Physics 262: Statistical Physics
Fall 2008
Problem Set 2 due: Wed Oct 1
Reading:
Parthria, Chapter 1; Lifshitz+Pitaevski, Chapters 1,2
1. The classical canonical ensemble is defined by the phase space density
ρ
=
1
Z
c
exp(

βH
)
(1)
where the Hamiltonian
H
is a function of the particle coordinates and momenta, and
Z
c
=
Z
d
3
N
pd
3
N
q
h
3
N
N
!
exp(

βH
)
(2)
By foliating the integral over equal energy surfaces, we argued in class that
Z
c
=
Z
∞
0
dE
Δ
exp (

β
(
E

TS
(
E
)))
≈
f
exp

β
(
E

T
S
)
(3)
where
S
is the entropy,
E
is the average energy, and
f
is given by
f
=
√
2
πk
B
T
2
C
V
Δ
(4)
(a) Establish that in the thermodynamic limit
lim
N
→∞
ln
f
N
= 0
(5)
(b) Using the above identity, show that in the thermodynamic limit
ln
Z
c
=

β
(
E

TS
)
(6)
holds exactly as
N
→ ∞
, with corrections which are smaller by a factor of ln
N/N
.
This is the reason the value of
f
is unimportant. Also, I have dropped the bars
over
E
, because the fluctuations of
E
are negligible, and
E
=
E
with probability
1 in the thermodynamic limit. Eq. (6) is a key result which we will use frequently
in this course.
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 Summer '19
 Physics, Work, Statistical Mechanics, Zc, ln Zc