Phase space and partition function 269N=N 1 +N 2 , V=V 1 +V 2. (6.4.1)
In order to carry out the derivation of the ensemble distribution function, we will need
to consider explicitly the dependence of the Hamiltonian on particle number, usually
appearing as the upper limit of sums in the kinetic and potential energies. Therefore,
letH 1 (x 1 ,N 1 ) be the Hamiltonian of system 1 andH(x 2 ,N 2 ) be the Hamiltonian of
system 2. As usual, we will take the total Hamiltonian to be
H(x,N) =H 1 (x 1 ,N 1 ) +H 2 (x 2 ,N 2 ). (6.4.2)Consider first the simpler case in which systems 1 and 2 do not exchange particles.
The overall canonical partition function in this limit is
Q(N,V,T) =
1
N!h^3 N∫
dx 1∫
dx 2 e−β[H^1 (x^1 ,N^1 )+H^2 (x^2 ,N^2 )]=
N 1 !N 2!
N!
1
N 1 !h^3 N^1∫
dx 1 e−βH^1 (x^1 ,N^1 )1
N 2 !h^3 N^2∫
dx 2 e−βH^2 (x^2 ,N^2 )=
N 1 !N 2!
N!
Q 1 (N 1 ,V 1 ,T)Q 2 (N 2 ,V 2 ,T), (6.4.3)
whereQ 1 (N 1 ,V 1 ,T) andQ 2 (N 2 ,V 2 ,T) are the canonical partition functions of systems
1 and 2, respectively, at the common temperatureT.
When the systems are allowed to exchange particles, the right side of eqn. (6.4.3)
represents one specific choice ofN 1 particles for system 1 andN 2 =N−N 1 particles
for systems 2. In order to account for particle number variationsin systems 1 and 2, the
true partition function must contain a sum over all possible values ofN 1 andN 2 on the
right side of eqn. (6.4.3) subject to the restriction thatN 1 +N 2 =N. This restriction
is accounted for by summing onlyN 1 orN 2 over the range [0,N]. For concreteness,
we will carry out the sum overN 1 and setN 2 =N−N 1. Additionally, we need to
weight each term in the sum by a degeneracy factorg(N 1 ,N 2 ) =g(N 1 ,N−N 1 ) that
accounts for the number of distinct configurations that exist forparticular values of
N 1 andN 2. Thus, the partition function for varying particle numbers is
Q(N,V,T) =
∑N
N 1 =0g(N 1 ,N−N 1 )N 1 !(N−N 1 )!
N!
×Q 1 (N 1 ,V 1 ,T)Q 2 (N−N 1 ,V−V 1 ,T), (6.4.4)
where we have used the fact thatV 1 +V 2 =V.
We now determine the degeneracy factorg(N 1 ,N−N 1 ). For theN 1 = 0 term,
g(0,N) represents the number of ways in which system 1 can have 0 particles and
system 2 can have allNparticles. There is only one way to create such a configuration,
henceg(0,N) = 1. ForN 1 = 1,g(1,N−1) represents the number of ways in which
system 1 can have one particle and system 2 can have (N−1) particles. Since there are
Nways to choose that one particle to place in system 1, it follows thatg(1,N−1) =N.