A Hamilton-Jacobi Formalism for Thermodynamics

TL;DR

This paper develops a Hamilton-Jacobi formalism for classical thermodynamics, linking thermodynamic variables to a dynamical system framework and extending it to black hole thermodynamics in General Relativity.

Contribution

It introduces a Hamilton-Jacobi approach to thermodynamics, providing a new geometric perspective and applying it to black hole thermodynamics.

Findings

Thermodynamics can be formulated as a Hamilton-Jacobi system.

The formalism applies to systems like van der Waals gases and magnets.

Black hole thermodynamics is described using this Hamilton-Jacobi framework.

Abstract

We show that classical thermodynamics has a formulation in terms of Hamilton-Jacobi theory, analogous to mechanics. Even though the thermodynamic variables come in conjugate pairs such as pressure/volume or temperature/entropy, the phase space is odd-dimensional. For a system with n thermodynamic degrees of freedom it is (2n+1)-dimensional. The equations of state of a substance pick out an n-dimensional submanifold. A family of substances whose equations of state depend on n parameters define a hypersurface of co-dimension one. This can be described by the vanishing of a function which plays the role of a Hamiltonian. The ordinary differential equations (characteristic equations) defined by this function describe a dynamical system on the hypersurface. Its orbits can be used to reconstruct the equations of state. The `time' variable associated to this dynamics is related to, but is not identical to, entropy. After developing this formalism on well-grounded systems such as the van der Waals gases and the Curie-Weiss magnets, we derive a Hamilton-Jacobi equation for black hole thermodynamics in General Relativity. The cosmological constant appears as a constant of integration in this picture.