The Relationship between Tsallis Statistics, the Fourier Transform, and Nonlinear Coupling

Abstract

Tsallis statistics (or q-statistics) in nonextensive statistical mechanics is a one-parameter description of correlated states. In this paper we use a translated entropic index: $1-q\to q$ . The essence of this translation is to improve the mathematical symmetry of the q-algebra and make q directly proportional to the nonlinear coupling. A conjugate transformation is defined $\hat{q}\equiv \frac{-2q}{2+q}$ which provides a dual mapping between the heavy-tail q-Gaussian distributions, whose translated q parameter is between $-2<q<0$, and the compact-support q-Gaussians, between $0<q<\infty$ . This conjugate transformation is used to extend the definition of the q-Fourier transform to the domain of compact support. A conjugate q-Fourier transform is proposed which transforms a q-Gaussian into a conjugate $\hat{q}$ -Gaussian, which has the same exponential decay as the Fourier transform of a power-law function. The nonlinear statistical coupling is defined such that the conjugate pair of q-Gaussians have equal strength but either couple (compact-support) or decouple (heavy-tail) the statistical states. Many of the nonextensive entropy applications can be shown to have physical parameters proportional to the nonlinear statistical coupling.

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• T. Dauxois' "Non-Gaussian distributions under scrutiny" under scrutiny