Statistical Description of Magnetic Domains in the Two Dimensional Ising Model
Lukierska-Walasek Krystyna 1, Topolski Krzysztof 2
ul. Z. Szafrana 4a, 65-516 Zielona Góra, Poland
e-mail: klukie@proton.if.uz.zgora.pl
Institute of Mathematics, Wrocław University
Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
e-mail: topolski@math.uni.wroc.pl
Received:
Received: 23 March 2010; accepted: 8 July 2010; published online: 27 September 2010
DOI: 10.12921/cmst.2010.16.02.173-176
OAI: oai:lib.psnc.pl:726
Abstract:
The Zipf-Mandelbrot power law and its connection with the inhomogeneity of the system has been used. We describe the statistical distributions of the domain masses in the Ising model near the phase transition induced by the temperature. For the large domain masses we observe the characteristic irregularities. The statistical distribution near the critical point appears to be of the Pareto type.
Key words:
Ising model, phase transition, the Zipf-Mandelbrot power law
References:
[1] E. Ising, Beitrag zur Theorie des Ferromagnetismus. Z. Physik 31, 253-258 (1925).
[2] D. Stauffer, Introduction to percolation theory. Taylor and Francis, London and Philadelphia (1985).
[3] R. Cont, J.P. Bouchand, Herd behavior and aggregate fluctuations in financial markets. Macroeconomics Dynamics 4, 170-196 (2000).
[4] K. Sznajd-Weron, J. Sznajd, Opinion evolution in closed community. Int. Mod. Phys. C 11, 1157-1165 (2000).
[5] C.M. Fortuin, P.W. Kasteleyn, On the random cluster model. I. Introduction and relation to other models. Physica 57, 536-564 (1972).
[6] R. Swendsen, J. Wang, Non-universal critical dynamics in Monte Carlo simulation. Phys. Rev. Lett. 58, 86-88 (1987).
[7] J.P. Bouchand, More Lévy distributions in physics. In: M.F. Shlesinger, G.M. Zasławsky, V. Frisch (ed.). Proc. Int. Workshop on Lévy Flights and Related Topics in Physics (Nice, France, 27-30 June 1994), Springer, 239-250 (1995).
[8] M. Kratz, S.I. Resnick, The qq-estimator and heavy tails. Stochastic Models 12, 699-724 (1996) and J. Beirlant, Y. Goegebeeur, J. Teugels, J. Segers, Statistics of Extremes, Theory and Applications. Wiley, Chichester, UK (2004).
[9] W. Janke, A.M.J. Schakel, Fractal structure of spin clusters and domain walls in the two-dimensional Ising model. Phys. Rev. E 71, 036703-036710 (2005).
[10] E. Dobierzewska-Mozrzymas, P. Biegański, E. Pieciul, J. Wójcik, Statistical description of systems on the basis of the Mandelbrot law: discontinuous metal films on dielectric substrates. J. Phys. Condens. Matter 11, 5561-5568 (1999).
[11] S. Liberman, F. Brouers, P. Gadenne, Levy’s distributions of local fields intensities in metal-dielectric systems. Physica B 279, 56-58 (2000).
[12] A. Sicilia, J.J. Arenzon, I. Dierking, A.J. Brag, L.F. Cugliandolo, J. Martinez-Perdiguero, I. Alonso, I.C. Pintre, Experimental test of curvature-driven dynamics in the phase ordering in two dimensional liquid crystal. Phys. Rev. Lett. 101, 197801-197804 (2008).
The Zipf-Mandelbrot power law and its connection with the inhomogeneity of the system has been used. We describe the statistical distributions of the domain masses in the Ising model near the phase transition induced by the temperature. For the large domain masses we observe the characteristic irregularities. The statistical distribution near the critical point appears to be of the Pareto type.
Key words:
Ising model, phase transition, the Zipf-Mandelbrot power law
References:
[1] E. Ising, Beitrag zur Theorie des Ferromagnetismus. Z. Physik 31, 253-258 (1925).
[2] D. Stauffer, Introduction to percolation theory. Taylor and Francis, London and Philadelphia (1985).
[3] R. Cont, J.P. Bouchand, Herd behavior and aggregate fluctuations in financial markets. Macroeconomics Dynamics 4, 170-196 (2000).
[4] K. Sznajd-Weron, J. Sznajd, Opinion evolution in closed community. Int. Mod. Phys. C 11, 1157-1165 (2000).
[5] C.M. Fortuin, P.W. Kasteleyn, On the random cluster model. I. Introduction and relation to other models. Physica 57, 536-564 (1972).
[6] R. Swendsen, J. Wang, Non-universal critical dynamics in Monte Carlo simulation. Phys. Rev. Lett. 58, 86-88 (1987).
[7] J.P. Bouchand, More Lévy distributions in physics. In: M.F. Shlesinger, G.M. Zasławsky, V. Frisch (ed.). Proc. Int. Workshop on Lévy Flights and Related Topics in Physics (Nice, France, 27-30 June 1994), Springer, 239-250 (1995).
[8] M. Kratz, S.I. Resnick, The qq-estimator and heavy tails. Stochastic Models 12, 699-724 (1996) and J. Beirlant, Y. Goegebeeur, J. Teugels, J. Segers, Statistics of Extremes, Theory and Applications. Wiley, Chichester, UK (2004).
[9] W. Janke, A.M.J. Schakel, Fractal structure of spin clusters and domain walls in the two-dimensional Ising model. Phys. Rev. E 71, 036703-036710 (2005).
[10] E. Dobierzewska-Mozrzymas, P. Biegański, E. Pieciul, J. Wójcik, Statistical description of systems on the basis of the Mandelbrot law: discontinuous metal films on dielectric substrates. J. Phys. Condens. Matter 11, 5561-5568 (1999).
[11] S. Liberman, F. Brouers, P. Gadenne, Levy’s distributions of local fields intensities in metal-dielectric systems. Physica B 279, 56-58 (2000).
[12] A. Sicilia, J.J. Arenzon, I. Dierking, A.J. Brag, L.F. Cugliandolo, J. Martinez-Perdiguero, I. Alonso, I.C. Pintre, Experimental test of curvature-driven dynamics in the phase ordering in two dimensional liquid crystal. Phys. Rev. Lett. 101, 197801-197804 (2008).
