Hamilton-Jacobi approach to thermodynamic transformations

TL;DR

This paper develops a Hamilton-Jacobi framework for thermodynamic transformations, modeling the phase space as a contact manifold and demonstrating that the approach effectively describes equilibrium state changes, with applications to ideal gases.

Contribution

It introduces a novel Hamilton-Jacobi formulation for thermodynamics, linking contact geometry with thermodynamic transformations and analyzing the invariance of the dynamical description.

Findings

Hamilton-Jacobi theory describes thermodynamic transformations on equilibrium manifolds.

The principal function choice does not affect the transformation dynamics.

Characteristic curves on the equilibrium space fully characterize thermodynamic evolution.

Abstract

In this note, we formulate and study a Hamilton-Jacobi approach for describing thermodynamic transformations. The thermodynamic phase space assumes the structure of a contact manifold with the points representing equilibrium states being restricted to certain submanifolds of this phase space. We demonstrate that Hamilton-Jacobi theory consistently describes thermodynamic transformations on the space of externally controllable parameters or equivalently, the space of equilibrium states. It turns out that in the Hamilton-Jacobi description, the choice of the principal function is not unique but, the resultant dynamical description for a given transformation remains the same irrespective of this choice. Some examples involving thermodynamic transformations of the ideal gas are discussed where the characteristic curves on the space of equilibrium states completely describe the dynamics. The geometric Hamilton-Jacobi formulation which has emerged recently is also discussed in the context of thermodynamics.