Adaptive Biasing Force Implementation and Accuracy

We previously showed that the free energy can be computed using (4.33). This is done by binning the force and computing the average force along the interval. 

Assume that we bin the interval of interest for and that we have collected nk (N ) samples in bin k after N steps in a molecular dynamics simulation. We use those samples to compute a running average of the force acting along  

In general, when very few samples are available the force Fk(N) will not be an accurate approximation of d/l/d . Large variations in Fj (N) may lead to nonequilibrium effects and systematic bias of the calculation. Mathematically, this can be expressed by introducing a perturbation zLif fq, p, N), which is a function of the number of steps N. At N = 1 if we average over all possible initial configurations, abbreviated by subscript 0, we obtain 

For intermediate values of N, a perturbation A (q, p, N) can be defined such that the weight in the average is exp(-/9( + A f)). In this case a systematic bias may be introduced and dA/d / (Fk(N))0. In addition incorrect estimates of dA/d can lead to short-lived free energy barriers in the initial steps of the simulation. 

In order to control those initial nonequilibrium effects, a ramp function can be added which reduces the variations from one step to the next of the external force applied in a given bin. The external force applied to the system can be chosen equal to 

---