Statistical Mechanics - Physics at Oregon State University

52 CHAPTER 3. VARIABLE NUMBER OF PARTICLES
, this combination
is often abbreviated by defining β = 1
kBT . Using β simplifies many superscripts
to simple products in stead of ratios. Of course, this β should not be confused
with the critical exponent describing the disappearance of the order parameter
Since the temperature in many formulas appears as 1
kBT
near the critical temperature! Also, the chemical potential often occurs in the
combination e µ
k B T . This exponential is called the absolute activity (often de-
noted by λ = e µ
k B T ) or the fugacity (often denoted by z = e µ
k B T ). The notation
in the latter case is not the same in all books, unlike the conventional abbreviation
β, which is used almost uniformly. In terms of β, one finds immediately
that
∂Z
∂β µ,V
since U =< ɛs >. This gives
=
(µns − ɛs)e β(µns−ɛs)
= Z(µN − U) (3.41)
∂ log(Z)
U = µN −
∂β
s
µ,V
=
µ
β
∂ ∂
−
∂µ ∂β
log(Z) (3.42)
The recipe for calculations at specified values of µ, T, and V is now clear. Calculate
the grand partition function Z(µ, T, V ) and then the number of particles,
internal energy, etc. The construct the grand potential. There is a better and
direct way, however, as one might expect.
Since the grand potential is the correct measure of energy available to do
work at constant V, µ and T, it should be related to the grand partition function.
The correct formula can be guessed directly when we remember that:
∂Ω
∂µ T,V
= −N (3.43)
After comparing this partial derivative with the expression for N in 3.40 we
conclude that:
Ω = −kBT log(Z) + f(T, V ) (3.44)
where the function f(T,V) is still unknown. We will show that it is zero. This
last formula can be inverted to give an expression for log(Z). Inserting this
expression in the formula for U leads to
U =
µ
β
∂ ∂
−
∂µ ∂β
U = Ω − f − µ
µ
log(Z) =
β
∂ ∂
−
∂µ ∂β
(β(f − Ω)) (3.45)
U = Ω − f + µ ∂
∂
− β (f − Ω) (3.46)
∂µ ∂β
∂Ω
− β
∂µ β,V
∂f
+ β
∂β µ,V
∂Ω
∂β µ,V
(3.47)