Legendre symmetry and first order phase transitions of homogeneous systems

TL;DR

This paper characterizes first order phase transitions as symmetry-breaking equilibrium processes in thermodynamic phase space, introduces a generalized Hamiltonian framework for thermodynamics, and discusses conditions for equilibrium in systems with homogeneous potentials.

Contribution

It provides a novel contact Hamiltonian formulation for thermodynamic processes and clarifies the role of potential homogeneity in phase transition equilibrium conditions.

Findings

First order phase transitions break Legendre symmetry.

Equilibrium corresponds to the zeroth level of homogeneous potentials.

Homogeneity of order one is necessary for equilibrium in generalized thermodynamics.

Abstract

In this work we give a characterisation of first order phase transitions as equilibrium processes on the thermodynamic phase space for which the Legendre symmetry is broken. Furthermore, we consider generalised theories of thermodynamics, where the potential is a homogeneous function of any order $\beta$ and we propose a (contact) Hamiltonian formulation of equilibrium processes. Indeed we prove that equilibrium corresponds to the zeroth levels of such function. Using these results we infer that the description in equilibrium of first order phase transitions is possible only when the potential is a homogeneous function of order one, unless a generalised Zeroth Law is postulated in order to allow for equilibrium between sub-parts of the system at different values of the intensive quantities. Finally, we show the example of the Tolman-Ehrenfest effect.