The Weighted Histogram Analysis Method (WHAM)

There are situations in which it is necessary to combine samples obtained under different simulation conditions to determine properties (e.g. averages or PMFs) of the system under a single condition (which may not even be one of the conditions that were actually simulated). One example is the determination of a PMF to obtain accurate measures of a free energy barrier. If the barrier is substantially higher than the minima, then a standard canonical simulation will provide little sampling at the barrier, and thus may not be able to accurately estimate the barrier height. An alternative strategy is to use umbrella sampling with a series of biasing potentials to confine the system to small regions of the PMF. Together, these provide adequate sampling of the entire reduced coordinate of interest (see, for example, references 1,2). Since we know the biasing potential, we can “unbias” the samples of each simulation. Suppose that the underlying potential of interest is U(p) and we apply a biasing potential Vi(x) to the i-th simulation, where x(p) is the reduced coordinate with respect to which we are interested in obtaining the PMF, i.e. we obtain a canonical sample distributed according to exp[−β(U(p)+Vi(x(p))]. To unbias the estimated population P (x = xj) we can multiply the estimated probability by exp[βVi(xj)]. To obtain the full unbiased distribution, we need to combine the data from all of the simulations. If the sampling were extremely thorough, then