Pathway model, superstatistics, Tsallis statistics, and a generalized measure of entropy

Abstract

Abstract. The pathway model of Mathai (2005) is shown to be inferable from the maximization of a certain generalized entropy measure. This entropy is a variant of the generalized entropy of order α, considered in Mathai and Rathie (1975), and it is also associated with Shannon, Boltzmann-Gibbs, Rényi, Tsallis, and Havrda-Charvát entropies. The generalized entropy measure introduced here is also shown to have interesting statistical properties and it can be given probabilistic interpretations in terms of inaccuracy measure, expected value, and information content in a scheme. Particular cases of the pathway model are shown to be Tsallis statistics (Tsallis, 1988) and superstatistics introduced by Beck and Cohen (2003). The pathway model's connection to fractional calculus is illustrated by considering a fractional reaction equation.

Generalized statistical mechanics for superstatistical systems

“…More generally, one can prove a superstatistical generalization of fluctuation theorems [4], develop a variational principle for the large-energy asymptotics of general superstatistics [3], proceed to generalized entropies for general superstatistics [5,40,43], let the q-values in equation (2.1) fluctuate as well [7] and prove superstatistical versions of a central limit theorem [8]. There are also relations with fractional reaction equations [45], random matrix theory [20,21,35], networks [36] and path integrals [9]. Very useful for practical applications is a superstatistical approach to time-series analysis [2,24,27].…”

On the q-Laplace Transform and Related Special Functions

“…Mathai and Rathie [17] considered various generalizations of the Shannon entropy measure and describe various properties, including additivity, the characterization theorem, etc. Mathai and Haubold [18] introduced a new generalized entropy measure, which is a generalization of the Shannon entropy measure. For a multinomial population P = (p 1 , .…”

Certain Fractional Integral Inequalities Associated With Pathway Fractional Integral Operators

“…Recently many authors have presented a number of interesting integral inequalities of Pólya and Szegö type by using the Riemann-Liouville fractional integral operator (see [4,23]). Nair [22] introduced and investigated a new fractional integral operator through the idea of pathway model given by Mathai [17] (and further studied by Mathai and Haubold [18,19]). Here, motivated essentially by the above works, we aim at establishing certain (presumably) new Pólya-Szegö type inequalities associated with the pathway fractional integral operator.…”