Overlapping distribution method

The Overlapping distribution method was introduced by Charles H. Bennett^[1] for estimating chemical potential.

Theory

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

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

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

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

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

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

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

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

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

since

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

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

now define two functions:

${\displaystyle f_{0}(\Delta U)=\ln p_{0}(\Delta U)-{\frac {\beta \Delta U}{2}}f_{1}(\Delta U)=\ln p_{1}(\Delta U)+{\frac {\beta \Delta U}{2}}}$

thus that

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

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