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PHYSICS 210A : STATISTICAL PHYSICS
HW ASSIGNMENT #5
(1) Consider a system with single particle density of states g(ε) = A ε Θ(ε) Θ(W −ε), which
is linear on the interval [0, W ] and vanishes outside this interval. Find the second virial
coefficient for both bosons and fermions. Plot your results as a function of dimensionless
temperature t = kB T /W .
(2) Consider a two-dimensional system with dispersion ε(k) = A|k|3/2 obeying photon
statistics.
(a) Derive the analog of Stefan’s law.
(b) In an infinite two-dimensional plane, consider a circular region of radius R0 centered
at the origin. Its surface temperature is T0 . Find the steady-state surface temperature
of a circular region of radius R1 whose center lies a distance a from the origin.
(3) Consider an infinite linear chain of identical atoms described by the potential energy
function
U=
∞
X
1
K(n
4
′
n,n =−∞
− n′ ) un − un′
2
,
where K(n − n′ ) = K(n′ − n) depends only on the relative distance |n − n′ |. Find the
phonon dispersion and examine its long wavelength limit. Show that if K(m) ∝ |m|−p for
large |m| then the long-wavelength dispersion in the vicinity of the zone center k = 0 is
linear in the crystal momentum k if p > 3 but for 1 < p < 3 one has ω(k) ∝ |k|(p−1)/2 .
(4) Consider a set of N noninteracting S =
1
2
fermions in a one-dimensional harmonic
oscillator potential. The oscillator frequency is ω. For kB T ≪ ~ω, find the lowest order
nontrivial contribution to the heat capacity C(T ), using the ordinary canonical ensemble.
The calculation depends on whether N is even or odd, so be careful! Then repeat your
calculation for S = 32 .
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