Spring 2006
PHY 203: Introduction to Statistical Mechanics
Assignment #8
Due at the beginning of class on Wednesday, 4/05/06
Problems:
1. Isentropic expansion of a photon gas
Consider the gas of photons of the thermal equilibrium radiation in a cube of volume V at
temperature T . Let the cavity volume increase; the radiation pressure performs work during
the expansion, and the temperature of the radiation will drop. From the result for the entropy
we know that T V 1/3 is constant in such an expansion.
(a) Assume that the temperature of the cosmic black-body radiation was decoupled from
the temperature of the matter when both were at 3000 K. (The current temperature
of the photon gas in the universe – the cosmic microwave background is about 2.8K.)
What was the radius of the universe at that time, compared to now? If the radius has
increased linearly with time, at what fraction of the present age of the universe did the
decoupling take place? [5 points]
(b) Show that the work done by the photons during the expansion is
W = (π 2 /15h̄3 c3 )Vi (kTi )3 k(Ti − Tf ),
where the subscripts refer to the initial and final states. [10 points]
NOTE: The temperature of decoupling is lower than we would naively estimate for the transition from a gas of electrons and protons to a gas of neutral atoms because our estimate of
kT equal to the binding energy of hydrogen does not take into account the establishment of
equilibrium in the full system consisting of hydrogen, electrons, protons, and the photon gas.
See, for example, Chapter 7, Section (i) of Physical Cosmology by P.J.E. Peebles.
2. Ideal Bose gas in Flatland
Consider an ideal Bose gas confined to a region of area A in two dimensions.
(a) Express the number of particles in the excited states, Ne , and the number of particles in
the ground state, N0 , in terms of z, T and A, and show that the system does not exhibit
Bose-Einstein condensation unless T → 0 K. [15 points]
(b) Refine your argument to show that, if the area A and the total number of particles N
are held fixed and we require both Ne and N0 to of order N , we do achieve condensation
when
1
h2
T ≡
2
mkl ln N
p
where l ∼ A/N is the mean interparticle distance in the system. Of course, if both A
and N → ∞, keeping l fixed, then the desired T does go to zero. [10 points]
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3. Ideal Fermi gas in Flatland
Show that, in two dimensions, the specific heat CV (N, T ) of an ideal Fermi gas is identical
with the specific heat of an ideal Bose gas, for all N and T . [20 points]
Hint: It will suffice to show that, for given N and T , the thermal energies of the two system
differ at most by a constant. For this, first show that the fugacities, zF and zB , of the two
systems are mutually related:
(1 + zF )(1 − zB ) = 1
i.e. zB =
zF
1 + zF
Next, show that the functions f2 (zF ) and g2 (zB ) are also related:
ZzF
f2 (zF ) =
0
ln(1 + z
zF
dz = g2
z
1 + zF
+
1 2
ln (1 + zF )
2
It is now straightforward to show that EF (N, T ) = EB (N, T ) + const, the constant being
EF (N, 0).
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