## The Annals of Applied Probability

### Analysis of phase transitions in the mean-field Blume–Emery–Griffiths model

#### Abstract

In this paper we give a complete analysis of the phase transitions in the mean-field Blume–Emery–Griffiths lattice-spin model with respect to the canonical ensemble, showing both a second-order, continuous phase transition and a first-order, discontinuous phase transition for appropriate values of the thermodynamic parameters that define the model. These phase transitions are analyzed both in terms of the empirical measure and the spin per site by studying bifurcation phenomena of the corresponding sets of canonical equilibrium macrostates, which are defined via large deviation principles. Analogous phase transitions with respect to the microcanonical ensemble are also studied via a combination of rigorous analysis and numerical calculations. Finally, probabilistic limit theorems for appropriately scaled values of the total spin are proved with respect to the canonical ensemble. These limit theorems include both central-limit-type theorems, when the thermodynamic parameters are not equal to critical values, and noncentral-limit-type theorems, when these parameters equal critical values.

#### Article information

Source
Ann. Appl. Probab. Volume 15, Number 3 (2005), 2203-2254.

Dates
First available in Project Euclid: 15 July 2005

http://projecteuclid.org/euclid.aoap/1121433782

Digital Object Identifier
doi:10.1214/105051605000000421

Mathematical Reviews number (MathSciNet)
MR2152658

Zentralblatt MATH identifier
1113.82017

#### Citation

Ellis, Richard S.; Otto, Peter T.; Touchette, Hugo. Analysis of phase transitions in the mean-field Blume–Emery–Griffiths model. Ann. Appl. Probab. 15 (2005), no. 3, 2203--2254. doi:10.1214/105051605000000421. http://projecteuclid.org/euclid.aoap/1121433782.

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