Contact Symmetries and Hamiltonian Thermodynamics

TL;DR

This paper explores the application of contact and metric geometry to thermodynamics, analyzing gauge symmetries, phase transitions, and the geometric nature of entropy production and fluctuations.

Contribution

It provides a novel geometric framework for understanding thermodynamic processes, phase transitions, and entropy production using contact Hamiltonian dynamics.

Findings

Legendre symmetry breaking characterizes first order phase transitions.

Thermodynamic process length measures entropy production.

Contact Hamiltonian dynamics models thermodynamic processes and fluctuations.

Abstract

It has been shown that contact geometry is the proper framework underlying classical thermodynamics and that thermodynamic fluctuations are captured by an additional metric structure related to Fisher's Information Matrix. In this work we analyze several unaddressed aspects about the application of contact and metric geometry to thermodynamics. We consider here the Thermodynamic Phase Space and start by investigating the role of gauge transformations and Legendre symmetries for metric contact manifolds and their significance in thermodynamics. Then we present a novel mathematical characterization of first order phase transitions as equilibrium processes on the Thermodynamic Phase Space for which the Legendre symmetry is broken. Moreover, we use contact Hamiltonian dynamics to represent thermodynamic processes in a way that resembles the classical Hamiltonian formulation of conservative mechanics and we show that the relevant Hamiltonian coincides with the irreversible entropy production along thermodynamic processes. Therefore, we use such property to give a geometric definition of thermodynamically admissible fluctuations according to the Second Law of thermodynamics. Finally, we show that the length of a curve describing a thermodynamic process measures its entropy production.