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 σ-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.
Similar content being viewed by others
References
D. Ruelle, Statistical Mechanics: Rigorous Results (Benjamin, Reading, 1969).
T. Dauxois, S. Ruffo, E. Arimondo, and M. Wilkens (Eds.), Dynamics and Thermodynamics of Systems with Long-Range Interactions, Lecture Notes in Physics 602 (Springer, Berlin, 2002).
L. Caiani, L. Casetti, C. Clementi, and M. Pettini, Geometry of dynamics, Lyapunov exponents, and phase transitions, Phys. Rev. Lett. 79:4361–4364 (1997).
L. Casetti, M. Pettini, and E. G. D. Cohen, Geometric approach to Hamiltonian dynamics and statistical mechanics, Phys. Rep. 337:237–341 (2000).
L. Casetti, E. G. D. Cohen, and M. Pettini, Exact result on topology and phase transitions at any finite N, Phys. Rev. E 65:036112 [4 pages] (2002).
L. Angelani, L. Casetti, M. Pettini, G. Ruocco, and F. Zamponi, Topological signature of first-order phase transitions in a mean-field model, Europhys. Lett. 62:775–781 (2003).
L. Casetti, M. Pettini, and E. G. D. Cohen, Phase transitions and topology changes in configuration space, J. Stat. Phys. 111:1091–1123 (2003).
R. Franzosi and M. Pettini, Theorem on the origin of phase transitions, Phys. Rev. Lett. 92:060601 [4 pages] (2004).
P. Grinza and A. Mossa, Topological origin of the phase transition in a model of DNA denaturation, Phys. Rev. Lett. 92:158102 [3 pages] (2004).
L. Angelani, L. Casetti, M. Pettini, G. Ruocco, and F. Zamponi, Topology and phase transitions: From an exactly solvable model to a relation between topology and thermodynamics, Phys. Rev. E 71:036152 [12 pages] (2005).
M. Kastner, Topological approach to phase transitions and inequivalence of statical ensembles, Physica A 359, 447–454 (2006).
R. Franzosi, M. Pettini, and L. Spinelli, Topology and phase transitions I: Theorem on a necessary relation, math-ph/0505057.
F. Baroni and L. Casetti, Topological conditions for discrete symmetry breaking and phase transitions, J. Phys. A: Math. Gen. (to appear).
T. H. Berlin and M. Kac, The spherical model of a ferromagnet, Phys. Rev. 86:821–835 (1952).
G. S. Joyce, Critical properties of the spherical model, in: C. Domb and M. S. Green (Eds.), Phase Transitions and Critical Phenomena, Vol. 2 (Academic Press, London, 1972).
G. S. Joyce, Spherical model with long-range ferromagnetic interactions, Phys. Rev. 146:349–358 (1966).
C. C. Yan and G. H. Wannier, Observations on the spherical model of a ferromagnet, J. Math. Phys. 6:1833–1838 (1965).
A. Sokal and A. O. Starinets, Pathologies of the large-N limit for RP N−1, CP N−1, QP N−1 and mixed isovector/isotensor σ-models, Nucl. Phys. B 601:425–502 (2001).
T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model, Phys. Rev. 87:410–419 (1952).
T. Matsubara and H. Matsuda, A lattice model of liquid helium, I, Prog. Theor. Phys. 16:569–582 (1956).
H. E. Stanley, Spherical model as the limit of infinite spin dimensionality, Phys. Rev. 176:718–722 (1968).
M. Kac and C. J. Thompson, Spherical model and the infinite spin dimensionality limit, Phys. Norvegica 5:163–168 (1971).
M. Weigel and W. Janke, Numerical extension of CFT amplitude universality to three-dimensional systems, Physica A 281, 287–294 (2000).
G. Kohring and R. E. Shrock, Generalized isotropic-nematic phase transitions: critical behavior of 3D P N models, Nucl. Phys. B 285:504–518 (1987).
M. E. Fisher, The nature of critical points, in: W. E. Brittin (Ed.), Lectures in Theoretical Physics, Vol. VII, Part c. (University of Colorado Press, Boulder, 1965)
C. N. Yang and T. D. Lee, Statistical theory of equations of state and phase transitions. I. Theory of condensation, Phys. Rev. 87:404–409 (1952).
H. Behringer, Microcanonical entropy for small magnetizations, J. Phys. A: Math. Gen. 37:1443–1458 (2004).
M. Pleimling, H. Behringer, and A. Hüller, Microcanonical scaling in small systems, Phys. Lett. A 328:432–436 (2004).
M. Kastner, work in progress.
F. Bouchet and J. Barré, Classification of phase transitions and ensemble inequivalence, in systems with long range interactions, J. Stat. Phys. 118:1073–1105 (2005).
R. S. Ellis, K. Haven, and B. Turkington, Large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensembles, J. Stat. Phys. 101:999–1064 (2000).
J. Barré, F. Bouchet, T. Dauxois, and S. Ruffo, Large deviation techniques applied to systems with long-range interactions, J. Stat. Phys. 119:677–713 (2005).
A. C. Ribeiro Teixeira and D. A. Stariolo, Topological hypothesis on phase transitions: The simplest case, Phys. Rev. E 70:016113 [7 pages] (2004).
M. Kastner, Unattainability of a purely topological criterion for the existence of a phase transition for non-confining potentials, Phys. Rev. Lett. 93:150601 [4 pages] (2004).
D. A. Garanin, R. Schilling, and A. Scala, Saddle index properties, singular topology, and its relation to thermodynamic singularities for a φ4 mean-field model, Phys. Rev. E 70:036125 [9 pages] (2004).
A. Andronico, L. Angelani, G. Ruocco, and F. Zamponi, Topological properties of the mean-field φ4 model, Phys. Rev. E 70:041101 [14 pages] (2004).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kastner, M., Schnetz, O. On the Mean-Field Spherical Model. J Stat Phys 122, 1195–1214 (2006). https://doi.org/10.1007/s10955-005-8031-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-005-8031-9