Microcanonical entropy of the spherical model with nearest-neighbour interactions

TL;DR

This paper analytically computes the microcanonical entropy of the spherical model with nearest-neighbour interactions, revealing phase transitions and ensemble equivalence in the thermodynamic limit.

Contribution

It provides the first analytical derivation of the microcanonical entropy for this model, showing ensemble equivalence despite long-range constraints.

Findings

Entropy is concave, indicating ensemble equivalence.

Two phase transition lines identified in the (e,m)-plane.

Microcanonical phase diagram compared to canonical one.

Abstract

For the spherical model with nearest-neighbour interactions, the microcanonical entropy s(e,m) is computed analytically in the thermodynamic limit for all accessible values of the energy e and the magnetization m per spin. The entropy function is found to be concave (albeit not strictly concave), implying that the microcanonical and the canonical ensembles are equivalent, despite the long-range nature of the spherical constraint the spins have to obey. Two transition lines are identified in the (e,m)-plane, separating a paramagnetic phase from a ferromagnetic and an antiferromagnetic one. The resulting microcanonical phase diagram is compared to the more familiar canonical one.