Abstract
Searching for the dynamical foundations of 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.
Highlights
Havrda-Charvát [1] / Daróczy [2] / Cressie-Read [3,4] / Tsallis [5,6] entropy is single parameter family of functionals, which have attracted some interest in the Statistical Mechanics community over the last 25 years
The Ricci curvature in the direction of e1, which as was stated above is assumed to be tangent to the geodesic c passing by P, is a symmetric bilinear form related to the Ricci tensor by
The definition of the Ricci tensor or Ricci curvature uses explicitly two distinct facets of Riemannian manifolds: their metric and a measure
Summary
Havrda-Charvát [1] / Daróczy [2] / Cressie-Read [3,4] / Tsallis [5,6] entropy is single parameter family of functionals, which have attracted some interest in the Statistical Mechanics community over the last 25 years. Entropy 2015, 17 ρ : Ω → R, and is assumed to be absolutely continuous everywhere on Ω with respect to the Lebesgue measure (volume) dvolΩ is In both of the above expressions q ∈ R, a recent work has suggested [7] the possibility of q ∈ C. To determine the microscopic dynamical systems whose collective behaviour is encoded by the entropic functional under consideration. The dynamical behaviour of such systems is described by their evolution in their configuration or phase space Such a space is a manifold M endowed with a Riemannian metric g.
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