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

nonextensive statistical mechanics is a one-parameter description of correlated states. In this paper we use a translated entropic index: 1-q 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 2 2 ˆ q q q − + ≡ which provides a dual mapping between the heavy-tail q-Gaussian distributions, whose translated q parameter is between 2 0 q − < < , and the compact-support q-Gaussians, between 0 q < < ∞. 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 conjugatê 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.