Functions of Mittag-Leffler and Fox: The Pathway Model to Tsallis Statistics and Beck-Cohen Superstatistics

In reaction rate theory, in production-destruction type models and in reaction-diffusion problems when the total derivatives are replaced by fractional derivatives the solutions are obtained in terms of Mittag-Leffler functions and their generalizations. When fractional calculus enters into the picture the solutions of these problems, usually available in terms of generalized hypergeometric functions, switch to Mittag-Leffler functions and their generalizations into Wright functions and subsequently into Fox functions. In this paper, connections are established among generalized Mittag-Leffler functions, Mathai's pathway model, Tsallis statistics, Beck-Cohen superstatistics, and among corresponding entropic measures. The Mittag-Leffler function, for large values of the parameter, approaches a power-law. For values of the parameter close to zero, the Mittag-Leffler function behaves like a stretched exponential. The Mittag-Leffler function is a generalization of the exponential function and represents a deviation from the exponential paradigm whenever it shows up in solution of physical problems. The paper elucidates the relation between analytic representations of the q-exponential function that is fundamental to Tsallis statistics, Mittag-Leffler, Wright, and Fox functions, respectively, utilizing Mellin-Barnes integral representations.