Physics: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 [math]\displaystyle{ Q_{0} }[/math] and [math]\displaystyle{ Q_{1} }[/math] ,

from [math]\displaystyle{ F(N,V,T) = - k_{B}T \ln Q }[/math]

get the thermodynamic free energy difference is [math]\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})]}) }[/math]

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

[math]\displaystyle{ \Delta U = U_{1}(s^{N}) - U_{0}(s^{N}) }[/math]

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

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

where in [math]\displaystyle{ p_{1} }[/math] is a configurational part of a partition function

[math]\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}} }[/math] [math]\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) }[/math]

since

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

[math]\displaystyle{ \ln p_{1}(\Delta U) = \beta(\Delta F -\Delta U) + \ln p_{0}(\Delta U) }[/math]

now define two functions:

[math]\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} }[/math]

thus that

[math]\displaystyle{ f_{1}(\Delta U) = f_{0}(\Delta U) + \beta\Delta F }[/math]

and[math]\displaystyle{ \Delta F }[/math] can be obtained by fitting [math]\displaystyle{ f_{1} }[/math] and [math]\displaystyle{ f_{0} }[/math]