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 (x) 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

Composition law of κ\kappa-entropy for statistically independent systems

The intriguing and still open question concerning the composition law of $\kappa$-entropy $S_{\kappa}(f)=\frac{1}{2\kappa}\sum_i (f_i^{1-\kappa}-f_i^{1+\kappa})$ with $0<\kappa<1$ and $\sum_i f_i =1$ is here reconsidered and solved. It is shown that, for a statistical system described by the probability distribution $f=\{ f_{ij}\}$, made up of two statistically independent subsystems, described through the probability distributions $p=\{ p_i\}$ and $q=\{ q_j\}$, respectively, with $f_{ij}=p_iq_j$, the joint entropy $S_{\kappa}(p\,q)$ can be obtained starting from the $S_{\kappa}(p)$ and $S_{\kappa}(q)$ entropies, and additionally from the entropic functionals $S_{\kappa}(p/e_{\kappa})$ and $S_{\kappa}(q/e_{\kappa})$, $e_{\kappa}$ being the $\kappa$-Napier number. The composition law of the $\kappa$-entropy is given in closed form, and emerges as a one-parameter generalization of the ordinary additivity law of Boltzmann-Shannon entropy recovered in the $\kappa \rightarrow 0$ limit.Comment: 14 page

Extended Poisson-Kac theory: A unifying framework for stochastic processes with finite propagation velocity

Stochastic processes play a key role for mathematically modeling a huge variety of transport problems out of equilibrium. To formulate models of stochastic dynamics the mainstream approach consists in superimposing random fluctuations on a suitable deterministic evolution. These fluctuations are sampled from probability distributions that are prescribed a priori, most commonly as Gaussian or Levy. While these distributions are motivated by (generalised) central limit theorems they are nevertheless unbounded. This property implies the violation of fundamental physical principles such as special relativity and may yield divergencies for basic physical quantities like energy. It is thus clearly never valid in real-world systems by rendering all these stochastic models ontologically unphysical. Here we solve the fundamental problem of unbounded random fluctuations by constructing a comprehensive theoretical framework of stochastic processes possessing finite propagation velocity. Our approach is motivated by the theory of Levy walks, which we embed into an extension of conventional Poisson-Kac processes. Our new theory possesses an intrinsic flexibility that enables the modelling of many different kinds of dynamical features, as we demonstrate by three examples. The corresponding stochastic models capture the whole spectrum of diffusive dynamics from normal to anomalous diffusion, including the striking Brownian yet non Gaussian diffusion, and more sophisticated phenomena such as senescence. Extended Poisson-Kac theory thus not only ensures by construction a mathematical representation of physical reality that is ontologically valid at all time and length scales. It also provides a toolbox of stochastic processes that can be used to model potentially any kind of finite velocity dynamical phenomena observed experimentally.Comment: 25 pages, 5 figure

Group entropies, correlation laws and zeta functions

The notion of group entropy is proposed. It enables to unify and generalize many different definitions of entropy known in the literature, as those of Boltzmann-Gibbs, Tsallis, Abe and Kaniadakis. Other new entropic functionals are presented, related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium, when the dynamics is weakly chaotic. The associated thermostatistics are discussed. The mathematical structure underlying our construction is that of formal group theory, which provides the general structure of the correlations among particles and dictates the associated entropic functionals. As an example of application, the role of group entropies in information theory is illustrated and generalizations of the Kullback-Leibler divergence are proposed. A new connection between statistical mechanics and zeta functions is established. In particular, Tsallis entropy is related to the classical Riemann zeta function.Comment: to appear in Physical Review

Using the κ-exponential to deform Gumbel and Gompertz probability distributions

Here we propose the use of the κ-exponential to deform Gumbel and the Gompertz distributions. The κ-exponential is a function of κ-statistics, a statistics which has been developed by G. Kaniadakis, Politecnico di Torino, in the framework of special relativity, and used for statistical analyses involving power law tailed distributions. The exponentiated κ-Gumbel function is also considered, to have a further deformation of the probability distribution