Ricci curvature, isoperimetry and a non-additive entropy

Abstract

Searching for the dynamical foundations of the Havrda-Charvát/Daróczy/Cressie-Read/Tsallis non-additive entropy, we come across a covariant quantity called, alternatively, a generalized Ricci curvature, an $N$-Ricci curvature or a Bakry-Émery-Ricci curvature in the configuration/phase space of a system. We explore some of the implications of this tensor and its associated curvature and present a connection with the non-additive entropy under investigation. We present an isoperimetric interpretation of the non-extensive parameter and comment on further features of the system that can be probed through this tensor.

arXiv comment31 pages, No figures, Standard LaTeX. Based, in part, on a talk delivered at SigmaPhi 2014, 7-11 July 2014, Rhodes, Greece

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• Riemannian submersions for q-entropies