Statistical Mechanics - Physics at Oregon State University

7.8. EFFECTS OF THE POTENTIAL ENERGY. 153
In a more general case, the Hamiltonian for the classical particles contains
a potential energy U, and we have
H =
N
i=1
p 2 i
2m + U(r1, r2, · · · , rN ) (7.44)
In many cases the potential energy obeys a given scaling relation. For example,
assume that
U(λr1, · · · , λrN) = λ γ U(r1, · · · , rN) (7.45)
If the interactions are pure Coulomb interactions, then γ = −1. The partition
function in this case is
Z =
1
N!h3N
With β = 1
kBT
1
N!
Z = 1
N!
√ 2mkBT π
h
3N
p2
− 2mk dpe B T
this is equal to
√ 2mkBT π
h
3N
3N
V
· · ·
In other words, in terms of V and T we have
Z(T, V, N) = 1
N!
d 3 r1 · · · d 3 rNe − U(r 1 ,..,r N )
k B T (7.46)
d 3
r1 · · · d
V
3 rN e −U(β 1 γ r1,..,β 1 γ rN )
=
3N −
β γ
V β 3 d
γ
3
x1 · · ·
V β 3 γ
d 3 xN e −U(x1,···,xN )
√ 2mkBπ
h
3N
k 3N
γ
B
(7.47)
1 1 3N(
T 2 + γ ) 3 −
g(N, V T γ ) (7.48)
where g(N,x) is the result of the integration over all spatial coordinates. The
Helmholtz free energy F = −kBT log Z has therefore only one term which
depends on volume:
√
2mkBπ
F (T, V, N) = kBT log(N!) − 3NkBT log
−
h
3NkBT
log(kB)−
γ
3NkBT ( 1 1
+
2 γ ) log(T ) − kBT
3 −
log(g(N, V T γ )) (7.49)
The pressure follows from −
∂F
∂V . The equation of state takes the general
N,T
form
3
3
−1+
pT γ −
= f(N, V T γ ) (7.50)