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The invaded cluster algorithm

Yongsoo Choi

Published Jan 1, 1997• 1 citation
OpenAlex

Abstract

A new class of Monte Carlo methods is introduced. The approach is devised to be especially useful for studying critical phenomena and phase transitions. The method can sample the critical point without a priori knowledge of the critical temperature. The method is first verified for Ising and Potts models. It is shown that the method also efficiently simulates the coexistence region near first-order transitions. The method is applied to dilute Ising models for studying disorder effects on critical phenomena. The dynamic properties of the method are investigated numerically. Finally a parallel version of the method is described and high-precision numerical results for the critical point of three-dimensional Ising models are presented.

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Invaded cluster algorithm for Potts models

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The invaded cluster algorithm, a method for simulating phase transitions, is described in detail. Theoretical, albeit nonrigorous, justification of the method is presented and the algorithm is applied to Potts models in two and three dimensions. The algorithm is shown to be useful for both first-order and continuous transitions and evidently provides an efficient way to distinguish between these possibilities. The dynamic properties of the invaded cluster algorithm are studied. Numerical evidence suggests that the algorithm has no critical slowing for Ising models. \textcopyright{} 1996 The American Physical Society.

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Jonathan L. Machta, Y. S. Choi, A. Lücke +2 more

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Cluster Monte Carlo and network flow algorithms for critical phenomena

Ilija Dukovski

Phase transitions and critical phenomena in several spin models are studied. These models include a two and three dimensional XY model and Ising models in structured and random field. The methods of study are cluster Monte Carlo simulations and network flow optimization algorithms. The XY model and a related soft spin model belonging to the O(2) universality class are studied using a modified version of the Invaded Cluster algorithm. The static critical properties of the models and the dynamic characteristics of the algorithm are studied. The critical exponent η for the O(2) universality class in two and three dimensions is measured. The Invaded Cluster algorithm for the XY model does not show any critical slowing for global variables. Benzene adsorption in a nanoporous solid is modeled with an Ising model with both internal and external magnetic fields. The first order phase transition and the critical properties of the system are studied. The system is simulated using the two replica cluster algorithm. The critical temperature of the system is found to decrease with increasing difference of the values of the internal fields. It is found that the system undergoes a phase transition for a relatively wide, experimentally obtainable, region of values of its parameters. The random field Ising model in three dimensions is studied at zero temperature. The nature of randomness induced phase transition is studied using a numerical network flow optimization method. It is found that the latent heat at the phase transition vanishes in the limit of infinite system size which can be taken as a proof of a second order phase transition.

•Jan 1, 2001•0 citations

Parallel-tempering cluster algorithm for computer simulations of critical phenomena

Elmar Bittner, Wolfhard Janke E

In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature range around the critical point. By combining the parallel-tempering algorithm with cluster updates and an adaptive routine to find the temperature window of interest, we introduce a flexible and powerful method for systematic investigations of critical phenomena. As a result, we gain one to two orders of magnitude in the performance for two- and three-dimensional Ising models in comparison with the recently proposed Wang-Landau recursion for cluster algorithms based on the multibondic algorithm, which is already a great improvement over the standard multicanonical variant.

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New Monte Carlo technique for studying phase transitions

Alan M. Ferrenberg, Robert H. Swendsen

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Physical Review Letters•Dec 5, 1988•2,675 citations

Monte Carlo simulation of the critical behavior near the percolation threshold with invaded cluster algorithm

Maria Shlyakhtich, Pavel Vladimirovich Prudnikov

Abstract We investigated the critical behavior of the disordered three-dimensional Ising model with an impurity concentration close to the threshold of impurity percolation p = pc = 0.688 313 500 on high-performance computing systems. The system was updated using Metropolis algorithm and invaded cluster algorithm from initial configurations. The values of dynamic critical exponent z and static critical exponents β and v were obtained in this paper.

Journal of Physics Conference Series•Jan 1, 2021•1 citation
A new class of Monte Carlo methods is introduced. The approach is devised to be especially useful for studying critical phenomena and phase transitions. The method can sample the critical point without a priori knowledge of the critical temperature. The method is first verified for Ising and Potts models. It is shown that the method also efficiently simulates the coexistence region near first-order transitions. The method is applied to dilute Ising models for studying disorder effects on critical phenomena. The dynamic properties of the method are investigated numerically. Finally a parallel version of the method is described and high-precision numerical results for the critical point of three-dimensional Ising models are presented.
The invaded cluster algorithm, a method for simulating phase transitions, is described in detail. Theoretical, albeit nonrigorous, justification of the method is presented and the algorithm is applied to Potts models in two and three dimensions. The algorithm is shown to be useful for both first-order and continuous transitions and evidently provides an efficient way to distinguish between these possibilities. The dynamic properties of the invaded cluster algorithm are studied. Numerical evidence suggests that the algorithm has no critical slowing for Ising models. \textcopyright{} 1996 The American Physical Society.
Phase transitions and critical phenomena in several spin models are studied. These models include a two and three dimensional XY model and Ising models in structured and random field. The methods of study are cluster Monte Carlo simulations and network flow optimization algorithms. The XY model and a related soft spin model belonging to the O(2) universality class are studied using a modified version of the Invaded Cluster algorithm. The static critical properties of the models and the dynamic characteristics of the algorithm are studied. The critical exponent η for the O(2) universality class in two and three dimensions is measured. The Invaded Cluster algorithm for the XY model does not show any critical slowing for global variables. Benzene adsorption in a nanoporous solid is modeled with an Ising model with both internal and external magnetic fields. The first order phase transition and the critical properties of the system are studied. The system is simulated using the two replica cluster algorithm. The critical temperature of the system is found to decrease with increasing difference of the values of the internal fields. It is found that the system undergoes a phase transition for a relatively wide, experimentally obtainable, region of values of its parameters. The random field Ising model in three dimensions is studied at zero temperature. The nature of randomness induced phase transition is studied using a numerical network flow optimization method. It is found that the latent heat at the phase transition vanishes in the limit of infinite system size which can be taken as a proof of a second order phase transition.
In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature range around the critical point. By combining the parallel-tempering algorithm with cluster updates and an adaptive routine to find the temperature window of interest, we introduce a flexible and powerful method for systematic investigations of critical phenomena. As a result, we gain one to two orders of magnitude in the performance for two- and three-dimensional Ising models in comparison with the recently proposed Wang-Landau recursion for cluster algorithms based on the multibondic algorithm, which is already a great improvement over the standard multicanonical variant.
We present a new method for using the data from Monte Carlo simulations that can increase the efficiency by 2 or more orders of magnitude. A single Monte Carlo simulation is sufficient to obtain complete thermodynamic information over the entire scaling region near a phase transition. The accuracy of the method is demonstrated by comparison with exact results for the d=2 Ising model. New results for d=2 eight-state Potts model are also presented. The method is generally applicable to statistical models and lattice-gauge theories.