156 Canonical ensemble
g(n)(r 1 ,...,rn) =N!
(N−n)!ρn〈n
∏i=1δ(ri−r′i)〉
r′ 1 ,...,r′N. (4.6.11)
Of course, the most important cases of eqn. (4.6.11) are the firstfew integers forn.
Ifn= 1, for example,g(1)(r) =V ρ(1)(r), whereρ(1)(r)dris the probability of finding
a particle in dr. For a perfect crystal,ρ(1)(r) is a periodic function, but in a liquid,
due to isotropy,ρ(1)(r) is independent ofr 1. Thus, sinceρ(1)(r) = (1/V)g(1)(r), and
ρ(1)(r) is a probability
∫
drρ(1)(r) = 1 =
1
V
∫
drg(1)(r). (4.6.12)However,g(1)(r) is also independent ofrfor an isotropic system, in which case, eqn.
(4.6.12) implies thatg(1)(r) = 1.
4.6.1 The pair correlation function and the radial distribution function
The casen= 2 is of particular interest. The functiong(2)(r 1 ,r 2 ) that results when
n= 2 is used in eqn. (4.6.11) is called thepair correlation function.
g(2)(r 1 ,r 2 ) =1
Z
N(N−1)
ρ^2∫
D(V)dr 3 ···drNe−βU(r^1 ,r^2 ,r^3 ...,rN)=
N(N−1)
ρ^2〈δ(r 1 −r′ 1 )δ(r 2 −r′ 2 )〉r′
1 ,...,r′N. (4.6.13)
Although eqn. (4.6.13) suggests thatg(2) depends onr 1 andr 2 individually, in a
homogeneous system such as a liquid, we anticipate thatg(2)actually depends only
on the relative position between two particles. Thus, it is useful to introduce a change
of variables to center-of-mass and relative coordinates of particles 1 and 2:
R=
1
2
(r 1 +r 2 ) r=r 1 −r 2. (4.6.14)The inverse of this transformation is
r 1 =R+1
2
r r 2 =R−1
2
r, (4.6.15)and its Jacobian is unity: dRdr= dr 1 dr 2. Defining ̃g(2)(r,R) =g(2)(R+r/ 2 ,R−r/2),
we find that
̃g(2)(r,R) =N(N−1)
ρ^2 Z∫
D(V)dr 3 ···drNe−βU(R+(^12) r,R− (^12) r,r 3 ,...,rN)
=
N(N−1)
ρ^2〈
δ(
R+
1
2
r−r′ 1)
δ(
R−
1
2
r−r′ 2)〉
r′ 1 ,...,r′N.(4.6.16)
In a homogeneous system, the location of the particle pair, determined by the center-of-
mass coordinateR, is of little interest since, on average, the distribution of particles