Abstract
Finite squareL×L Ising lattices with ferromagnetic nearest neighbor interaction are simulated using the Swendsen-Wang cluster algorithm. Both thermal properties (internal energyU, specific heatC, magnetization 〈|M|〉, susceptibilityχ) and percolation cluster properties relating to the “physical clusters,” namely the Fortuin-Kasteleyn clusters (percolation probability 〈P ∞〉, percolation susceptibilityχ p, cluster size distributionn l) are evaluated, paying particular attention to finite-size effects. It is shown that thermal properties can be expressed entirely in terms of cluster properties, 〈P ∞〉 being identical to 〈|M|〉 in the thermodynamic limit, while finite-size corrections differ. In contrast,χ p differs fromχ even in the thermodynamic limit, since a fluctuation in the size of the percolating net contributes toχ, but not toχ p. NearT c the cluster size distribution has the scaling properties as hypothesized by earlier phenomenological theories. We also present a generalization of the Swendsen-Wang algorithm allowing one to cross over continuously to the Glauber dynamics.
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References
M. E. Fisher,Physics 3:267 (1967).
C. S. Kiang and D. Stauffer,Z. Phys. 235:130 (1970); A. Eggington, C. S. Kiang, D. Stauffer, and G. H. Walker,Phys. Rev. Lett. 26:820 (1971); D. Stauffer, C. S. Kiang, and G. H. Walker,J. Stat. Phys. 3:323 (1971).
K. Binder,Ann. Phys. 98:390 (1976).
A. Coniglio and W. Klein,J. Phys. A 13:2775 (1980).
A. Coniglio, C. R. Nappi, F. Peruggi, and L. Russo,Commun. Math. Phys. 51:315 (1976).
C.-K. Hu,Phys. Rev. B 29:5103 (1984).
D. Stauffer,Introduction to Percolation Theory (Taylor and Francis, London, 1985).
B. M. McCoy and T. T. Wu,The Two-Dimensional Ising Model (Harvard University Press, Cambridge, Massachusetts, 1968).
E. Stoll, K. Binder, and T. Schneider,Phys. Rev. B 6:2777 (1972).
K. Binder and D. Stauffer,J. Stat. Phys. 6:49 (1972).
H. Müller-Krumbhaar,Phys. Lett. 48A:459 (1974).
D. W. Heermann and D. Stauffer,Z. Physik B 44:339 (1981).
H. Müller-Krumbhaar and E. P. Stoll,J. Chem. Phys. 65:4294 (1974).
A. Marro and J. Toral,Physica 122A:563 (1983).
M. Nauenberg and J. Cambier, inFractals in Physics, L. Pietronero and E. Tosatti, eds. (North-Holland, Amsterdam, 1986), p. 421.
M. F. Sykes and D. S. Gaunt,J. Phys. A 9:2131 (1976).
K. Binder, D. Stauffer, and H. Müller-Krumbhaar,Phys. Rev. B 12:5261 (1975); R. Kretschmer, K. Binder, and D. Stauffer,J. Stat. Phys. 15:267 (1976).
K. Binder and D. W. Heermann, inScaling Phenomena in Disordered Systems, R. Pynn and A. Skjeltorp, eds. (Plenum Press, New York, 1985), p. 227.
K. Binder (ed.),Monte Carlo Methods in Statistical Physics, 2nd ed. (Springer, Berlin, 1986).
D. W. Heermann,Computer Simulation Methods in Theoretical Physics (Springer, Berlin, 1976).
K. Binder (ed.),Applications of the Monte Carlo Method in Statistical Physics, 2nd ed. (Springer, Berlin, 1987).
K. Binder and D. W. Heermann,Monte Carlo Simulation in Statistical Physics: An Introduction (Springer, Berlin, 1988).
R. H. Swendsen and J. S. Wang,Phys. Rev. Lett. 58:86 (1987).
P. W. Kasteleyn and C. M. Fortuin,J. Phys. Soc. Japan 26(Suppl.):11 (1969); C. M. Fortuin and P. W. Kasteleyn,Physica 57:536 (1972); C. M. Fortuin,Physica 58:393,59:545 (1972).
M. D'Onorio de Meo, Diplomarbeit, Universität Mainz (1988), unpublished.
J. Hoshen and R. Kopelman,Phys. Rev. B 27:3438 (1976).
S. Wansleben,Comput. Phys. Commun. 43:9 (1987).
N. Ito and Y. Kanada,Supercomputer 25:31 (1988).
A. H. Burkitt and D. W. Heermann,Comput. Phys. Commun. 54:201 (1989).
K. Binder,Z. Physik B 43:119 (1981).
A. Margolina and H. J. Herrmann,Phys. Lett. 104A:295 (1984).
M. E. Fisher, inCritical Phenomena, M. S. Green, ed. (Academic Press, New York, 1971), p. 1.
M. E. Fisher and M. N. Barber,Phys. Rev. Lett. 28:1518 (1972).
M. N. Barber, inPhase Transitions and Critical Phenomena, Vol. 8, C. Domb and J. L. Lebowitz, eds. (Academic Press, New York, 1983), p. 145.
K. Binder,Ferroelectrics 43:73 (1987).
A. Milchev, K. Binder, and D. W. Heermann,Z. Physik B 63:527 (1986).
R. Gissler, D. W. Heermann, and A. H. Burkitt, work in progress.
R. G. Edwards and A. D. Sokal,Phys. Rev. D 38:2009 (1988).
N. Ito, M. Taiji, and M. Suzuki,J. Phys. (Paris) C8(Suppl. 12):1397 (1988).
D. W. Heermann and A. H. Burkitt,Physica A 162:210 (1990).
D. W. Heermann, A. H. Burkitt, and J. Kertesz, to be published.
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De Meo, M.D., Heermann, D.W. & Binder, K. Monte Carlo study of the ising model phase transition in terms of the percolation transition of “physical clusters”. J Stat Phys 60, 585–618 (1990). https://doi.org/10.1007/BF01025984
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DOI: https://doi.org/10.1007/BF01025984