Bennett acceptance ratio

The Bennett acceptance ratio method (sometimes abbreviated to 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(State_y) − U(State_x) is the difference in potential energy, β = 1/kT (T is the temperature in kelvins, while k is the Boltzmann constant), and ${\displaystyle M(x)\equiv \min(e^{-x},1)}$ 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 ${\displaystyle \left\langle \cdots \right\rangle }$.

...
Wikipedia