Statistical Mechanics - Physics at Oregon State University

54 CHAPTER 3. VARIABLE NUMBER OF PARTICLES
ɛs − µns
S = kB
kBT P rob(s) + kB log(Z)
P rob(s) (3.56)
s
where we used the fact that the sum of all probabilities is one. This then leads
to:
µns − ɛs
S = −kB
kBT − kB
log(Z) P rob(s) (3.57)
or
s
S = −kB P rob(s) log(P rob(s)) (3.58)
s
just as we had in the canonical ensemble. Note, however, that the similarity is
deceptive. The summation is now over a much larger set of states, including
all possible values of N. So how can the result be the same? The answer is
found easily when we introduce the independent variables. The result of 3.58
is S(T, V, µ), while for the canonical case we get S(T, V, N). Like in thermodynamics,
we are doing calculus of many variables, and keeping track of the
independent variables is again important!
The fluctuations in the number of particles follow from a calculation of the
average square of the number of particles, just as in the case of the energy in
the canonical ensemble. We start with
< N 2 >=
n 2 sP rob(s) (3.59)
−2 1
which is equal to β Z
we find
or
∂N
∂µ T,V
kBT
2
∂ Z
∂µ 2
s
. Using with < N >=
−1 1
= β
Z
∂N
∂µ T,V
2 ∂ Z
∂µ 2
−1 1
− β
Z2 s
s
−1 1
nsP rob(s) = β Z
2 ∂Z
∂µ
=< n 2 s > − < ns > 2 = (∆N) 2
∂Z
∂µ
(3.60)
(3.61)
All the partial derivatives are at constant T and V. This formula shows that N is
a monotonically increasing function of µ, and hence µ a monotonically increasing
function of N, at constant T and V. This also followed from the stability criteria
in thermodynamics. The quantity is a response function, just like
∂N
∂µ
CV . The formula above again shows that this type of response function is
proportional to the square root of the fluctuations in the related state variable.
If the fluctuations in the number of particles in a given system are large, we
need only a small change in chemical potential to modify the average number
of particles. In other words, if intrinsic variations in N at a given value of µ
T,V