Thermodynamic systems as extremal hypersurfaces

TL;DR

This paper models thermodynamic systems as extremal hypersurfaces using geometrothermodynamics, applying variational principles and Nambu-Goto-like actions to describe equilibrium states and processes.

Contribution

It introduces a Legendre invariant geometric framework for thermodynamics, deriving new solutions and interpreting thermodynamic processes as extremal hypersurfaces.

Findings

Geodesic curves represent quasi-static processes.

Derived a Legendre invariant metric for ideal and van der Waals gases.

Obtained new solutions for thermodynamic systems.

Abstract

We apply variational principles in the context of geometrothermodynamics. The thermodynamic phase space ${\cal T}$ and the space of equilibrium states ${\cal E}$ turn out to be described by Riemannian metrics which are invariant with respect to Legendre transformations and satisfy the differential equations following from the variation of a Nambu-Goto-like action. This implies that the volume element of ${\cal E}$ is an extremal and that ${\cal E}$ and ${\cal T}$ are related by an embedding harmonic map. We explore the physical meaning of geodesic curves in ${\cal E}$ as describing quasi-static processes that connect different equilibrium states. We present a Legendre invariant metric which is flat (curved) in the case of an ideal (van der Waals) gas and satisfies Nambu-Goto equations. The method is used to derive some new solutions which could represent particular thermodynamic systems.