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Overlapping distribution method

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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}{\int ds^{N}\exp}})}

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\delta (U_{1}-U_{0}-\Delta U)}{Q_{1}}}} = Q 0 Q 1 exp ⁡ ( − β Δ U ) ∫ d s N exp ⁡ ( − β U 0 ) δ ( U 1 − U 0 − Δ U ) Q 0 = Q 0 Q 1 exp ⁡ ( − β Δ U ) p 0 ( Δ U ) {\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

Δ F = − k B T ln ⁡ ( Q 1 / Q 0 ) {\displaystyle \Delta F=-k_{B}T\ln(Q_{1}/Q_{0})}


ln ⁡ p 1 ( Δ U ) = β ( Δ F − Δ U ) + ln ⁡ p 0 ( Δ U ) {\displaystyle \ln p_{1}(\Delta U)=\beta (\Delta F-\Delta U)+\ln p_{0}(\Delta U)}


now define two functions:

f 0 ( Δ U ) = ln ⁡ p 0 ( Δ U ) − β Δ U 2 f 1 ( Δ U ) = ln ⁡ p 1 ( Δ U ) + β Δ U 2 {\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

f 1 ( Δ U ) = f 0 ( Δ U ) + β Δ F {\displaystyle f_{1}(\Delta U)=f_{0}(\Delta U)+\beta \Delta F}

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

References

  1. Bennett, C.H. (1976). "Efficient Estimation of Free Energy Differences from Monte Carlo Data". Journal of Computational Physics. 22 (2): 245–268. Bibcode:1976JCoPh..22..245B. doi:10.1016/0021-9991(76)90078-4.
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