We give general formulations of the multidimensional multicanonical algorithm, simulated tempering, and replica-exchange method. We generalize the original potential energy function E0 by adding any physical quantity V of interest as a new energy term. These multidimensional generalized-ensemble algorithms then perform a random walk not only in E0 space but also in V space. Among the three algorithms, the replica-exchange method is the easiest to perform because the weight factor is just a product of regular Boltzmann-like factors, while the weight factors for the multicanonical algorithm and simulated tempering are not a priori known. We give a simple procedure for obtaining the weight factors for these two latter algorithms, which uses a short replica-exchange simulation and the multiple-histogram reweighting techniques. As an example of applications of these algorithms, we have performed a two-dimensional replica-exchange simulation and a two-dimensional simulated-tempering simulation using an α -helical peptide system. From these simulations, we study the helix-coil transitions of the peptide in gas phase and in aqueous solution.
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