Overlapping distribution method

Theory

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

from 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 ) ] )

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 )

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

where in 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 = 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 )

since

Δ F = − k B T ln ⁡ ( Q 1 / Q 0 )

ln ⁡ p 1 ( Δ U ) = β ( Δ F − Δ U ) + ln ⁡ p 0 ( Δ 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

thus that

f 1 ( Δ U ) = f 0 ( Δ U ) + β Δ F

and Δ F can be obtained by fitting f 1 and f 0