Journal of the Korean Data and Information Science Society

The Korean Data and Information Science Society (한국데이터정보과학회)
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Simulating phase transition phenomena of the unitary cell model

  • Kim, Dong-Hoh (Department of Applied Mathematics, Sejong University)
  • Published : 2009.01.31
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Abstract

Lattice process models are used to explain phase transitions in statistical mechanics, a branch of physics. The Ising model, a specific form of lattice process model, was proposed by Ising in 1925. Since then, variants of the Ising model such as the Potts model and the unitary cell model have been proposed. Like the Ising model, it is believed that the more general models exhibit phase transitions on the critical surface, which is based on the mathematical equation. In statistical sense, phase transitions can be simulated through Markov Chain Monte Carlo (MCMC). We applied Swendsen-Wang algorithm, a block Gibbs algorithm, to a general lattice process models and we simulate phase transition phenomena of the unitary cell model.

Keywords

References

  1. Aguilar, A. and Braun, E. (1991a). Exact solution of a general two-dimensional Ising model: the partition function. Physica A, 170, 643-662. https://doi.org/10.1016/0378-4371(91)90011-Z
  2. Aguilar, A. and Braun, E. (1991b). The specific heat of a general two-dimensional Ising model. Physica A, 178, 551-560. https://doi.org/10.1016/0378-4371(91)90037-D
  3. Baxter, R. J. (1982). Exactly solved models in statistical mechanics, Academic Press, New York.
  4. Begum, M.-N. and Ali, M. M. (2004). Application of Bayesian computational techniques in estimation of posterior distributional properties of lognormal distribution. Journal of the Korean Data & Information Science Society, 15, 227-237.
  5. Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussion). Journal of the Royal Statistical Society, Series B, 36, 192-236.
  6. Besag, J. and Green, P. J. (1993). Spatial statistics and Bayesian computation. Journal of the Royal Statistical Society, Series B, 55, 25-37.
  7. Cressie, N. (1993). Statistics for spatial data, Wiley, New York.
  8. Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721-741. https://doi.org/10.1109/TPAMI.1984.4767596
  9. Kim, D. (2000). Likelihood inference for lattice spatial processes, Ph.D. Thesis, School of Statistics, University of Minnesota.
  10. Lee, K.-E. (2004). Bayesian variable selection in the proportional hazard model. Journal of the Korean Data & Information Science Society, 15, 605-616.
  11. Potts, R. B. (1952). Some generalized order-disorder transformations. Proceedings of the Cambridge Philosophical Society, 48, 106-109.
  12. Swendsen, R. H. and Wang, J. S. (1987). Nonuniversal critical dynamics in monte carlo simulations. Physical Review Letters, 58, 86-88. https://doi.org/10.1103/PhysRevLett.58.86