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