On the mean-field spherical model

Abstract

Exact solutions are obtained for the mean-field spherical model, with or without an external magnetic field, for any finite or infinite number N of degrees of freedom, both in the microcanonical and in the canonical ensemble. The canonical result allows for an exact discussion of the loci of the Fisher zeros of the canonical partition function. The microcanonical entropy is found to be nonanalytic for arbitrary finite N. The mean-field spherical model of finite size N is shown to be equivalent to a mixed isovector/isotensor sigma-model on a lattice of two sites. Partial equivalence of statistical ensembles is observed for the mean-field spherical model in the thermodynamic limit. A discussion of the topology of certain state space submanifolds yields insights into the relation of these topological quantities to the thermodynamic behavior of the system in the presence of ensemble nonequivalence.

21 pages, 5 figures

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Cited by in corpus (24)

• The large deviation approach to statistical mechanics
• 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
• Thermodynamic limits of dynamic cooling
• Partial equivalence of statistical ensembles and kinetic energy
• Non-analytic microscopic phase transitions and temperature oscillations in the microcanonical ensemble: An exactly solvable 1d-model for evaporation
• Quantum phase transitions without thermodynamic limits
• Phase Transitions from Saddles of the Potential Energy Landscape
• Nonanalyticities of entropy functions of finite and infinite systems
• Nonanalyticities of the entropy induced by saddle points of the potential energy landscape
• Extended gaussian ensemble solution and tricritical points of a system with long-range interactions
• Kinetic energy and microcanonical nonanalyticities in finite and infinite systems
• Topological approach to phase transitions and inequivalence of statistical ensembles
• Topological Theory of Phase Transitions
• Effects of energy extensivity on the quantum phases of long-range interacting systems
• Solvable model of a self-gravitating system
• The Geometric Theory of Phase Transitions
• Microcanonical entropy of the spherical model with nearest-neighbour interactions
• Necessary and sufficient conditions for ${\mathbb{Z}}_2$-symmetry-breaking phase transitions
• An Introduction to Large Deviations and Equilibrium Statistical Mechanics for Turbulent Flows
• Estimation of local microcanonical averages in two lattice mean-field models using coupling techniques
• Infinite volume Gibbs states and metastates of the random field mean-field spherical model
• Supersymmetric Quantum Spherical Spins