Efficient, Multiple-Range Random Walk Algorithm to Calculate the Density of States

Abstract

We present a new Monte Carlo algorithm that produces results of high accuracy with reduced simulational effort. Independent random walks are performed (concurrently or serially) in different, restricted ranges of energy , and the resultant density of states is modified continuously to produce locally flat histograms. This method permits us to directly access the free energy and entropy, is independent of temperature, and is efficient for the study of both 1st order and 2nd order phase transitions. It should also be useful for the study of complex systems with a rough energy landscape. 64.60.Cn, 05.50.+q, 02.70.Lq Computer simulation has become an essential tool in condensed matter physics [1], particularly for the study of phase transitions and critical phenomena. The workhorse for the past half-century has been the Metropolis importance sampling algorithm, but more recently new, efficient algorithms have begun to play a role in allowing simulation to achieve the resolution which is needed to accurately locate and characterize phase transitions. For example, cluster flip algorithms, beginning with the seminal work of Swendsen and Wang [2], have been used to reduce critical slowing down near 2nd order transitions. Similarly, the multicanonical ensemble method [3] was introduced to overcome the tunneling barrier between coexisting phases at 1st order transitions, and this approach also has utility for systems with a rough energy landscape [4][5][6]. In both situations, histogram reweighting techniques [7] can be applied in the analysis to increase the amount of information that can be gleaned from simulational data, but the applicability of reweighting is severely limited in large systems by the statistical quality of the "wings" of the histogram. This latter effect is quite important in systems with competing interactions for which short range order effects might occur over very broad temperature ranges or even give rise to frustration that produces a very complicated energy landscape and limit the efficiency of other methods.In this paper, we introduce a new, general, efficient Monte Carlo algorithm that offers substantial advantages over existing approaches. Unlike conventional Monte Carlo methods that directly generate a canonical distribution at a given temperature g(E)e −E/KBT , our approach is to estimate the density of states g(E) accurately via a random walk which produces a flat histogram in energy space. The method can be further enhanced by performing multiple random walks, each for a different range of energy, either serially or in parallel fashion. The resultant pieces of the density of states can be joined together and used to produce canonical averages for the calculation of thermodynamic quantities at essentially any temperature. We will apply our algorithm to the 2-dim ten state Potts model and Ising model which have 1st-and 2nd-order phase transitions, respectively, to demonstrate the efficiency and accuracy of the method.Our algorithm is based on the observation that if we perform a random w...

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