Euclidean and Riemannian Geometrical Approaches to Non-Extensive Thermo-Statistical Mechanics

TL;DR

This paper introduces a novel geometrical framework for thermo-statistical mechanics, contrasting it with previous formalisms and emphasizing its relevance to non-extensive thermodynamics, offering new insights and questions in the field.

Contribution

It presents a new geometrical approach using Riemannian and Euclidean geometries, enhancing understanding of non-extensive thermodynamics and providing a new investigation tool.

Findings

Highlights the physical ideas translated into geometrical language

Contrasts with previous geometrical formalisms in thermodynamics

Proposes new questions and insights in thermo-statistical mechanics

Abstract

I introduce a new geometrical approach to thermo--statistical mechanics. Here I highlight the main physical ideas, and how do they translate into geometrical language. I contrast the present approach with previous thermo--statistical--geometrical formalisms, (pseudo-)Riemannian [Weinhold 1975; Ruppeiner 1979] as well as Euclidean [Gilmore 1984; Gross& Votyakov 1999] or Euclidean--looking [Levine 1986]. I point out the relevance of this approach within the contexts of non--extensive statistical thermodynamics [Tsallis 1988, 2000]. I show how the language of Riemannian geometry can be used as a powerful investigation tool,leading to several new and profound questions, and some temptative answers, on thermo--statistical mechanics.