PHYSICS 140B : STATISTICAL PHYSICS
WINTER 2012 FINAL EXAM
(1) Consider an Ising model on a square lattice with Hamiltonian
Ĥ = −J
X X′
S i σj ,
i∈A j∈B
where the sum is over all nearest-neighbor pairs, such that i is on the A sublattice and j is
on the B sublattice (this is the meaning of the prime on the j sum), as depicted in Fig. 1.
The A sublattice spins take values Si ∈ {−1, 0, +1}, while the B sublattice spins take values
σj ∈ {−1, +1}.
(a) Make the mean field assumptions hSi i = mA for i ∈ A and hσj i = mB for j ∈ B. Find
the mean field free energy F (T, N, mA , mB ). Adimensionalize as usual, writing θ ≡
kB T /zJ (with z = 4 for the square lattice) and f = F/zJN . Then write f (θ, mA , mB ).
[10 points]
Figure 1: The square lattice and its A and B sublattices.
(b) Write down the two mean field equations (one for mA and one for mB ). [10 points]
(c) Expand the free energy f (θ, mA , mB ) up to fourth order in the order parameters mA
and mB . You may find the following useful:
x2 x4
ln 2 cosh x = ln 2 +
−
+ O(x6 ) ,
2
12
x2 x4
ln 1 + 2 cosh x = ln 3 +
−
+ O(x6 ) .
3
36
[10 points]
(d) Show that the part of f (θ, mA , mB ) which is quadratic in mA and mB may be written
as a quadratic form, i.e.
M11 M12
mA
1
f (θ, mA, mB ) = f0 + 2 mA mB
+ O m4A , m4B ,
mB
M21 M22
where the matrix M is symmetric, with components Maa′ which depend on θ. The
critical temperature θc is identified as the largest value of θ for which det M (θ) = 0.
Find θc and explain why this is the correct protocol to determine it. [5 points]
1
(2) Consider a two-dimensional gas of particles with dispersion ε(k) = Jk2 , where k is the
wavevector. The particles obey photon statistics, so µ = 0 and the equilibrium distribution
is given by
1
.
f 0 (k) = ε(k)/k T
B
e
−1
(a) Writing f = f 0 + δf , solve for δf (k) using the steady state Boltzmann equation in the
relaxation time approximation,
v·
∂f 0
δf
=− .
∂r
τ
Work to lowest order in ∇T . Remember that v =
1 ∂ε
~ ∂k
is the velocity. [15 points]
(b) Show that j = −λ ∇T , and find an expression for λ. Represent any integrals you
cannot evaluate as dimensionless expressions. [10 points]
(c) Show that jε = −κ ∇T , and find an expression for κ. Represent any integrals you
cannot evaluate as dimensionless expressions. [10 points]
(3) Provide clear, accurate, and substantial answers for each of the following:
(a) For the cluster γ shown in Fig. 2, identify the symmetry factor sγ , the lowest order
virial coefficient Bj to which γ contributes, and write an expression for the cluster
integral bγ (T ) in terms of the Mayer function f (r). [6 points]
Figure 2: The connected cluster γ for problem 3a.
(b) Sketch what the pair distribution function g(r) should look like for a gas of hard
spheres of diameter a, and discuss its salient features. [6 points]
(c) What is the Maxwell construction? [6 points]
(d) Provide explicit examples of models which have a discrete and a continuous global
symmetry, and identify the respective symmetry groups. [6 points]
(e) Explain the principle of detailed balance. [6 points]
(4) Which two of Gustav Mahler’s symphonies open in the key of D major?
[1000 quatloos extra credit]
2