Topology and Phase Transitions I. Preliminary Results

Abstract

In this first paper, we demonstrate a theorem that establishes a first step toward proving a necessary topological condition for the occurrence of first or second order phase transitions: we prove that the topology of certain submanifolds of configuration space must necessarily change at the phase transition point. The theorem applies to smooth, finite-range and confining potentials V bounded below, describing systems confined in finite regions of space with continuously varying coordinates. The relevant configuration space submanifolds are both the level sets {Σ_v := V_N^{-1} (v)}_{v \in R} of the potential function V_N and the configuration space submanifolds enclosed by the Σ_{v} defined by {M_v := V_N^{-1} ((-\infty,v])}_{v \in R}, which are labeled by the potential energy value v, and where N is the number of degrees of freedom. The proof of the theorem proceeds by showing that, under the assumption of diffeomorphicity of the equipotential hypersurfaces {Σ_v}_{v \in R}, as well as of the ${M_v}_{v \in R}, in an arbitrary interval of values for \vb=v/N, the Helmoltz free energy is uniformly convergent in N to its thermodynamic limit, at least within the class of twice differentiable functions, in the corresponding interval of temperature. This preliminary theorem is essential to prove another theorem - in paper II - which makes a stronger statement about the relevance of topology for phase transitions.

35 pages. This is an improved version of math-ph/0305032. Added minor chances: Title, Abstract, Introduction

References in corpus (9)

• Geometric approach to Hamiltonian dynamics and statistical mechanics
• Geometry of dynamics, Lyapunov exponents and phase transitions
• A Theorem on the origin of Phase Transitions
• Topological Signature of First Order Phase Transitions
• Phase transitions and topology changes in configuration space
• Topology and phase transitions: a paradigmatic evidence
• Topological aspects of geometrical signatures of phase transitions
• Topological origin of the phase transition in a mean-field model
• Topology and phase transitions: from an exactly solvable model to a relation between topology and thermodynamics

Cited by in corpus (33)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Phase transitions and configuration space topology
• A Theorem on the origin of Phase Transitions
• Phase Transitions in Small Systems: Microcanonical vs. Canonical Ensembles
• Persistent Homology analysis of Phase Transitions
• Min-type Morse theory for configuration spaces of hard spheres
• Computational topology for configuration spaces of hard disks
• On the mean-field spherical model
• Exploring the energy landscape of XY models
• Phase Transitions from Saddles of the Potential Energy Landscape
• Microcanonical Entropy and Dynamical Measure of Temperature for Systems with Two First Integrals
• Nonanalyticities of the entropy induced by saddle points of the potential energy landscape
• Energy landscape and phase transitions in the self-gravitating ring model
• Topological Approach to Microcanonical Thermodynamics and Phase Transition of Interacting Classical Spins
• Topological approach to phase transitions and inequivalence of statistical ensembles
• Topological Theory of Phase Transitions
• On a microcanonical relation between continuous and discrete spin models
• Phase transitions in simplified models with long-range interactions
• Geometrical aspects in the analysis of microcanonical phase-transitions
• Confinement in non-Abelian lattice gauge theory via persistent homology
• Geometric Microcanonical Thermodynamics for Systems with First Integrals
• Phase transitions and topology in 2+k XY mean-field models
• A simple topological model with continuous phase transition
• Density of states of continuous and discrete spin models: a case study
• Models with symmetry-breaking phase transitions triggered by dumbbell-shaped equipotential surfaces
• The topological hypothesis for discrete spin models
• Configuration spaces of hard spheres
• Quotient Maps and Configuration Spaces of Hard Disks
• Magnetic Reversal Time in Open Long Range Systems
• Through synapses to spatial memory maps: a topological model
• Quantum Mechanics in symmetry language
• A geometric conjecture about phase transitions
• On the origin of Phase Transitions in the absence of Symmetry-Breaking