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 $\mathrm{\Delta }F=-k_{B}T\ln (Q_1/Q_0)=-k_{B}T\ln (\frac{\int d{s}^N\exp [-\beta U_1(s^N)]}{\int d{s}^N\exp [-\beta U_0(s^N)]})$

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

$\mathrm{\Delta }U=U_1(s^N)-U_0(s^N)$

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

$p_1(\mathrm{\Delta }U)=\frac{\int d{s}^N\exp (-\beta U_1)\delta (U_1-U_0-\mathrm{\Delta }U)}{Q_1}$

where in $p_1$ is a configurational part of a partition function

$p_1(\mathrm{\Delta }U)=\frac{\int d{s}^N\exp (-\beta U_1)\delta (U_1-U_0-\mathrm{\Delta }U)}{Q_1}=\frac{\int d{s}^N\exp [-\beta (U_0+\mathrm{\Delta }U)]\delta (U_1-U_0-\mathrm{\Delta }U)}{Q_1}$ $=\frac{Q_0}{Q_1}\exp (-\beta \mathrm{\Delta }U)\frac{\int d{s}^N\exp (-\beta U_0)\delta (U_1-U_0-\mathrm{\Delta }U)}{Q_0}=\frac{Q_0}{Q_1}\exp (-\beta \mathrm{\Delta }U)p_0(\mathrm{\Delta }U)$

since

$\mathrm{\Delta }F=-k_{B}T\ln (Q_1/Q_0)$

$\ln p_1(\mathrm{\Delta }U)=\beta (\mathrm{\Delta }F-\mathrm{\Delta }U)+\ln p_0(\mathrm{\Delta }U)$

now define two functions:

$f_0(\mathrm{\Delta }U)=\ln p_0(\mathrm{\Delta }U)-\frac{\beta \mathrm{\Delta }U}{2}{f}_1(\mathrm{\Delta }U)=\ln p_1(\mathrm{\Delta }U)+\frac{\beta \mathrm{\Delta }U}{2}$

thus that

$f_1(\mathrm{\Delta }U)=f_0(\mathrm{\Delta }U)+\beta \mathrm{\Delta }F$

and$\mathrm{\Delta }F$ can be obtained by fitting $f_1$ and $f_0$