The Geometric Theory of Phase Transitions

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 ${\varphi }^4$ and Ginzburg-Landau-like models.

References in corpus (35)

• A Dynamical Approach to Temperature
• Geometric approach to Hamiltonian dynamics and statistical mechanics
• Riemannian theory of Hamiltonian chaos and Lyapunov exponents
• Geometry of dynamics, Lyapunov exponents and phase transitions
• Phase transitions and configuration space topology
• A Theorem on the origin of Phase Transitions
• Microcanonical entropy inflection points: Key to systematic understanding of transitions in finite systems
• Phase Transitions in Small Systems: Microcanonical vs. Canonical Ensembles
• A Geometric, Dynamical Approach to Thermodynamics
• The microcanonical thermodynamics of finite systems: The microscopic origin of condensation and phase separations; and the conditions for heat flow from lower to higher temperatures
• Phase transitions detached from stationary points of the energy landscape
• Geometry of dynamics and phase transitions in classical lattice phi^4 theories
• Topology and phase transitions: a paradigmatic evidence
• Classification of Phase Transitions by Microcanonical Inflection-Point Analysis
• Energy landscape analysis of the two-dimensional nearest-neighbor ϕ^4 model
• Topological aspects of geometrical signatures of phase transitions
• Hamiltonian dynamics of the two-dimensional lattice phi^4 model
• Non-analytic microscopic phase transitions and temperature oscillations in the microcanonical ensemble: An exactly solvable 1d-model for evaporation
• Topology and phase transitions: from an exactly solvable model to a relation between topology and thermodynamics
• Phase transitions induced by saddle points of vanishing curvature
• On the mean-field spherical model
• Exploring the energy landscape of XY models
• Nonanalyticities of the entropy induced by saddle points of the potential energy landscape
• Nonanalyticities of entropy functions of finite and infinite systems
• Kinetic energy and microcanonical nonanalyticities in finite and infinite systems
• Exact microcanonical statistical analysis of transition behavior in Ising chains and strips
• Microcanonical entropy for classical systems
• Geometrical aspects in the analysis of microcanonical phase-transitions
• Coherent Riemannian-geometric description of Hamiltonian order and chaos with Jacobi metric
• Hamiltonian chaos and differential geometry of configuration space-time
• Four Lectures on Scalar Curvature
• Phase Transitions in the Two-Dimensional Ising Model from the Microcanonical Perspective
• A microcanonical entropy correcting finite-size effects in small systems
• A Simple Formula for Scalar Curvature of Level Sets in Euclidean Spaces
• Configurational microcanonical statistical mechanics from Riemannian geometry of equipotenital level sets

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