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 Q₀ and Q₁ ,

from F(N,V,T) = − k_BTlnQ

get the thermodynamic free energy difference is $\Delta F = -k_{B}T ln (Q_{1}/Q_{0}) = - k_{B} T ln (\frac{\int ds^{N}exp[-\beta U_{1}(s^{N})]}{\int ds^{N}exp[-\beta U_{0}(s^{N})]})$

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

ΔU = U₁(s^N) − U₀(s^N)

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

$p_{1}(\Delta U) = \frac{\int ds^{N}exp(-\beta U_{1})\delta(U_{1}-U_{0}-\Delta U)}{q_{1}}$

where in p₁ is a configurational part of a partition function

$p_{1}(\Delta U) = \frac{\int ds^{N}exp(-\beta U_{1})\delta(U_{1}-U_{0}-\Delta U)}{q_{1}} = \frac{\int ds^{N}exp[-\beta(U_{0}+\Delta U)]\delta(U_{1}-U_{0}-\Delta U)}{q_{1}}$ $= \frac{q_{0}}{q_{1}} exp (-\beta \Delta U) \frac{\int ds^{N}exp(-\beta U_{0})\delta(U_{1}-U_{0}-\Delta U)}{q_{0}} = \frac{q_{0}}{q_{1}} exp (- \beta \Delta U) p_{0}(\Delta U)$

since

ΔF = − k_BTln(q₁ / q₀)

lnp₁(ΔU) = β(ΔF − ΔU) + lnp₀(ΔU)

now define two functions:

$f_{0}(\Delta U) = ln p_{0}(\Delta U) - \frac{\beta\Delta U}{2} f_{1}(\Delta U) = ln p_{1}(\Delta U) + \frac{\beta\Delta U}{2}$

thus that

f₁(ΔU) = f₀(ΔU) + βΔF

andΔF can be obtained by fitting f₁ and f₀

Notes

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