The mean-field phi4-model: entropy, analyticity, and configuration space topology

Abstract

A large deviation technique is applied to the mean-field phi4-model, providing an exact expression for the configurational entropy s(v,m) as a function of the potential energy v and the magnetization m. Although a continuous phase transition occurs at some critical energy v_c, the entropy is found to be a real analytic function in both arguments, and it is only the maximization over m which gives rise to a nonanalyticity in s(v)=sup_m s(v,m). This mechanism of nonanalyticity-generation by maximization over one variable of a real analytic function is restricted to systems with long-range interactions and has--for continuous phase transitions--the generic occurrence of classical critical exponents as an immediate consequence. Furthermore, this mechanism can provide an explanation why, contradictory to the so-called topological hypothesis, the phase transition in the mean-field phi4-model need not be accompanied by a topology change in the family of constant-energy submanifolds.

10 pages, 5 figures

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Cited by in corpus (21)

• The large deviation approach to statistical mechanics
• Statistical mechanics and dynamics of solvable models with long-range interactions
• Thermodynamics and dynamics of systems with long-range interactions
• Phase transitions and configuration space topology
• Phase space gaps and ergodicity breaking in systems with long range interactions
• Negative magnetic susceptibility and nonequivalent ensembles for the mean-field ${\varphi }^4$ spin model
• Microcanonical solution of the mean-field ${\varphi }^4$ model: comparison with time averages at finite size
• Energy Landscape of the Finite-Size Mean-field 3-Spin Spherical Model
• Phase Transitions from Saddles of the Potential Energy Landscape
• Exploring the energy landscape of XY models
• Nonanalyticities of the entropy induced by saddle points of the potential energy landscape
• Microcanonical phase diagrams of short-range ferromagnets
• Relaxation dynamics of stochastic long-range interacting systems
• Phase transitions in simplified models with long-range interactions
• Methods for calculating nonconcave entropies
• Topology of configuration space of the mean-field phi^4 model by Morse theory
• Models with symmetry-breaking phase transitions triggered by dumbbell-shaped equipotential surfaces
• When topology triggers a phase transition
• Phase transitions triggered by dumbbell equipotential hypersurfaces
• Collective coordinate descriptions of a kink in a driven-damped ${\varphi }^4$ model
• Simplified energy landscape of the ${\varphi }^4$ model and the phase transition