Theory of Adaptive Optimization for Umbrella Sampling

[Image: see text] We present a theory of adaptive optimization for umbrella sampling. With the analytical bias force constant obtained from the constrained thermodynamic length along the reaction coordinate, the windows are distributed to optimize the overlap between neighbors. Combining with the replica exchange method, we propose a method of adaptive window exchange umbrella sampling. The efficiency gain in sampling by the present method originates from the optimal window distribution, in which windows are concentrated to the region where the free energy is steep, as well as consequently improved random walk.