A Hamilton–Jacobi formalism for thermodynamics

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+12n+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.