Abstract
Autocorrelation times for thermodynamic quantities atT C are calculated from Monte Carlo simulations of the site-diluted simple cubic Ising model, using the Swendsen-Wang and Wolff cluster algorithms. Our results show that for these algorithms the autocorrelation timesdecrease when reducing the concentration of magnetic sites from 100% down to 40%. This is of crucial importance when estimating static properties of the model, since the variances of these estimators increase with autocorrelation time. The dynamical critical exponents are calculated for both algorithms, observing pronounced finite-size effects in the energy autocorrelation data for the algorithm of Wolff. We conclude that, when applied to the dilute Ising model, cluster algorithms become even more effective than local algorithms, for whichincreasing autocorrelation times are expected.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. H. Swendsen and J. S. Wang, Nonuniversal critical dynamics in Monte Carlo simulations,Phys. Rev. Lett. 58:86–88 (1987).
U. Wolff, Comparison between cluster Monte Carlo algorithms in the Ising model,Phys. Lett. B 228:379–382 (1989).
P. Tamayo, R. C. Brower, and W. Klein, Single-cluster Monte Carlo dynamics for the Ising model,J. Stat. Phys. 58:1083–1094,60:899 (1990).
P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena,Rev. Mod. Phys. 49:435–479 (1977).
M. D'Onorio De Meo, D. W. Heermann, and K. Binder, Monte Carlo studies of the Ising model phase transition in terms of the percolation transition of physical clusters,J. Stat. Phys. 60:585–618 (1990).
W. Klein, T. Ray, and P. Tamayo, Scaling ansatz for Swendsen-Wang dynamics,Phys. Rev. Lett. 62:163–165 (1989).
J. S. Wang, Critical dynamics of the Swendsen-Wang algorithm in the three-dimensional Ising model,Physica A 164:240–244 (1990).
D. W. Heermann and A. N. Burkitt, System size dependence of the autocorrelation time for the Swendsen-Wang Ising model,Physica A 162:210–214 (1990).
T. S. Ray, P. Tamayo, and W. Klein, Mean-field study of the Swendsen-Wang dynamics,Phys. Rev. A 39:5949–5953 (1989).
P. Tamayo and W. Klein, Critical dynamics and global conservation laws,Phys. Rev. Lett. 63:2757–2759 (1989).
U. Wolff, Critical slowing down, in LATTICE '89 Symposium on Lattice Field Theory, Capri, Italy; Bielefeld Preprint BI-TP 89/35 (November 1989).
A. D. Sokal, New numerical algorithms for critical phenomena, inComputer Studies in Condensed Matter Physics (Springer, Berlin, 1988), pp. 6–18.
J. S. Wang and R. H. Swendsen, Cluster Monte Carlo algorithms,Physica A 167:565–579 (1990).
M. B. Priestley,Spectral Analysis and Time Series (Academic Press, London, 1981).
N. Madras and A. D. Sokal, The Pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk,J. Stat. Phys. 50:109–186 (1988).
W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1988).
E. Stanley,Introduction to Phase Transitions and Critical Phenomena (Clarendon Press, Oxford, 1971).
A. Hankey and H. E. Stanley, Systematic application of generalized homogeneous functions to static scaling, dynamic scaling and universality,Phys. Rev. B 6:3515–3542 (1972).
M. N. Barber, Finite-size scaling, inPhase Transitions and Critical Phenomena, Vol. 8 (Academic Press, London, 1983).
S. Wansleben and D. P. Landau, Dynamical critical exponent of the 3D Ising model,J. Appl. Phys. 61:3968–3970 (1987).
S. Wansleben and D. P. Landau, Monte Carlo investigation of critical dynamics in the 3D Ising model,Phys. Rev. B 43:6006–6014 (1991).
B. I. Halperin, Rigorous inequalities for the spin-relaxation function in the kinetic Ising model,Phys. Rev. B 8:4437–4440 (1973).
X. J. Li and A. D. Sokal, Rigorous lower bound on the dynamic critical exponents of the Swendsen-Wang algorithm,Phys. Rev. Lett. 63:827–830 (1989).
S. Kirkpatrick and E. Stoll, A very fast shift-register sequence random number generator,J. Computational Phys. 40:517–526 (1981).
J. C. Le Guillou and J. Zinn-Justin, Critical exponents from field theory,Phys. Rev. B 21:3976–3998 (1980).
J. S. Wang and D. Chowdhury, The critical behaviour of the three-dimensional dilute Ising model: Universality and the Harris criterion,J. Phys. (Paris)50:2905 (1989).
G. Jug, Critical behaviour of disordered spin systems in two and three dimensions,Phys. Rev. B 27:609–612 (1983).
I. O. Mayer, Critical exponents of the dilute Ising model from four-loop expansions,J. Phys. A 22:2815–2823 (1989).
A. Coniglio, Thermal phase transition of the dilutes-state Potts andn-vector models at the percolation threshold,Phys. Rev. Lett. 46:250–253 (1981).
J. Adler, Y. Meir, A. Aharony, and A. B. Harris, Series study of percolation moments in general dimension,Phys. Rev. Lett. 46:250–253 (1981).
R. B. Stinchcombe, Dilute magnetism, inPhase Transitions and Critical Phenomena, Vol. 7 (Academic Press, London, 1983).
H. O. Heuer, Dynamic scaling of disordered Ising systems, Preprint, 25.09.1992.
H. O. Heuer, Critical slowing down in local dynamics simulation,J. Phys. A 25:L567–L573 (1992).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hennecke, M., Heyken, U. Critical dynamics of cluster algorithms in the dilute Ising model. J Stat Phys 72, 829–844 (1993). https://doi.org/10.1007/BF01048034
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01048034