Abstract
We investigate the relationship between the mixing times of the Glauber dynamics of a statistical mechanical system with its thermodynamic equilibrium structure. For this we consider the mean-field Blume-Capel model, one of the simplest statistical mechanical models that exhibits the following intricate phase transition structure: within a two-dimensional parameter space there exists a curve at which the model undergoes a second-order, continuous phase transition, a curve where the model undergoes a first-order, discontinuous phase transition, and a tricritical point which separates the two curves. We determine the interface between the regions of slow and rapid mixing. In order to completely determine the region of rapid mixing, we employ a novel extension of the path coupling method, successfully proving rapid mixing even in the absence of contraction between neighboring states.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blume, M.: Theory of the first-order magnetic phase change in UO2. Phys. Rev. 141, 517–524 (1966)
Blume, M., Emery, V.J., Griffiths, R.B.: Ising model for the λ transition and phase separation in He3–He4 mixtures. Phys. Rev. A 4, 1071–1077 (1971)
Bubley, R., Dyer, M.: Path coupling: a technique for proving rapid mixing in Markov chains. In: Proceedings of the 38th Annual Symposium on Foundations of Computer Science, pp. 223–231 (1997)
Brémaud, P.: Markov Chains, Gibbs Fields, Monte Carlo Simulation, and Queues. Texts in Applied Mathematics, vol. 31. Springer, New York (1999)
Capel, H.W.: On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting. Physica 32, 966–988 (1966)
Capel, H.W.: On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting II. Physica 33, 295–331 (1967)
Capel, H.W.: On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting III. Physica 37, 423–441 (1967)
Costeniuc, M., Ellis, R.S., Touchette, H.: Complete analysis of phase transitions and ensemble equivalence for the Curie-Weiss-Potts model. J. Math. Phys. 46, 063301 (2005)
Ding, J., Lubetzky, E., Peres, Y.: The mixing time evolution of Glauber dynamics for the mean-field Ising model. Commun. Math. Phys. 289, 725–764 (2009)
Ellis, R.S.: Entropy, Large Deviations and Statistical Mechanics. Springer, New York (1985). Reprinted in 2006 in Classics in Mathematics
Ellis, R.S., Haven, K., Turkington, B.: Large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensembles. J. Stat. Phys. 101, 999–1064 (2000)
Ellis, R.S., Otto, P.T., Touchette, H.: Analysis of phase transitions in the mean-field Blume-Emery-Griffiths model. Ann. Appl. Probab. 15, 2203–2254 (2005)
Ellis, R.S., Machta, J., Otto, P.T.: Asymptotic behavior of the magnetization near critical and tricritical points via Ginzburg-Landau polynomials. J. Stat. Phys. 133, 101–129 (2008)
Ellis, R.S., Machta, J., Otto, P.T.: Asymptotic behavior of the finite-size magnetization as a function of the speed of approach to criticality. Ann. Appl. Probab. 20, 2118–2161 (2010)
Levin, D., Peres, Y., Wilmer, E.: Markov Chains and Mixing Times. Am. Math. Soc., Providence (2009)
Levin, D., Luczak, M., Peres, Y.: Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability. Probab. Theory Relat. Fields 146 (2010)
Wu, F.Y.: The Potts model. Rev. Mod. Phys. 54 235–268 (1982)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kovchegov, Y., Otto, P.T. & Titus, M. Mixing Times for the Mean-Field Blume-Capel Model via Aggregate Path Coupling. J Stat Phys 144, 1009 (2011). https://doi.org/10.1007/s10955-011-0286-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10955-011-0286-8