PHYSICS 140A : STATISTICAL PHYSICS
PRACTICE FINAL EXAM
105 POINTS TOTAL
(1) A pair of spins is described by the Hamiltonian
ĥ = −J S σ − µ0 H (S + σ) ,
where S takes values −1, 0, or +1 (three possibilities) and σ takes values ±1 but not 0 (two
possibilities).
(a) Find the partition function Z(T, H). [10 points]
(b) Find the magnetization M (T, H). [10 points]
(c) Find the zero field magnetic susceptibility, χ(T ) = (∂M/∂H)H=0 . [10 points]
(d) Provide a physical interpretation of your result for χ(T ) in the limits J → 0 and
J → ∞. [5 points]
(2) A three-dimensional (d = 3) system has excitations with dispersion ε(k) = A |k|4/3 .
There is no internal degeneracy (g = 1), and the excitations are noninteracting.
(a) Find the density of states g(ε) for this excitation branch. [15 points]
(b) If the excitations obey photon statistics (i.e. µ = 0), find CV (T ). [10 points]
(c) If the excitations obey Bose-Einstein statistics, show that the system undergoes BoseEinstein condensation. Find the critical temperature Tc (n), where n is the number
density of the excitations. [10 points]
(3) Consider a two-dimensional (d = 2) gas of relativistic particles with dispersion
ε(p) =
p
p2 c2 + m2 c4 .
The particles are classical, i.e. they obey Maxwell-Boltzmann statistics.
(a) Find the single particle partition function ζ(T, A) (A is the area of the system). You
may find the substitution p = mc sinh θ to be helpful at some stage. [10 points]
(b) Find the entropy S(T, A, N ). [10 points]
(c) Find the heat capacity (at constant area) per particle, c̃ (T ) = CA /N . [10 points]
(d) Which is the Dulong-Petit limit: mc2 ≫ kB T or mc2 ≪ kB T , and why? Verify that
your c̃ (T ) has the correct asymptotic behavior. [5 points]
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