Giorgio Kaniadakis

Non-linear kinetics underlying generalized statistics

The purpose of the present effort is threefold. Firstly, it is shown that there exists a principle, that we call kinetical interaction principle (KIP), underlying the non-linear kinetics in particle systems, independently on the picture (Kramers, Boltzmann) used to describe their time evolution. Secondly, the KIP imposes the form of the generalized entropy associated to the system and permits to obtain the particle statistical distribution, both as stationary solution of the non-linear evolution equation and as the state which maximizes the generalized entropy. Thirdly, the KIP allows, on one hand, to treat all the classical or quantum statistical distributions already known in the literature in a unifying scheme and, on the other hand, suggests how we can introduce naturally new distributions. Finally, as a working example of the approach to the non-linear kinetics here presented, a new non-extensive statistics is constructed and studied starting from a one-parameter deformation of the exponential function holding the relation f(−x)f(x)=1.

Statistical mechanics in the context of special relativity. II

The special relativity laws emerge as one-parameter (light speed) generalizations of the corresponding laws of classical physics. These generalizations, imposed by the Lorentz transformations, affect both the definition of the various physical observables (e.g., momentum, energy, etc.), as well as the mathematical apparatus of the theory. Here, following the general lines of [Phys. Rev. E 66, 056125 (2002)], we show that the Lorentz transformations impose also a proper one-parameter generalization of the classical Boltzmann-Gibbs-Shannon entropy. The obtained relativistic entropy permits us to construct a coherent and self-consistent relativistic statistical theory, preserving the main features of the ordinary 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. Furthermore, this statistical mechanics can be obtained as the stationary case of a generalized kinetic theory governed by an evolution equation obeying the H theorem and reproducing the Boltzmann equation of the ordinary kinetics in the classical limit.

Generalized statistics and solar neutrinos

Abstract The generalized Tsallis statistics produces a distribution function appropriate to describe the interior solar plasma, thought as a stellar polytrope, showing a tail depleted with respect to the Maxwell-Boltzmann distribution and reduces to zero at energies greater than about 20 k B T . The Tsallis statistics can theoretically support the distribution suggested in the past by Clayton and collaborators, which shows also a depleted tail, to explain the solar neutrino counting rate.

New approach to optical analysis of absorbing thin solid films.

A powerful new technique is reported which enables realistic calculation of the optical energy gap of absorbing thin solid films by an analysis of measured transmittance and reflectance spectra in the fundamental absorption region. At the same time a new analytical method allows the thickness of films to be evaluated by measurements of transmittance only.

H-theorem and generalized entropies within the framework of nonlinear kinetics

Abstract In the present effort we consider the most general nonlinear 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 nonlinear 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 κ -exponential exp {κ} (x)=( 1+κ 2 x 2 +κx) 1/κ recently proposed, which reduces to the standard exponential as the deformation parameter κ approaches to zero and presents the relevant power law asymptotic behaviour exp {κ} (x) ∼ x→±∞ |2κx| ±1/|κ| . The κ -kinetics obeys the H-theorem and in the case of Brownian particles, admits as stationary state the distribution f = Z −1 exp { κ } [−( βmv 2 /2− μ )] which can be obtained also by maximizing the entropy S κ =∫d n v [c(κ)f 1+κ +c(−κ)f 1−κ ] with c ( κ )=− Z κ /[2 κ (1+ κ )] after properly constrained.

Two-parameter deformations of logarithm, exponential, and entropy: A consistent framework for generalized statistical mechanics

A consistent generalization of statistical mechanics is obtained by applying the maximum entropy principle to a trace-form entropy and by requiring that physically motivated mathematical properties are preserved. The emerging differential-functional equation yields a two-parameter class of generalized logarithms, from which entropies and power-law distributions follow: these distributions could be relevant in many anomalous systems. Within the specified range of parameters, these entropies possess positivity, continuity, symmetry, expansibility, decisivity, maximality, concavity, and are Lesche stable. The Boltzmann-Shannon entropy and some one-parameter generalized entropies already known belong to this class. These entropies and their distribution functions are compared, and the corresponding deformed algebras are discussed.

Maximum entropy principle and power-law tailed distributions

AbstractIn ordinary statistical mechanics the Boltzmann-Shannon entropy is related to the Maxwell-Bolzmann distribution pi by means of a twofold link. The first link is differential and is offered by the Jaynes Maximum Entropy Principle. Indeed, the Maxwell-Boltzmann distribution is obtained by maximizing the Boltzmann-Shannon entropy under proper constraints. The second link is algebraic and imposes that both the entropy and the distribution must be expressed in terms of the same function in direct and inverse form. Indeed, the Maxwell-Boltzmann distribution pi is expressed in terms of the exponential function, while the Boltzmann-Shannon entropy is defined as the mean value of -ln (pi). In generalized statistical mechanics the second link is customarily relaxed. Of course, the generalized exponential function defining the probability distribution function after inversion, produces a generalized logarithm Λ(pi). But, in general, the mean value of -Λ(pi) is not the entropy of the system. Here we reconsider the question first posed in [Phys. Rev. E 66, 056125 (2002) and 72, 036108 (2005)], if and how is it possible to select generalized statistical theories in which the above mentioned twofold link between entropy and the distribution function continues to hold, such as in the case of ordinary statistical mechanics. Within this scenario, apart from the standard logarithmic-exponential functions that define ordinary statistical mechanics, there emerge other new couples of direct-inverse functions, i.e. generalized logarithms Λ(x) and generalized exponentials Λ-1(x), defining coherent and self-consistent generalized statistical theories. Interestingly, all these theories preserve the main features of ordinary statistical mechanics, and predict distribution functions presenting power-law tails. Furthermore, the obtained generalized entropies are both thermodynamically and Lesche stable.

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.

A new one-parameter deformation of the exponential function

Recently, in Kaniadakis (Physica A 296 (2001) 405), a new one-parameter deformation for the exponential function exp{κ}(x)=(1+κ2x2+κx)1/κ; exp{0}(x)=exp(x), which presents a power-law asymptotic behaviour, has been proposed. The statistical distribution f=Z−1exp{κ}[−β(E−μ)], has been obtained both as stable stationary state of a proper nonlinear kinetics and as the state which maximizes a new entropic form. In the present contribution, starting from the κ-algebra and after introducing the κ-analysis, we obtain the κ-exponential exp{κ}(x) as the eigenstate of the κ-derivative and study its main mathematical properties.

Theoretical foundations and mathematical formalism of the power-law tailed statistical distributions

We present the main features of the mathematical theory generated by the κ-deformed exponential function exp κ(χ) = (√ 1 κ²χ² κχ)¹/κ, with 0 ≤ κ < 1, developed in the last twelve years, which turns out to be a continuous one parameter deformation of the ordinary mathematics generated by the Euler exponential function. The κ-mathematics has its roots in special relativity and furnishes the theoretical foundations of the κ-statistical mechanics predicting power law tailed statistical distributions which have been observed experimentally in many physical, natural and artificial systems. After introducing the κ-algebra we present the associated κ-differential and κ-integral calculus. Then we obtain the corresponding κ-exponential and κ-logarithm functions and give the κ-version of the main functions of the ordinary mathematics