Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 015, 14 pages      arXiv:1807.00873      https://doi.org/10.3842/SIGMA.2019.015

A Geometric Approach to the Concept of Extensivity in Thermodynamics

Miguel Ángel García-Ariza
Instituto de Ciencias, Benemérita Universidad Autónoma de Puebla, 72750, Puebla, Pue., Mexico

Received May 24, 2018, in final form February 22, 2019; Published online March 02, 2019

Abstract
This paper presents a rigorous treatment of the concept of extensivity in equilibrium thermodynamics from a geometric point of view. This is achieved by endowing the manifold of equilibrium states of a system with a smooth atlas that is compatible with the pseudogroup of transformations on a vector space that preserve the radial vector field. The resulting geometric structure allows for accurate definitions of extensive differential forms and scaling, and the well-known relationship between both is reproduced. This structure is represented by a global vector field that is locally written as a radial one. The submanifolds that are transversal to it are embedded, and locally defined by extensive functions.

Key words: homogeneous functions; extensive variables; equilibrium thermodynamics.

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