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We investigate the theoretical foundations of the simulated tempering methodand use our findings to design efficient algorithms. Employing a largedeviation argument first used for replica exchange molecular dynamics [Plattneret al., J. Chem. Phys. 135:134111 (2011)], we demonstrate that the mostefficient approach to simulated tempering is to vary the temperature infinitelyrapidly. In this limit, we can replace the equations of motion for thetemperature and physical variables by averaged equations for the latter alone,with the forces rescaled according to a position-dependent function defined interms of temperature weights. The averaged equations are similar to those usedin Gao's integrated-over-temperature method, except that we show that it isbetter to use a continuous rather than a discrete set of temperatures. We givea theoretical argument for the choice of the temperature weights as thereciprocal partition function, thereby relating simulated tempering toWang-Landau sampling. Finally, we describe a self-consistent algorithm forsimultaneously sampling the canonical ensemble and learning the weights duringsimulation. This algorithm is tested on a system of harmonic oscillators aswell as a continuous variant of the Curie-Weiss model, where it is shown toperform well and to accurately capture the second-order phase transitionobserved in this model.

Simulated Tempering Method in the Infinite Switch Limit with Adaptive Weight Learning