From multidimensional replica-exchange method to multidimensional multicanonical algorithm and simulated tempering

We discuss multidimensional generalizations of multicanonical algorithm, simulated tempering, and replica-exchange method. We generalize the original potential-energy function E-0 by adding any physical quantity V of interest as a new energy term with a coupling constant lambda. We then perform a multidimensional multicanonical simulation where a random walk in E-0 and V spaces is realized. We can alternately perform a multidimensional simulated-tempering simulation where a random walk in temperature T and parameter lambda is realized. The results of the multidimensional replica-exchange simulations can be used to determine the weight factors for these multidimensional multicanonical and simulated-tempering simulations.