10 pages, minor change

Quantum Orthogonal Planes: ISO_{q,r}(N) and SO_{q,r}(N) -- Bicovariant Calculi and Differential Geometry on Quantum Minkowski Space

We construct differential calculi on multiparametric quantum orthogonal planes in any dimension N. These calculi are bicovariant under the action of the full inhomogeneous (multiparametric) quantum group ISO_{q,r}(N), and do contain dilatations. If we require bicovariance only under the quantum orthogonal group SO_{q,r}(N), the calculus on the q-plane can be expressed in terms of its coordinates x^a, differentials dx^a and partial derivatives \partial_a without the need of dilatations, thus generalizing known results to the multiparametric case. Using real forms that lead to the signature (n+1,m) with m = n-1, n, n+1 , we find ISO_{q,r}(n+1, m) and SO_{q,r}(n+1,m) bicovariant calculi on the multiparametric quantum spaces. The particular case of the quantum Minkowski space ISO_{q,r}(3,1)/SO_{q,r}(3,1) is treated in detail. The conjugated partial derivatives \partial_a* can be expressed as linear combinations of the \partial_a. This allows a deformation of the phase-space where no additional operators (besides x^a and p_a) are needed.Comment: LaTeX, 36 pages. Considered more real forms, added some explicit formulas, used simpler definition of hermitean momenta. To be published in European Phys. Jou.

Deformed logarithms and entropies

By solving a differential-functional equation inposed by the MaxEnt principle we obtain a class of two-parameter deformed logarithms and construct the corresponding two-parameter generalized trace-form entropies. Generalized distributions follow from these generalized entropies in the same fashion as the Gaussian distribution follows from the Shannon entropy, which is a special limiting case of the family. We determine the region of parameters where the deformed logarithm conserves the most important properties of the logarithm, and show that important existing generalizations of the entropy are included as special cases in this two-parameter class.Comment: Presented at next2003, Second Sardinian International Conference on News and Expectations in Thermostatistics, Villasimius (Cagliari) Italy, 21st-28th September 2003. In press Physica A (2004). Elsevier LaTeX macros, 11 pages, 1 figur