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

TL;DR

This paper explores the mathematical relationship between Tsallis statistics, Fourier transforms, and nonlinear coupling, introducing a conjugate transformation that links different q-Gaussian distributions and extends the q-Fourier transform's applicability.

Contribution

It introduces a conjugate transformation between heavy-tail and compact-support q-Gaussians, extending the q-Fourier transform and linking nonextensive entropy to nonlinear statistical coupling.

Findings

Defined a conjugate transformation $ ilde{q}$ for q-Gaussians.

Extended the q-Fourier transform to compact support distributions.

Linked nonlinear statistical coupling to physical parameters in entropy applications.

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.