The Weighted Histogram Analysis Method (WHAM), an extension of Ferrenberg and Swendsen's Multiple Histogram Technique, has been applied for the first time on complex biomolecular Hamiltonians. The method is presented here as an extension of the Umbrella Sampling method for free‐energy and Potential of Mean Force calculations. This algorithm possesses the following advantages over methods that are currently employed: (1) It provides a built‐in estimate of sampling errors thereby yielding objective estimates of the optimal location and length of additional simulations needed to achieve a desired level of precision; (2) it yields the “best” value of free energies by taking into account all the simulations so as to minimize the statistical errors; (3) in addition to optimizing the links between simulations, it also allows multiple overlaps of probability distributions for obtaining better estimates of the free‐energy differences. By recasting the Ferrenberg–Swendsen Multiple Histogram equations in a form suitable for molecular mechanics type Hamiltonians, we have demonstrated the feasibility and robustness of this method by applying it to a test problem of the generation of the Potential of Mean Force profile of the pseudorotation phase angle of the sugar ring in deoxyadenosine. © 1992 by John Wiley & Sons, Inc.

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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 the d 2, eight-state Potts model are also presented. The method is generally applicable to statistical models and lattice gauge theories.

A new Monte Carlo method is presented for simulations of systems with quenched random interactions. The approach greatly reduces the long correlation times characteristic of standard methods, allowing the investigation of lower temperatures with less computer time than previously necessary.PACS numbers: 05.50. +q, 75.lO.Hk Over the past decade, there have been continuing controversies about the properties of systems with quenched random interactions, such as spin-glasses 1-5 and random-field models. 6 "" 8 Extensive work has been carried out to answer some of the questions by Monte Carlo simulations. While these efforts have been partially successful, they have been greatly hampered by the extremely long relaxation times that are characteristic of systems with frustrated interactions. Similar difficulties are also found in some engineering applications involving optimization subject to conflicting constraints. 9 In this paper, we present a new approach to Monte Carlo computer simulations, which provides rapid relaxation times, making possible the study of equilibrium properties with relatively modest amounts of computer time.In constructing a Markov process for Monte Carlo simulations, two conditions should be met: A sequence of transitions with nonzero probability must connect any two configurations, and the condition of detailed balance must be satisfied. The standard Monte Carlo algorithm satisfies both conditions, ensuring that equilibrium will be achieved eventually (although not necessarily within budget limitations).By retaining standard Monte Carlo methods as part of the new simulation, we satisfy the first condition and provide a fast process for relaxation of local fluctuations. Additional processes satisfying detailed balance are then introduced to reduce relaxation times for large fluctuations. This strategy has been shown to be effective when the form of the important large-scale fluctuations is known. 10 The method described below provides for the automatic recognition of important fluctuations in a spin-glass.To illustrate our method, we will discuss its application to the Ising spin-glass,where s t takes on the values ± 1 and the factor -I/ICBT has been absorbed into the coupling constant K. The Bgj are dimensionless variables, which describe the quenched, random interactions. Instead of simulating different temperatures sequentially, we treat several independent "replicas" of this system at different values of K simultaneously. By including a replica index, /i, we can describe the entire set of replicas with a single Hamiltonian, modifying Eq. (1) to become fR'I,I t B l jK M s l M sj'\ (2) n (ij)Information is transferred by the "mixing" of two neighboring replicas, by use of new variables, t^n\ defined at each site: 5/» + 1 >-*/»ty»>. (3)The pair of replicas can now be represented by the variables {s^} and {t^} and the part of the Hamiltonian describing this pair of replicas becomes //pair -I Bij lK in) + K {n + X V"V W) W"V"^ < 4 > Changing sf n) while holding */ n) fixed is equivalent to ch...