Overlapping distribution method

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

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

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

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

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

where in ${\displaystyle p_1}$ is a configurational part of a partition function

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

since

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

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

now define two functions:

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

thus that

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

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