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(State) &minus; U(State) is the difference in potential energy, β = 1/kT (T is the temperature in kelvins, while k is the Boltzmann constant), and  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 
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 = F &minus; F) on moving between the two super states be calculated from sampling in both ensembles? The kinetic energy part in the free energy is equal between states so can be ignored. Also the Gibbs free energy corresponds to the NpT ensemble.