1
formula
Sterling
a
From the
lg N
logx
graphical
following
N
I
n
N
Ig N
representation
Ely N
show
that
611
ÉÉÉÉ
b We have the
following
identity
f dx é
PINT 1
N
Pdxexp
Notethat we have an
the dominant
x
Nlyx dx
x
of
exponential
large number N
a
contribution comes from the peak
fix
x
dx
of
Nlyx
As an approximation expand near this peak
Then
N
fx
I
f Xo
I
I
exp I
I f
f exo
xo
as
x Xo
x xo5
eft
Use this idea to prove that
Ing N
N IngN
N
I ly ZIN
OCT
and
2 Microcanonical
ensemble
En
harmonic oscillators
of
nt of
Consider a microcanonical
ensemble
which has energy levels given
nhW
E
a
by
Showthat the
ne
Ew
each
subtracted
Z
Nt Elt w
me
oscillators
with zero point energy
microstates
of
given by
N harmonic
w
energy
E
is
1
N 1
b Use Sterling's formula to show that for NS 1
Ste
c
KB
n in
Express the inverse temperature
as a function
Can it be
d
wInNhÉ
E
EIN
negative
Ftw
V E
tÉ
PI't
of energy per oscillator
Show that the
VCE
Nh
average energy level
satisfies
1
the
Bose Einstein distribution
of