Topological approach to phase transitions and inequivalence of statistical ensembles

Abstract

The relation between thermodynamic phase transitions in classical systems and topology changes in their state space is discussed for systems in which equivalence of statistical ensembles does not hold. As an example, the spherical model with mean field-type interactions is considered. Exact results for microcanonical and canonical quantities are compared with topological properties of a certain family of submanifolds of the state space. Due to the observed ensemble inequivalence, a close relation is expected to exist only between the topological approach and one of the statistical ensembles. It is found that the observed topology changes can be interpreted meaningfully when compared to microcanonical quantities.

arXiv comment9 pages, 1 figure

References in corpus (14)

• Geometric approach to Hamiltonian dynamics and statistical mechanics
• Geometry of dynamics, Lyapunov exponents and phase transitions
• Large deviation techniques applied to systems with long-range interactions
• 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 II. Theorem on a necessary relation
• Topology and Phase Transitions I. Preliminary Results
• Topological Origin of the Phase Transition in a Model of DNA Denaturation
• Topology and phase transitions: from an exactly solvable model to a relation between topology and thermodynamics
• Unattainability of a purely topological criterion for the existence of a phase transition for non-confining potentials
• Phase transitions as topology changes in configuration space: an exact result
• On the mean-field spherical model
• The topological hypothesis on phase transitions: the simplest case

Cited by in corpus (13)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Phase transitions and configuration space topology
• Phase transitions detached from stationary points of the energy landscape
• The mean-field phi4-model: entropy, analyticity, and configuration space topology
• Continuous phase transitions with a convex dip in the microcanonical entropy
• Energy landscape analysis of the two-dimensional nearest-neighbor ϕ^4 model
• On the mean-field spherical model
• Quantum phase transitions without thermodynamic limits
• Exploring the energy landscape of XY models
• Nonanalyticities of entropy functions of finite and infinite systems
• Topological Theory of Phase Transitions
• Microcanonical entropy of the spherical model with nearest-neighbour interactions
• Density of states of the $XY$ model: an energy landscape approach