270 Grand canonical ensemble
WhenN 1 = 2, we need to place two particles in system 1. The first particle can be
chosen inNways, while the second can be chosen in (N−1) ways, which seems to lead
to a productN(N−1) ways that this configuration can be created. However, choosing
particle 1, for example, as the first particle to put into system 1 andparticle 2 as the
second particle leads to the same physical configuration as choosing particle 2 as the
first particle and particle 1 as the second. Thus, the degeneracy factorg(2,N−2) is
actuallyN(N−1)/2. In general,g(N 1 ,N 1 −1) is nothing more than the number of
ways of placingN“labeled” objects into 2 containers, which is just the well-known
binomial coefficient
g(N 1 ,N 1 −N) =N!
N 1 !(N−N 1 )!
. (6.4.5)
We can check eqn. (6.4.5) against the specific examples we analyzed:
g(0,N) =N!
0!N!
= 1
g(1,N−1) =N!
1!(N−1)!
=N
g(2,N−2) =N!
2!(N−2)!
=
N(N−1)
2
. (6.4.6)
Interestingly, the degeneracy factor exactly cancels theN 1 !(N−N 1 )!/N! appearing in
eqn. (6.4.4). This cancellation is not unexpected since, as we recall, the latter factor was
included as a “fudge factor” to correct for the fact that classical particles are always
distinguishable, and we need our results to be consistent with the indistinguishable
nature of the particles (recall Section 3.5.1). Thus, allNconfigurations in which one
particle is in system 1 are physically the same, and so forth. Inserting eqn. (6.4.5) into
eqn. (6.4.4) gives
Q(N,V,T) =
∑N
N 1 =0Q 1 (N 1 ,V 1 ,T)Q 2 (N−N 1 ,V−V 1 ,T). (6.4.7)
Now the total phase space distribution functionf(x,N) =e−βH(x,N)
N!h^3 NQ(N,V,T)(6.4.8)
satisfies the normalization condition
∫
dxf(x,N) = 1, (6.4.9)
since it is just a canonical distribution. However, the phase space distribution of system
1, obtained by integrating over x 2 according to