The Geometric Theory of Phase Transitions

TL;DR

This paper introduces a geometric framework for understanding phase transitions in Hamiltonian systems by linking thermodynamic properties to the curvature of energy level sets, providing a new perspective on microscopic and collective behaviors.

Contribution

It develops a geometric theory connecting energy level set properties to phase transitions, offering a novel approach to classify and analyze PTs in finite and infinite systems.

Findings

Establishes an exact equivalence between entropy derivatives and geometric curvature structures.

Shows non-analyticities in entropy derivatives correspond to geometric non-analyticities of energy level sets.

Validates the theory through analysis of $$ and Ginzburg-Landau-like models.

Abstract

We develop a geometric theory of phase transitions (PTs) for Hamiltonian systems in the microcanonical ensemble. This theory allows to reformulate Bachmann's classification of PTs for finite-size systems in terms of geometric properties of the energy level sets (ELSs) associated to the Hamiltonian function. Specifically, by defining the microcanonical entropy as the logarithm of the ELS's volume equipped with a suitable metric tensor, we obtain an exact equivalence between thermodynamics and geometry. In fact, we show that any derivative of entropy with respect to the energy variable can be associated to a specific combination of geometric curvature structures of the ELSs which, in turn, are precise combinations of the potential function derivatives. In this way, we establish a direct connection between the microscopic description provided by the Hamiltonian and the collective behavior which emerges in a PT. Finally, we also analyze the behavior of the ELSs' geometry in the thermodynamic limit, showing that non-analyticities of the energy-derivatives of the entropy are caused by non-analyticities of certain geometric properties of the ELSs around the transition point. Finally, we validate the theory studying the PTs that occur in the $\phi^4$ and Ginzburg-Landau-like models.