When topology triggers a phase transition

Abstract

Two mathematical mechanisms, responsible for the generation of a thermodynamic singularity, are individuated. For a class of short-range, confining potentials, a topology change in some family of configuration space submanifolds is the only possible such mechanism. Two examples of systems in which the phase transition is not accompanied by a such topology change are discussed. The first one is a model with long-range interactions, namely the mean-field phi^4-model, the second example is a one-dimensional system with a non-confining potential energy function. For both these systems, the thermodynamic singularity is generated by a maximization over one variable (or one discrete index) of a smooth function, although the context in which the maximization occurs is very different.

Talk given at the Next-SigmaPhi conference in Kolymbari, Crete, Greece, August 13-18, 2005

References in corpus (7)

• Geometric approach to Hamiltonian dynamics and statistical mechanics
• A Theorem on the origin of Phase Transitions
• Phase transitions and topology changes in configuration space
• The mean-field phi4-model: entropy, analyticity, and configuration space topology
• Unattainability of a purely topological criterion for the existence of a phase transition for non-confining potentials
• Saddle index properties, singular topology, and its relation to thermodynamical singularities for a phi^4 mean field model
• On the relationship between phase transitions and topological changes in one dimensional models

Cited by in corpus (4)

• Phase transitions and configuration space topology
• Phase transitions and topology in 2+k XY mean-field models
• The topological hypothesis for discrete spin models
• Simplified energy landscape of the ${\varphi }^4$ model and the phase transition