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When studying high-dimensional dynamical systems such as macromolecules, quantum systems, and polymers, a prime concern is the identification of the most probable states and their stationary probabilities or free energies. Often, these systems have metastable regions or phases, prohibiting the estimation of the stationary probabilities by direct simulation. Efficient sampling methods such as umbrella sampling, metadynamics, and conformational flooding have been developed that perform a number of simulations where the system's potential is biased so as to accelerate the rare barrier crossing events. A joint free energy profile or stationary density can then be obtained from these biased simulations with the weighted histogram analysis method. This approach (a) requires a few essential order parameters to be defined in which the histogram is set up, and (b) assumes that each simulation is in global equilibrium. Both assumptions make the investigation of high-dimensional systems with previously unknown energ...

Optimal Estimation of Free Energies and Stationary Densities from Multiple Biased Simulations