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We discuss multi-dimensional generalizations of multicanonical algorithm,simulated tempering, and replica-exchange method. We generalize the originalpotential energy function $E_0$ by adding any physical quantity $V$ of interestas a new energy term with a coupling constant $\lambda$. We then perform amulti-dimensional multicanonical simulation where a random walk in $E_0$ and$V$ space is realized. We can alternately perform a multi-dimensional simulatedtempering simulation where a random walk in temperature $T$ and parameter$\lambda$ is realized. The results of the multi-dimensional replica-exchangesimulations can be used to determine the weight factors for thesemulti-dimensional multicanonical and simulated tempering simulations.

Multicanonical Algorithm, Simulated Tempering, Replica-Exchange Method, and All That