The Adaptive Biasing Force Method

We now describe a different approach which is simpler than the method of constraints and also very efficient. It does not require running a constrained simulation and can be performed entirely with a single molecular dynamics run. 

One reason for the inefficiencies of constraint methods is that they may prevent an efficient sampling of the set (x) = . This is illustrated by Fig. 4.2. It is common that many pathways separated by high energy barriers exist to go from A to B. In constrained simulation, the system can get trapped in one of the pathways. In the most serious cases, this leads to quasi-nonergodic effect where only a part of the set (x) is effectively explored. In less serious cases, the convergence is quite slow. 

An approach that does not suffer from such problems is the ABF method. This method is based on computing the mean force on and then removing this force in order to improve sampling. This leads to uniform sampling along . The dynamics of corresponds to a random walk with zero mean force. Only the fluctuating part of the instantaneous force on remains. This method is quite simple to implement and leads to a very small statistical error and excellent convergence. 

Let us first provide an expression to compute the derivative of the free energy for unconstrained simulations. Then, we will discuss the calculation of the biasing force and the algorithmic implementation of the method. 

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