Ricci curvature, isoperimetry and a non-additive entropy

TL;DR

This paper investigates the connection between a generalized Ricci curvature tensor in phase space and the non-additive entropy, providing geometric insights and an isoperimetric interpretation of the non-extensive parameter.

Contribution

It introduces a covariant curvature tensor linked to non-additive entropy and explores its implications and geometric interpretation in the context of complex systems.

Findings

Establishes a connection between generalized Ricci curvature and non-additive entropy.

Provides an isoperimetric interpretation of the non-extensive parameter.

Suggests new geometric tools to analyze non-extensive thermodynamic systems.

Abstract

Searching for the dynamical foundations of the Havrda-Charv\'{a}t/Dar\'{o}czy/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-\'{E}mery-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.