Generalized-ensemble simulations and cluster algorithms

Abstract

The importance-sampling Monte Carlo algorithm appears to be the universally optimal solution to the problem of sampling the state space of statistical mechanical systems according to the relative importance of configurations for the partition function or thermal averages of interest. While this is true in terms of its simplicity and universal applicability, the resulting approach suffers from the presence of temporal correlations of successive samples naturally implied by the Markov chain underlying the importance-sampling simulation. In many situations, these autocorrelations are moderate and can be easily accounted for by an appropriately adapted analysis of simulation data. They turn out to be a major hurdle, however, in the vicinity of phase transitions or for systems with complex free-energy landscapes. The critical slowing down close to continuous transitions is most efficiently reduced by the application of cluster algorithms, where they are available. For first-order transitions and disordered systems, on the other hand, macroscopic energy barriers need to be overcome to prevent dynamic ergodicity breaking. In this situation, generalized-ensemble techniques such as the multicanonical simulation method can effect impressive speedups, allowing to sample the full free-energy landscape. The Potts model features continuous as well as first-order phase transitions and is thus a prototypic example for studying phase transitions and new algorithmic approaches. I discuss the possibilities of bringing together cluster and generalized-ensemble methods to combine the benefits of both techniques. The resulting algorithm allows for the efficient estimation of the random-cluster partition function encoding the information of all Potts models, even with a non-integer number of states, for all temperatures in a single simulation run per system size.

15 pages, 6 figures, proceedings of the 2009 Workshop of the Center of Simulational Physics, Athens, GA

References in corpus (9)

• An efficient, multiple range random walk algorithm to calculate the density of states
• Multicanonical Ensemble: A New Approach to Simulate First-order Phase Transitions
• A fast Monte Carlo algorithm for site or bond percolation
• Understanding and Improving the Wang-Landau Algorithm
• Multicanonical Recursions
• Multibondic Cluster Algorithm for Monte Carlo Simulations of First-Order Phase Transitions
• Partition function of two- and three-dimensional Potts ferromagnets for arbitrary values of q>0
• Cross-correlations in scaling analyses of phase transitions
• Random-cluster multi-histogram sampling for the q-state Potts model

Cited by in corpus (5)

• Massively parallel multicanonical simulations
• Critical Binder cumulant and universality: Fortuin-Kasteleyn clusters and order-parameter fluctuations
• Efficient simulation of the random-cluster model
• Microcanonical Determination of the Interface Tension of Flat and Curved Interfaces from Monte Carlo Simulations
• Dynamic connectivity algorithms for Monte Carlo simulations of the random-cluster model