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 {\displaystyle Q_{0}} and Q 1 {\displaystyle Q_{1}} , from F ( N , V , T ) = − k B T ln Q {\displaystyle F(N,V,T)=-k_{B}T\ln Q}
get the thermodynamic free energy difference is Δ F = − k B T ln ( Q 1 / Q 0 ) = − k B T ln ( ∫ d s N exp [ − β U 1 ( s N ) ] ∫ d s N exp [ − β U 0 ( s N ) ] ) {\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
Δ U = U 1 ( s N ) − U 0 ( s N ) {\displaystyle \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 ( Δ U ) = ∫ d s N exp ( − β U 1 ) δ ( U 1 − U 0 − Δ U ) Q 1 {\displaystyle 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 {\displaystyle p_{1}} is a configurational part of a partition function
p 1 ( Δ U ) = ∫ d s N exp ( − β U 1 ) δ ( U 1 − U 0 − Δ U ) Q 1 = ∫ d s N exp [ − β ( U 0 + Δ U ) ] δ ( U 1 − U 0 − Δ U ) Q 1 {\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}}}}
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