154 Canonical ensemble
expect peaks at particular values characteristic of the average structural motifs present
in the system, with the peak widths largely determined by the temperature, density,
etc. This argument suggests that the spatial distribution functions in a system contain
a considerable amount of information about the local structure and the fluctuations. In
this section, we will discuss the formulation of such distribution functions as ensemble
averages and relate these functions to the thermodynamics of the system.
We begin the discussion with the canonical partition function for a system ofN
identical particles interacting via a potentialU(r 1 ,...,rN).
Q(N,V,T) =1
N!h^3 N∫
dNp∫
D(V)dNrexp{
−β[N
∑
i=1p^2 i
2 m+U(r 1 ,...,rN)]}
. (4.6.1)
Since the momentum integrations can be evaluated independently, the partition func-
tion can also be expressed as
Q(N,V,T) =1
N!λ^3 N∫
D(V)dNre−βU(r^1 ,...,rN). (4.6.2)Note that in the Hamiltonian
H=
∑N
i=1p^2 i
2 m
+U(r 1 ,...,rN), (4.6.3)the kinetic energy term is a universal term that appears in all such Hamiltonians.
It is only the potentialU(r 1 ,...,rN) that determines the particular properties of the
system. In order to make this fact manifest in the partition function, we introduce the
configurational partition function
Z(N,V,T) =∫
D(V)dr 1 ···drNe−βU(r^1 ,...,rN), (4.6.4)in terms of which,Q(N,V,T) =Z(N,V,T)/(N!λ^3 N). Note that the ensemble average
of any coordinate-dependent functiona(r 1 ,...,rN) can be expressed as
〈a〉=1
Z
∫
D(V)dr 1 ···drNa(r 1 ,...,rN)e−βU(r^1 ,...,rN). (4.6.5)(Throughout the discussion, the arguments of the configurational partition function
Z(N,V,T) will be left off for notational simplicity.) From the form of eqn. (4.6.4),
we see that the probability of finding particle 1 in a small volume elementdr 1 about
the pointr 1 and particle 2 in a small volume element dr 2 about the pointr 2 ,..., and
particleNin a small volume element drNabout the pointrNis
P(N)(r 1 ,...,rN)dr 1 ···drN=1
Z
e−βU(r^1 ,...,rN)dr 1 ···drN. (4.6.6)Now suppose that we are interested in the probability of finding only the firstn < N
particles in small volume elements about the pointsr 1 ,...,rn, respectively, independent