The Annals of Applied Probability

Asymptotic behavior of the finite-size magnetization as a function of the speed of approach to criticality

Richard S. Ellis, Jonathan Machta, and Peter Tak-Hun Otto

Full-text: Open access

Abstract

The main focus of this paper is to determine whether the thermodynamic magnetization is a physically relevant estimator of the finite-size magnetization. This is done by comparing the asymptotic behaviors of these two quantities along parameter sequences converging to either a second-order point or the tricritical point in the mean-field Blume–Capel model. We show that the thermodynamic magnetization and the finite-size magnetization are asymptotic when the parameter α governing the speed at which the sequence approaches criticality is below a certain threshold α0. However, when α exceeds α0, the thermodynamic magnetization converges to 0 much faster than the finite-size magnetization. The asymptotic behavior of the finite-size magnetization is proved via a moderate deviation principle when 0 < α < α0 and via a weak-convergence limit when α > α0. To the best of our knowledge, our results are the first rigorous confirmation of the statistical mechanical theory of finite-size scaling for a mean-field model.

Article information

Source
Ann. Appl. Probab. Volume 20, Number 6 (2010), 2118-2161.

Dates
First available: 19 October 2010

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1287494556

Digital Object Identifier
doi:10.1214/10-AAP679

Zentralblatt MATH identifier
05829436

Mathematical Reviews number (MathSciNet)
MR2759730

Subjects
Primary: 60F10: Large deviations 60F05: Central limit and other weak theorems
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Keywords
Finite-size magnetization thermodynamic magnetization second-order phase transition first-order phase transition tricritical point moderate deviation principle large deviation principle scaling limit Blume–Capel model finite-size scaling

Citation

Ellis, Richard S.; Machta, Jonathan; Otto, Peter Tak-Hun. Asymptotic behavior of the finite-size magnetization as a function of the speed of approach to criticality. The Annals of Applied Probability 20 (2010), no. 6, 2118--2161. doi:10.1214/10-AAP679. http://projecteuclid.org/euclid.aoap/1287494556.


Export citation

References

  • [1] Antoni, M. and Ruffo, S. (1995). Clustering and relaxation in Hamiltonian long-range dynamics. Phys. Rev. E 52 2361–2374.
  • [2] Barber, M. N. (1983). Finite-size scaling. In Phase Transitions and Critical Phenomena (C. Domb and J. Lebowitz, eds.) 8 145–266. Academic Press, London.
  • [3] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [4] Blume, M. (1966). Theory of the first-order magnetic phase change in UO2. Phys. Rev. 141 517–524.
  • [5] Blume, M., Emery, V. J. and Griffiths, R. B. (1971). Ising model for the λ transition and phase separation in He3-He4 mixtures. Phys. Rev. A 4 1071–1077.
  • [6] Capel, H. W. (1966). On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting. Physica 32 966–988.
  • [7] Capel, H. W. (1967). On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting. II. Physica 33 295–331.
  • [8] Capel, H. W. (1967). On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting. III. Physica 37 423–441.
  • [9] Costeniuc, M., Ellis, R. S. and Otto, P. T.-H. (2007). Multiple critical behavior of probabilistic limit theorems in the neighborhood of a tricritical point. J. Stat. Phys. 127 495–552.
  • [10] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics (New York) 38. Springer, New York.
  • [11] Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York.
  • [12] Ellis, R. S. (1985). Entropy, Large Deviations, and Statistical Mechanics. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 271. Springer, New York. Reprinted in Classics in Mathematics in 2006.
  • [13] Ellis, R. S., Machta, J. and Otto, P. T.-H. (2008). Asymptotic behavior of the magnetization near critical and tricritical points via Ginzburg–Landau polynomials. J. Stat. Phys. 133 101–129.
  • [14] Ellis, R. S., Machta, J. and Otto, P. T.-H. (2008). Ginzburg–Landau polynomials and the asymptotic behavior of the magnetization near critical and tricritical points. Unpublished manuscript. Available at http://arxiv.org/abs/0803.0178.
  • [15] Ellis, R. S. and Newman, C. M. (1978). Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Gebiete 44 117–139.
  • [16] Ellis, R. S., Otto, P. T. and Touchette, H. (2005). Analysis of phase transitions in the mean-field Blume–Emery–Griffiths model. Ann. Appl. Probab. 15 2203–2254.
  • [17] Ellis, R. S. and Wang, K. (1990). Limit theorems for the empirical vector of the Curie–Weiss–Potts model. Stochastic Process. Appl. 35 59–79.
  • [18] Fortuin, C. M. and Kasteleyn, P. W. (1972). On the random-cluster model. I. Introduction and relation to other models. Physica 57 536–564.
  • [19] Grimmett, G. (2006). The Random-Cluster Model. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 333. Springer, Berlin.
  • [20] Plischke, M. and Bergersen, B. (2006). Equilibrium Statistical Physics, 3rd ed. World Scientific, Hackensack, NJ.
  • [21] Shiryaev, A. N. (1996). Probability, 2nd ed. Translated by R. P. Boas. Graduate Texts in Mathematics 95. Springer, New York.
  • [22] Stanley, H. E. (1971). Introduction to Phase Transitions and Critical Phenomena. Oxford Univ. Press, New York.