https://doi.org/10.1007/s100510070183
Violation of ensemble equivalence in the antiferromagnetic mean-field XY model
1
Laboratoire de PhysiqueUMR-CNRS 5672,
ENS Lyon, 46 allée d'Italie, 69364 Lyon Cedex 07, France
2
Dipartimento di Energetica "S. Stecco",
Università di Firenze, via S. Marta, 3, 50139 Firenze, Italy
Received:
1
February
2000
Published online: 15 August 2000
It is well known that long-range interactions pose serious
problems for the formulation of statistical mechanics. We show in
this paper that ensemble equivalence is violated in a simple
mean-field model of N fully coupled classical rotators with
repulsive interaction (antiferromagnetic XY model). While in the
canonical ensemble the rotators are randomly dispersed over
all angles, in the microcanonical ensemble a bi-cluster of
rotators separated by angle π, forms in the low energy limit. We
attribute this behavior to the extreme degeneracy of the ground
state. We obtain empirically an analytical formula for the
probability density function for the angle made by the rotator,
which compares extremely well with numerical data and should become
exact in the zero energy limit. At low energy, in the presence of
the bi-cluster, an extensive amount of energy is located in the
single harmonic mode, with the result that the energy temperature
relation is modified. Although still linear, 
, it has
the slope 
, instead of the canonical value
.
PACS: 05.20.-y – Classical statistical mechanics / 05.20.Gg – Classical ensemble theory / 05.45.-a – Nonlinear dynamics and nonlinear dynamical systems
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2000
