Overlapping distribution method

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The Overlapping distribution method was introduced by Charles H. Bennett[1] for estimating chemical potential.

[edit] Theory

For two N particle systems 0 and 1 with partition function Q0 and Q1 ,

from F(N,V,T) = − kBTlnQ

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(sN) − U0(sN)

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 = − kBTln(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

[edit] Notes

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