338 Free energy calculations
∂
∂sδ(q 1 −s) =−∂
∂q 1δ(q 1 −s). (8.7.6)Finally, we integrate by parts to obtain
1
P(s)dP
ds=
CN
Q(N,V,T)
∫
d^3 Npd^3 Nq[
∂
∂q 1 e−βH ̃(q,p)]
δ(q 1 −s)
〈δ(q 1 −s)〉=−
βCN
Q(N,V,T)∫
d^3 Npd^3 Nq∂
H ̃
∂q 1 e−βH ̃(q,p)δ(q 1 −s)〈δ(q 1 −s)〉=−
β
〈δ(q 1 −s)〉〈(
∂H ̃
∂q 1)
δ(q 1 −s)〉
. (8.7.7)
The last line defines a new ensemble average. Specifically, the average must be per-
formed subject to the conditionq 1 =s. Note, however, that this is not equivalent to
a mechanical constraint since the additional condition ̇q 1 = 0 is not imposed. This
new ensemble average will be denoted〈···〉conds. Hence, the derivative dP/dscan be
expressed as
1
P(s)dP
ds=−β〈
∂H ̃
∂q 1〉conds. (8.7.8)
Substituting eqn. (8.7.8) yields the free energy profile
A(q) =A(si) +∫qs(i)ds〈
∂H ̃
∂q 1〉conds. (8.7.9)
Noting that−〈∂H ̃/∂q 1 〉conds is the expression for the average of the generalized force
onq 1 whenq 1 =s, the integral represents the work done on the system in moving from
s(i)to an arbitrary final pointq. Since the conditional average implies a full simulation
at each fixed value ofq 1 , the thermodynamic transformation is carried out reversibly,
and eqn. (8.7.9) is consistent with the work–free–energy inequality.
Eqn. (8.7.9) provides insight into the underlying statistical mechanical expression
for the free energy. Technically, however, the need for a full canonical transformation
to generalized coordinates and conjugate momenta is inconvenient(see eqn. (1.4.16)).
A more useful expression results when we perform the momentum integrations before
introducing the transformation to generalized coordinates. Starting again with eqn.
(8.7.4), we integrate out the momenta to yield
1
P(s)dP
ds=
1
N!λ^3 NQ(N,V,T)∫
dNre−βU(r)∂s∂δ(f 1 (r)−s)
〈δ(f 1 (r)−s)〉. (8.7.10)
Next, we transform just the coordinates to generalized coordinatesqα=fα(r 1 ,...,rN).
However, because there is no corresponding momentum transformation, the Jacobian