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
Q
0
{\displaystyle Q_{0}}
and
Q
1
{\displaystyle Q_{1}}
,
from
F
(
N
,
V
,
T
)
=
−
k
B
T
ln
Q
{\displaystyle 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
)
]
)
{\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})]}})}
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
)
{\displaystyle \Delta 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
{\displaystyle p_{1}(\Delta U)={\frac {\int ds^{N}\exp(-\beta U_{1})\delta (U_{1}-U_{0}-\Delta U)}{Q_{1}}}}
where in
p
1
{\displaystyle 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
{\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}}}}
=
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
)
{\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)}
since
Δ
F
=
−
k
B
T
ln
(
Q
1
/
Q
0
)
{\displaystyle \Delta F=-k_{B}T\ln(Q_{1}/Q_{0})}
ln
p
1
(
Δ
U
)
=
β
(
Δ
F
−
Δ
U
)
+
ln
p
0
(
Δ
U
)
{\displaystyle \ln p_{1}(\Delta U)=\beta (\Delta F-\Delta U)+\ln p_{0}(\Delta 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
{\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}}}
thus that
f
1
(
Δ
U
)
=
f
0
(
Δ
U
)
+
β
Δ
F
{\displaystyle f_{1}(\Delta U)=f_{0}(\Delta U)+\beta \Delta F}
and
Δ
F
{\displaystyle \Delta F}
can be obtained by fitting
f
1
{\displaystyle f_{1}}
and
f
0
{\displaystyle f_{0}}
References [ ]