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PHY 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu  Spring 2013
Problem Set 6
Due Wednesday, April 17, in lecture
 Problem 1 [5 points]
The StefanBoltzmann law states that a black body at temperature T radiates power per unit surface area equal to σT^{4} where σ 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 the earth's temperature.
 Problem 2 [30 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 ("a" is labeling the full Nparticle state),
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 ε_{i}, many particle states are specified by the occupation numbers {n_{i}} and have energy E = ∑_{i} [ε_{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}, and p_{i}(n_{i}) is given by
p_{i}(n_{i}) = [e^{(εiµ)ni/kBT}]/w_{i}
where
w_{i} = ∑_{ni} [e^{(εiµ)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.
(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})]
(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.
 Problem 3 [30 points]
Consider an ideal quantum gas of noninteracting identical particles in the grand canonical ensemble. The gas is in equilibrium at temperature T and chemical potential μ. In the previous problem part (a) you found the number of particles n_{i} that occupy the singleparticle energy eigenstate i is statistically independent of the number of particles n_{j} in that occupy state j, and you found the probability distribution p_{i}(n_{i}) that there are exactly n_{i} particles in state i.
a) Using this p_{i}(n_{i}) rederive the average occupation number of particles <n_{i}> in state i, for a gas of bosons and a gas of fermions.
b) Using p_{i}(n_{i}), derive the occupation number fluctuations <n_{i}^{2}>  <n_{i}>^{2} for a gas of bosons and a gas of fermions.
c) If the total number of particles is N = ∑_{i}n_{i}, show that the fluctuation in N is give by,
<N^{2}>  <N>^{2} = ∑_{i} [ <n_{i}^{2}>  <n_{i}>^{2} ].
Are the flutucations in N for the quantum gas bigger or smaller than they are for the corresponding clasical gas, for a gas of bosons? for a gas of fermions?

