Topology and phase transitions: from an exactly solvable model to a relation between topology and thermodynamics

Abstract

The elsewhere surmised topological origin of phase transitions is given here new important evidence through the analytic study of an exactly solvable model for which both topology and thermodynamics are worked out. The model is a mean-field one with a k-body interaction. It undergoes a second order phase transition for k=2 and a first order one for k>2. This opens a completely new perspective for the understanding of the deep origin of first and second order phase transitions, respectively. In particular, a remarkable theoretical result consists of a new mathematical characterization of first order transitions. Moreover, we show that a "reduced" configuration space can be defined in terms of collective variables, such that the correspondence between phase transitions and topology changes becomes one-to-one, for this model. Finally, an unusual relationship is worked out between the microscopic description of a classical N-body system and its macroscopic thermodynamic behaviour. This consists of a functional dependence of thermodynamic entropy upon the Morse indexes of the critical points (saddles) of the constant energy hypersurfaces of the microscopic 2N-dimensional phase space. Thus phase space (and configuration space) topology is directly related to thermodynamics.

References in corpus (20)

• Geometric approach to Hamiltonian dynamics and statistical mechanics
• Saddles in the energy landscape probed by supercooled liquids
• 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
• Quasi-saddles as relevant points of the potential energy surface in the dynamics of supercooled liquids
• Topology and phase transitions: a paradigmatic evidence
• General features of the energy landscape in Lennard-Jones like model liquids
• Topological aspects of geometrical signatures of phase transitions
• Topological Origin of the Phase Transition in a Model of DNA Denaturation
• Topological origin of the phase transition in a mean-field model
• 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
• The topological hypothesis on phase transitions: the simplest case
• Hamiltonian dynamics and geometry of phase transitions in classical XY models
• Topological properties of the mean field phi^4 model
• Saddle index properties, singular topology, and its relation to thermodynamical singularities for a phi^4 mean field model
• Dynamics and geometric properties of the k-Trigonometric model
• Saddles and dynamics in a solvable mean-field model

Cited by in corpus (28)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Phase transitions and configuration space topology
• Phase Transitions in Small Systems: Microcanonical vs. Canonical Ensembles
• Persistent Homology analysis of Phase Transitions
• Partial equivalence of statistical ensembles and kinetic energy
• The mean-field phi4-model: entropy, analyticity, and configuration space topology
• Topology and Phase Transitions II. Theorem on a necessary relation
• Topology and Phase Transitions I. Preliminary Results
• Min-type Morse theory for configuration spaces of hard spheres
• Computational topology for configuration spaces of hard disks
• Non-analytic microscopic phase transitions and temperature oscillations in the microcanonical ensemble: An exactly solvable 1d-model for evaporation
• How does geometry affect quantum gases?
• Phase transitions induced by saddle points of vanishing curvature
• On the mean-field spherical model
• Quantum phase transitions without thermodynamic limits
• Exploring the energy landscape of XY models
• Nonanalyticities of the entropy induced by saddle points of the potential energy landscape
• Fermions on a torus knot
• 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
• Topological conditions for discrete symmetry breaking and phase transitions
• The Geometric Theory of Phase Transitions
• Mode-locking transitions in nano-structured weakly disordered lasers
• Phase transitions and topology in 2+k XY mean-field models
• The topological hypothesis for discrete spin models
• Quantum gases on a torus
• On the origin of Phase Transitions in the absence of Symmetry-Breaking