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Physics 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu  Spring 2005
Problem Set 5
Due Thursday, March 31, in lecture
 Problem 1 [10 points total]
Consider photons of a given energy = .
(a) If <n> is the average number of such photons in equilibrium
at temperature T, show that the fluctuation in the number of photons is
<n^{2}>  <n>^{2} =  (1/) (d<n>/d_{})
where _{} = 1/k_{B}T
[5 pts]
(b) Using the forumula for the equilibrium value of <n>,
apply the above result to determine the relative fluctuation in the number
of photons
[<n^{2}>  <n>^{2}]/<n>^{2}
Is this large or small? [5 pts]
 Problem 2 [5 points]
The StefanBoltzmann law states that a black body at
temperature T radiates power per unit surface area equal to
T^{4} where is a universal constant
independent of the material properties of the body.
Assuming that the sun and the earth are black bodies,
and that the earth is in thermal equilibrium with the
sun [i.e. energy absorbed = energy emitted] calclate
the temperature of the earth in terms of the temperature
of the sun. Look up the parameters you need in order
to compute a number for this estimate of earth's temperature.
 Problem 3 [25 points]
In the grand canonical ensemble, the probability to have a given state "a"
with total energy E_{a} and total number of particles N_{a}
is,
P_{a} =
[e^{(EaµNa)/kBT}]/L
where
L = _{a} [e^{(EaµNa)/kBT}]
is the grand canonical partition function.
(a) For a quantum ideal gas, with single particle states i of energy
_{i}, many particle states are specified by the occupation
numbers {n_{i}} and have energy E = _{i}
[_{i}n_{i}]. Show that the probability for the state with
occupations {n_{i}} is given by
P({n_{i}}) = _{i} [p_{i}(n_{i})]
where p_{i}(n_{i}) is the probability that single particle
state i has occupation n_{i}
p_{i}(n_{i}) =
[e^{(iµ)ni/kBT}]/w_{i}
where
w_{i} = _{ni} [e^{(iµ)ni/kBT}]
can be thought of as the partition function
for the single particle state i.
The above factorization says that the number of particles n_{i}
in state i, is independent of the number of particles
n_{j} in state j. [6 points]
(b) Using the above result, show that the Shannon definition of entropy can
be written as
S = k_{B} _{{ni}} [P({n_{i}}) ln
P({n_{i}})] = k_{B} _{i}
_{ni} [p_{i}(n_{i}) ln
p_{i}(n_{i})]
[7 points]
(c) Using the above result, show that the following expressions apply for
the entropies of an ideal gas of bosons and fermions, respectively
bosons: 
S = k_{B} _{i} [(1+<n_{i}>)
ln (1+<n_{i}>)  <n_{i}> ln
<n_{i}>] 
fermions: 
S = k_{B} _{i} [(1<n_{i}>)
ln (1<n_{i}>)  <n_{i}> ln
<n_{i}>] 
where <n_{i}> = _{ni}
[n_{i} p_{i}(n_{i})] is the average occupation
number of state i.
[12 points]
