150 Canonical ensemble
Q(N,V,T) =
1
h^6(
2 π
βhω) 3 N(
2 πm
β) 3
1
(N+ 1)^3 /^2
×
∫
dr 0 drN+1e−βmω(^2) (r 0 −rN+1) (^2) /(N+1)
. (4.5.35)
We now introduce a change of variables to the center-of-massR= (r 0 +rN+1)/2 of
the endpoint particles and their corresponding relative coordinates=r 0 −rN+1. The
Jacobian of the transformation is 1. With this transformation, we have
Q(N,V,T) =
1
h^6(
2 π
βhω) 3 N(
2 πm
β) 3
1
(N+ 1)^3 /^2
∫
dRdse−βmω(^2) s (^2) /(N+1)
.(4.5.36)
The integration overscan be performed over all space because the Gaussian rapidly
decays to 0. However, the integration over the center-of-massRis completely free and
must be restricted to the containing volumeV. The result of performing the last two
coordinate integrations is
Q(N,V,T) =
(
V
λ^3)(
2 π
βhω)3(N+1)
, (4.5.37)
whereλ=
√
βh^2 / 2 πm.
Now that we have seen how to perform the coordinate integrationsdirectly, let us
demonstrate how a change of integration variables in the partition function can simplify
the problem considerably. The use of variable transformations in a partition function
is a powerful technique that can lead to novel computational algorithms for solving
complex problems (Tuckermanet al., 1993; Zhuet al., 2002; Minaryet al., 2007).
Consider, once again, the polymer chain with fixed endpoints, so that the partition
function is given by eqn. (4.5.23), and consider a change of integration variables from
rktoukgiven by
uk=rk−krk+1+r
(k+ 1), (4.5.38)
where, again, the conditionrN+1=r′is implied. In order to express the harmonic
coupling in terms of the new variablesu 1 ,...,uN, we need the inverse of this transfor-
mation. Interestingly, if we simply solve eqn. (4.5.38) forrk, we obtain
rk=uk+
k
k+ 1rk+1+1
k+ 1r. (4.5.39)Note that eqn. (4.5.39) defines the inverse transformationrecursively, since knowledge
of howrk+1depends onu 1 ,...,uN allows the dependence ofrkonu 1 ,...,uN to be
determined. Consequently, the inversion process is “seeded” by starting with thek=N
term and working backwards tok= 1.