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
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
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?
Also the Gibbs free energy corresponds to the NpT ensemble.
Substituting for f the Metropolis function defined above (which satisfies the detailed balance condition), and setting C to zero, gives The advantage of this formulation (apart from its simplicity) is that it can be computed without performing two simulations, one in each specific ensemble.
Indeed, it is possible to define an extra kind of "potential switching" Metropolis trial move (taken every fixed number of steps), such that the single sampling from the "mixed" ensemble suffices for the computation.
Bennett explores which specific expression for ΔF is the most efficient, in the sense of yielding the smallest standard error for a given simulation time.
He shows that the optimal choice is to take Some assumptions needed for the efficiency are the following: The multistate Bennett acceptance ratio (MBAR) is a generalization of the Bennett acceptance ratio that calculates the (relative) free energies of several multi states.
It essentially reduces to the BAR method when only two super states are involved.
This method, also called Free energy perturbation (or FEP), involves sampling from state A only.
or This exact result can be obtained from the general BAR method, using (for example) the Metropolis function, in the limit
and Taylor expanding the second exact perturbation theory expression to the second order, one gets the approximation Note that the first term is the expected value of the energy difference, while the second is essentially its variance.
writing the potential energy as depending on a continuous parameter,
This can either be directly verified from definitions or seen from the limit of the above Gibbs-Bogoliubov inequalities when
The Bennett acceptance ratio method is implemented in modern molecular dynamics systems, such as Gromacs.