Extended state-space Monte Carlo methods

In this paper various extensions of the parallel-tempering algorithm are developed and their properties are analyzed. The algorithms are designed to alleviate quasiergodic sampling in systems which have rough energy landscapes by coupling individual Monte Carlo chains to form a composite chain. As with parallel tempering, the procedures are based upon extending the state space to include parameters to encourage sampling mobility. One of the drawbacks of the parallel-tempering method is the stochastic nature of the Monte Carlo dynamics in the auxiliary variables which extend the state space. In this work, the possibility of improving the sampling rate by designing deterministic methods of moving through the parameter space is investigated. The methods developed in this article, which are based upon a statistical quenching and heating procedure similar in spirit to simulated annealing, are tested on a simple two-dimensional spin system $(xy$ model) and on a model in vacuo polypeptide system. In the coupled Monte Carlo chain algorithms, we find that the net mobility of the composite chain is determined by the competition between the characteristic time of coupling between adjacent chains and the degree of overlap of their distributions. Extensive studies of all methods are carried out to obtain optimal sampling conditions. In particular, the most efficient parallel-tempering procedure is to attempt to swap configurations after very few Monte Carlo updates of the composite chains. Furthermore, it is demonstrated that, contrary to expectations, the deterministic procedure does not improve the sampling rate over that of parallel tempering.