A Geometric Approach to the Concept of Extensivity in Thermodynamics

TL;DR

This paper develops a geometric framework for understanding extensivity in equilibrium thermodynamics, using smooth manifolds and vector fields to formalize the concept and its properties.

Contribution

It introduces a geometric structure on the manifold of equilibrium states that rigorously captures the notion of extensivity and scaling in thermodynamics.

Findings

Defines extensive differential forms and scaling geometrically.

Reproduces the classical relationship between extensivity and scaling.

Provides a global vector field representing the scaling behavior.

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.