148 Canonical ensemble
Q(N,V,T,r,r′) =1
h^3 N∫
dNpdNrexp{
−β[N
∑
i=1p^2 i
2 m+
1
2
mω^2∑N
i=0(ri−ri+1)^2]}
. (4.5.23)
We will regard the particles as truly distinguishable so that no 1/N! is needed. The
Gaussian integrals over theNmomenta can be performed immediately, yielding
Q(N,V,T,r,r′) =1
h^3 N(
2 πm
β) 3 N/ 2 ∫
dNrexp[
−
1
2
βmω^2∑N
i=0(ri−ri+1)^2]
. (4.5.24)
The coordinate integrations can be performed straightforwardly, if tediously, by simply
integrating first overr 1 , then overr 2 ,... and recognizing the pattern that results after
n < Nsuch integrations have been performed. We will first follow this procedure, and
then we will show how a simple change of integration variables can be used to simplify
the integrations by uncoupling the harmonic interaction term.
Consider, first, the integration overr 1. Definingα=βmω^2 /2, and using the fact
thatV is much larger than the average nearest-neighbor particle distance to extend
the integration over all space, the integral that must be performed is
I 1 =∫
all spacedr 1 e−α[(r^1 −r)(^2) +(r 2 −r 1 ) (^2) ]
. (4.5.25)
Expanding the squares gives
I 1 = e−α(r(^2) +r (^22) )
∫
all spacedr 1 e−^2 α[r(^21) −r 1 ·(r+r 2 )]
. (4.5.26)
Now, we can complete the square to give
I 1 = e−α(r(^2) +r (^22) )
eα(r+r^2 )
(^2) / 2
∫
all spacedr 1 e−^2 α[r^1 −(r+r^2 )/2]2= e−α(r^2 −r)(^2) / 2 (π
2 α
) 3 / 2
. (4.5.27)
We can now proceed to ther 2 integration, which is of the form
I 2 =(π
2 α) 3 / 2 ∫
all spacedr 2 e−α(r^2 −r)(^2) / 2
e−α(r^3 −r^2 )
2
. (4.5.28)
Again, we begin by expanding the squares to yield
I 2 =
(π
2 α) 3 / 2
e−α(r(^2) +2r (^23) )/ 2
∫
all spacedr 2 e−^3 α[r(^22) − 2 r 2 ·(r+2r 3 )/3]/ 2
. (4.5.29)
Completing the square gives