Overlapping distribution method

The Overlapping distribution method was introduced by Charles H. Bennett for estimating chemical potential.

Theory
For two N particle systems 0 and 1 with partition function $$Q_{0}$$ and $$Q_{1}$$ ,

from $$ F(N,V,T) = - k_{B}T \ln Q $$

get the thermodynamic free energy difference is $$\Delta F = -k_{B}T \ln (Q_{1}/Q_{0}) = - k_{B} T \ln (\frac{\int ds^{N}\exp[-\beta U_{1}(s^{N})]}{\int ds^{N}\exp[-\beta U_{0}(s^{N})]})$$

For every configuration visited during this sampling of system 1 we can compute the potential energy U as a function of the configuration space, and the potential energy difference is

$$\Delta U = U_{1}(s^{N}) - U_{0}(s^{N})$$

Now construct a probability density of the potential energy from the above equation:

$$p_{1}(\Delta U) = \frac{\int ds^{N}\exp(-\beta U_{1})\delta(U_{1}-U_{0}-\Delta U)}{Q_{1}} $$

where in $$p_{1}$$ is a configurational part of a partition function

$$ p_{1}(\Delta U) = \frac{\int ds^{N}\exp(-\beta U_{1})\delta(U_{1}-U_{0}-\Delta U)}{Q_{1}} = \frac{\int ds^{N}\exp[-\beta(U_{0}+\Delta U)]\delta(U_{1}-U_{0}-\Delta U)}{Q_{1}}$$ $$= \frac{Q_{0}}{Q_{1}} \exp (-\beta \Delta U) \frac{\int ds^{N}\exp(-\beta U_{0})\delta(U_{1}-U_{0}-\Delta U)}{Q_{0}} = \frac{Q_{0}}{Q_{1}} \exp (- \beta \Delta U) p_{0}(\Delta U) $$

since

$$\Delta F = -k_{B}T \ln (Q_{1}/Q_{0})$$

$$\ln p_{1}(\Delta U) = \beta(\Delta F -\Delta U) + \ln p_{0}(\Delta U)$$

now define two functions:

$$f_{0}(\Delta U) = \ln p_{0}(\Delta U) - \frac{\beta\Delta U}{2}

f_{1}(\Delta U) = \ln p_{1}(\Delta U) + \frac{\beta\Delta U}{2} $$

thus that

$$f_{1}(\Delta U) = f_{0}(\Delta U) + \beta\Delta F$$

and$$ \Delta F$$ can be obtained by fitting $$f_{1}$$ and $$f_{0}$$