H-Theorem and Generalized Entropies Within the Framework of Non Linear Kinetics

In the present effort we consider the most general non linear particle kinetics within the framework of the Fokker-Planck picture. We show that the kinetics imposes the form of the generalized entropy and subsequently we demonstrate the H-theorem. The particle statistical distribution is obtained, both as stationary solution of the non linear evolution equation and as the state which maximizes the generalized entropy. The present approach allows to treat the statistical distributions already known in the literature in a unifying scheme. As a working example we consider the kinetics, constructed by using the $\kappa$-exponential $\exp_{_{\{\kappa\}}}(x)= (\sqrt{1+\kappa^2x^2}+\kappa x)^{1/\kappa}$ recently proposed which reduces to the standard exponential as the deformation parameter $\kappa$ approaches to zero and presents the relevant power law asymptotic behaviour $\exp_{_{\{\kappa\}}}(x){\atop\stackrel\sim x\to \pm \infty}|2\kappa x|^{\pm 1/|\kappa|}$. The $\kappa$-kinetics obeys the H-theorem and in the case of Brownian particles, admits as stationary state the distribution $f=Z^{-1}\exp_{_{\{\kappa\}}}[-(\beta mv^2/2-\mu)]$ which can be obtained also by maximizing the entropy $S_{\kappa}=\int d^n v [ c(\kappa)f^{1+\kappa}+c(-\kappa)f^{1-\kappa}]$ with $c(\kappa)=-Z^{\kappa}/ [2\kappa(1+\kappa)]$ after properly constrained.Comment: To appear in Phys. Lett.

The HH-theorem in κ\kappa-statistics: influence on the molecular chaos hypothesis

We rediscuss recent derivations of kinetic equations based on the Kaniadakis' entropy concept. Our primary objective here is to derive a kinetical version of the second law of thermodynamycs in such a $\kappa$-framework. To this end, we assume a slight modification of the molecular chaos hypothesis. For the $H_{\kappa}$-theorem, it is shown that the collisional equilibrium states (null entropy source term) are described by a $\kappa$-power law extension of the exponential distribution and, as should be expected, all these results reduce to the standard one in the limit $\kappa\to 0$.Comment: 4 pages, eqs. (18) and (22) have been corrected, to appear in Phys. Lett.

The relativistic statistical theory and Kaniadakis entropy: an approach through a molecular chaos hypothesis

We have investigated the proof of the $H$ theorem within a manifestly covariant approach by considering the relativistic statistical theory developed in [G. Kaniadakis, Phy. Rev. E {\bf 66}, 056125, 2002; {\it ibid.} {\bf 72}, 036108, 2005]. As it happens in the nonrelativistic limit, the molecular chaos hypothesis is slightly extended within the Kaniadakis formalism. It is shown that the collisional equilibrium states (null entropy source term) are described by a $\kappa$ power law generalization of the exponential Juttner distribution, e.g., $f(x,p)\propto (\sqrt{1+ \kappa^2\theta^2}+\kappa\theta)^{1/\kappa}\equiv\exp_\kappa\theta$, with $\theta=\alpha(x)+\beta_\mu p^\mu$, where $\alpha(x)$ is a scalar, $\beta_\mu$ is a four-vector, and $p^\mu$ is the four-momentum. As a simple example, we calculate the relativistic $\kappa$ power law for a dilute charged gas under the action of an electromagnetic field $F^{\mu\nu}$. All standard results are readly recovered in the particular limit $\kappa\to 0$.Comment: 7 pages; to be published in EPJ

Legendre structure of the thermostatistics theory based on the Sharma-Taneja-Mittal entropy

The statistical proprieties of complex systems can differ deeply for those of classical systems governed by Boltzmann-Gibbs entropy. In particular, the probability distribution function observed in several complex systems shows a power law behavior in the tail which disagrees with the standard exponential behavior showed by Gibbs distribution. Recently, a two-parameter deformed family of entropies, previously introduced by Sharma, Taneja and Mittal (STM), has been reconsidered in the statistical mechanics framework. Any entropy belonging to this family admits a probability distribution function with an asymptotic power law behavior. In the present work we investigate the Legendre structure of the thermostatistics theory based on this family of entropies. We introduce some generalized thermodynamical potentials, study their relationships with the entropy and discuss their main proprieties. Specialization of the results to some one-parameter entropies belonging to the STM family are presented.Comment: 11 pages, RevTex4; contribution to the international conference "Next Sigma Phi" on News, EXpectations, and Trends in statistical physics, Crete 200

A new one parameter deformation of the exponential function

Recently, in the ref. Physica A \bfm{296} 405 (2001), a new one parameter deformation for the exponential function $\exp_{_{\{{\scriptstyle \kappa}\}}}(x)= (\sqrt{1+\kappa^2x^2}+\kappa x)^{1/\kappa}; \exp_{_{\{{\scriptstyle 0}\}}}(x)=\exp (x)$, which presents a power law asymptotic behaviour, has been proposed. The statistical distribution $f=Z^{-1}\exp_{_{\{{\scriptstyle \kappa}\}}}[-\beta(E-\mu)]$, has been obtained both as stable stationary state of a proper non linear kinetics and as the state which maximizes a new entropic form. In the present contribution, starting from the $\kappa$-algebra and after introducing the $\kappa$-analysis, we obtain the $\kappa$-exponential $\exp_{_{\{{\scriptstyle \kappa}\}}}(x)$ as the eigenstate of the $\kappa$-derivative and study its main mathematical properties.Comment: 5 pages including 2 figures. Paper presented in NEXT2001 Meetin

k-Generalized Statistics in Personal Income Distribution

Starting from the generalized exponential function $\exp_{\kappa}(x)=(\sqrt{1+\kappa^{2}x^{2}}+\kappa x)^{1/\kappa}$, with $\exp_{0}(x)=\exp(x)$, proposed in Ref. [G. Kaniadakis, Physica A \textbf{296}, 405 (2001)], the survival function $P_{>}(x)=\exp_{\kappa}(-\beta x^{\alpha})$, where $x\in\mathbf{R}^{+}$, $\alpha,\beta>0$, and $\kappa\in[0,1)$, is considered in order to analyze the data on personal income distribution for Germany, Italy, and the United Kingdom. The above defined distribution is a continuous one-parameter deformation of the stretched exponential function $P_{>}^{0}(x)=\exp(-\beta x^{\alpha})$\textemdash to which reduces as $\kappa$ approaches zero\textemdash behaving in very different way in the $x\to0$ and $x\to\infty$ regions. Its bulk is very close to the stretched exponential one, whereas its tail decays following the power-law $P_{>}(x)\sim(2\beta\kappa)^{-1/\kappa}x^{-\alpha/\kappa}$. This makes the $\kappa$-generalized function particularly suitable to describe simultaneously the income distribution among both the richest part and the vast majority of the population, generally fitting different curves. An excellent agreement is found between our theoretical model and the observational data on personal income over their entire range.Comment: Latex2e v1.6; 14 pages with 12 figures; for inclusion in the APFA5 Proceeding

Kinetical Foundations of Non Conventional Statistics

After considering the kinetical interaction principle (KIP) introduced in ref. Physica A {\bf296}, 405 (2001), we study in the Boltzmann picture, the evolution equation and the H-theorem for non extensive systems. The $q$-kinetics and the $\kappa$-kinetics are studied in detail starting from the most general non linear Boltzmann equation compatible with the KIP.Comment: 11 pages, no figures. Contribution paper to the proseedings of the International School and Workshop on Nonextensive Thermodynamics and Physical Applications, NEXT 2001, 23-30 May 2001, Cagliari Sardinia, Italy (Physica A

A note on bounded entropies

The aim of the paper is to study the link between non additivity of some entropies and their boundedness. We propose an axiomatic construction of the entropy relying on the fact that entropy belongs to a group isomorphic to the usual additive group. This allows to show that the entropies that are additive with respect to the addition of the group for independent random variables are nonlinear transforms of the R\'enyi entropies, including the particular case of the Shannon entropy. As a particular example, we study as a group a bounded interval in which the addition is a generalization of the addition of velocities in special relativity. We show that Tsallis-Havrda-Charvat entropy is included in the family of entropies we define. Finally, a link is made between the approach developed in the paper and the theory of deformed logarithms.Comment: 10 pages, 1 figur

Towards a relativistic statistical theory

In special relativity the mathematical expressions, defining physical observables as the momentum, the energy etc, emerge as one parameter (light speed) continuous deformations of the corresponding ones of the classical physics. Here, we show that the special relativity imposes a proper one parameter continuous deformation also to the expression of the classical Boltzmann-Gibbs-Shannon entropy. The obtained relativistic entropy permits to construct a coherent and selfconsistent relativistic statistical theory [Phys. Rev. E {\bf 66}, 056125 (2002); Phys. Rev. E {\bf 72}, 036108 (2005)], preserving the main features (maximum entropy principle, thermodynamic stability, Lesche stability, continuity, symmetry, expansivity, decisivity, etc.) of the classical statistical theory, which is recovered in the classical limit. The predicted distribution function is a one-parameter continuous deformation of the classical Maxwell-Boltzmann distribution and has a simple analytic form, showing power law tails in accordance with the experimental evidence.Comment: Physica A (2006). Proof correction

κ\kappa-generalization of Gauss' law of error

Based on the $\kappa$-deformed functions ($\kappa$-exponential and $\kappa$-logarithm) and associated multiplication operation ($\kappa$-product) introduced by Kaniadakis (Phys. Rev. E \textbf{66} (2002) 056125), we present another one-parameter generalization of Gauss' law of error. The likelihood function in Gauss' law of error is generalized by means of the $\kappa$-product. This $\kappa$-generalized maximum likelihood principle leads to the {\it so-called} $\kappa$-Gaussian distributions.Comment: 9 pages, 1 figure, latex file using elsart.cls style fil