1549380323-Statistical Mechanics Theory and Molecular Simulation

Blue moon ensemble 339

of the transformation is not unity. LetJ(q) =|∂(r 1 ,...,rN)/∂(q 1 ,...,q 3 N)|denote this
Jacobian. Eqn. (8.7.10) then becomes

1

P(s)

dP

ds

=

1

N!λ^3 NQ(N,V,T)

∫

d^3 Nq J(q)e−βU ̃(q)∂s∂δ(q 1 −s)

〈δ(q 1 −s)〉

=

1

N!λ^3 NQ(N,V,T)

∫

d^3 Nqe−β(

U ̃(q)−kTlnJ(q))∂

∂sδ(q^1 −s)

〈δ(q 1 −s)〉

, (8.7.11)

where, in the last line, the Jacobian has been exponentiated. Changing the derivative
∂/∂sto∂/∂q 1 and performing the integration by parts as was done in eqn. (8.7.7),we
obtain

1

P(s)

dP

ds

=

1

N!λ^3 NQ(N,V,T)

∫

d^3 Nq∂q∂ 1 e−β(

U ̃(q)−kTlnJ(q))

δ(q 1 −s)

〈δ(q 1 −s)〉

=−

β

N!λ^3 NQ(N,V,T)

×

∫

d^3 Nq

[

∂U ̃

∂q 1 −kT

∂

∂q 1 lnJ(q)

]

e−β(

U ̃(q)−kTlnJ(q))

δ(q 1 −s)

〈δ(q 1 −s)〉

=−β

〈[

∂U ̃

∂q 1

−kT

∂

∂q 1

lnJ(q)

]〉cond

s

. (8.7.12)

Therefore, the free energy profile becomes

A(q) =A(s(i)) +

∫q

s(i)

ds

〈[

∂U ̃

∂q 1

−kT

∂

∂q 1

lnJ(q)

]〉cond

s

. (8.7.13)

The derivative ofU ̃, the transformed potential, can be computed form the original
potentialUusing the chain rule

∂U ̃

∂q 1

=

∑N

i=1

∂U

∂ri

·

∂ri

∂q 1

. (8.7.14)

Eqn. (8.7.13) can be applied straightforwardly to simple reaction coordinates for which
the full transformation to generalized coordinates is known. Let us return to the prob-
lem of computing conditional ensemble averages from constrained molecular dynamics.
We will use this discussion as a vehicle for introducing yet another expression forA(q)
that does not require a coordinate transformation at all.
Recall from Section 1.9 that the equations of motion for a system subject to a
single holonomic constraintσ(r 1 ,...,rN) = 0 are

r ̇i=

pi

mi