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