1549380323-Statistical Mechanics Theory and Molecular Simulation

Blue moon ensemble 341

√

g= e−w=

∑

i

1

mi

(

∂σ

∂ri

) 2

≡z(r). (8.7.20)

This metric factor arises frequently when holonomic constraints are imposed on a sys-
tem and is, therefore, given the special symbolz(r). The partition function generated
by eqns. (8.7.15) now becomes

Z=

∫

dNrdNpz(r)δ(H(r,p)−E)δ(σ(r))δ( ̇σ(r,p)). (8.7.21)

The energy-conservingδ-function,δ(H(p,r)−E) can be replaced by a canonical dis-
tribution exp[−βH(r,p)] simply by coupling eqns. (8.7.15) to a thermostat such as
the Nos ́e–Hoover chain thermostat (see Section 4.10), and eqn.(8.7.21) is replaced by

Z=

∫

dNrdNpz(r)e−βH(r,p)δ(σ(r))δ( ̇σ(r,p)). (8.7.22)

In order to compare eqn. (8.7.21) to eqn. (8.7.1), we need to perform the integration
over the momenta in order to clear the secondδ-function. Noting that

σ ̇(r,p) =

∑

i

∂σ

∂ri

·r ̇i=

∑

i

∂σ

∂ri

·

pi

mi

, (8.7.23)

the partition function becomes

Z=

∫

dNrdNpz(r)e−βH(r,p)δ(σ(r))δ

(

∑

i

∂σ

∂ri

·

pi

mi

)

. (8.7.24)

Fortunately, the secondδ-function is linear in the momenta and can be integrated over
relatively easily (Problem 8.4), yielding

Z∝

∫

dNrz^1 /^2 (r)e−βU(r)δ(σ(r)). (8.7.25)

Apart from prefactors irrelevant to the free energy, eqns. (8.7.25) and eqn. (8.7.1),
differ only by the factor ofz^1 /^2 (r). We have already seen that the conditional average
of any functionO(r) of the positions is

〈O(r)〉conds =

∫

dre−βU(r)O(r)δ(f 1 (r)−s)

〈δ(f 1 (r)−s)〉

. (8.7.26)

The above analysis suggests that the average ofO(r) in the ensemble generated by the
constrained dynamics is

〈O(r)〉constrs =

∫

dre−βU(r)z^1 /^2 (r)O(r)δ(f 1 (r)−s)

〈z^1 /^2 (r)δ(f 1 (r)−s)〉

, (8.7.27)

sinceσ(r) =f 1 (r)−s. Thus, the conditional average ofO(r) can be generated using the
constrained ensemble if, instead of computing the average ofO(r) in this ensemble, we