Bennett acceptance ratio

The Bennett acceptance ratio method (BAR) is an algorithm for estimating the difference in free energy between two systems (usually the systems will be simulated on the computer).It was suggested by Charles H. Bennett in 1976. The Bennett acceptance ratio method (BAR) is an algorithm for estimating the difference in free energy between two systems (usually the systems will be simulated on the computer).It was suggested by Charles H. Bennett in 1976. Take a system in a certain super (i.e. Gibbs) state. By performing a Metropolis Monte Carlo walk it is possible to sample the landscape of states that the system moves between, using the equation where ΔU = U(Statey) − U(Statex) is the difference in potential energy, β = 1/kT (T is the temperature in kelvins, while k is the Boltzmann constant), and M ( x ) ≡ min ( e − x , 1 ) {displaystyle M(x)equiv min(e^{-x},1)} is the Metropolis function.The resulting states are then sampled according to the Boltzmann distribution of the super state at temperature T.Alternatively, if the system is dynamically simulated in the canonical ensemble (also called the NVT ensemble), the resulting states along the simulated trajectory are likewise distributed.Averaging along the trajectory (in either formulation) is denoted by angle brackets ⟨ ⋯ ⟩ {displaystyle leftlangle cdots ight angle } . Suppose that two super states of interest, A and B, are given. We assume that they have a common configuration space, i.e., they share all of their micro states, but the energies associated to these (and hence the probabilities) differ because of a change in some parameter (such as the strength of a certain interaction). The basic question to be addressed is, then, how can the Helmholtz free energy change (ΔF = FB − FA) on moving between the two super states be calculated from sampling in both ensembles? Note that the kinetic energy part in the free energy is equal between states so can be ignored. Note also that the Gibbs free energy corresponds to the NpT ensemble. Bennett shows that for every function f satisfying the condition f ( x ) / f ( − x ) ≡ e − x {displaystyle f(x)/f(-x)equiv e^{-x}} (which is essentially the detailed balance condition), and for every energy offset C, one has the exact relationship where UA and UB are the potential energies of the same configurations, calculated using potential function A (when the system is in superstate A) and potential function B (when the system is in the superstate B) respectively. Substituting for f the Metropolis function defined above (which satisfies the detailed balance condition), and setting C to zero, gives