2008/11/23 by Kenric P. Nelson, Nelson, Kenric P., Sabir Umarov +1
Computer Science · Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Probability (math.PR) #cs.IT #math.IT #math.PR
paper · pdf · doi:10.48550/arxiv.0811.3777
arxiv created 2008/11/23 · arxiv updated 2009/12/01
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 → 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 q ≡ (- 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 < ∞ . 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 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.