A Geometric Approach to the Concept of Extensivity in Thermodynamics

M. Á. García-Ariza

Published 2018-07-02Version 1

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 functions with extensive differential. These submanifolds are a geometric generalization of the space of states of a closed system in equilibrium.