PHYSICS 140B : STATISTICAL PHYSICS
HW ASSIGNMENT #2
(1) Consider a model in which there are three possible states per site, which we can denote
by A, B, and V. The states A and B are for our purposes identical. The energies of A-A,
A-B, and B-B links are all identical and equal to W . The state V represents a vacancy, and
any link containing a vacancy, meaning A-V, B-V, or V-V, has energy 0.
(a) Suppose we write σ = +1 for A, σ = −1 for B, and σ = 0 for V. How would you write
a Hamiltonian for this system? Your result should be of the form
X
Ĥ =
E(σi , σj ) .
hiji
Find a simple and explicit function E(σ, σ 0 ) which yields the correct energy for each
possible bond configuration.
(b) Consider a triangle of three sites. Find the average total energy at temperature T .
There are 33 = 27 states for the triangle. You can just enumerate them all and find
the energies.
(c) For a one-dimensional ring of N sites, find the 3 × 3 transfer matrix R. Find the free
energy per site F (T, N )/N and the ground state entropy per site S(T, N )/N in the
N → ∞ limit for the cases W < 0 and W > 0. Interpret your results. The eigenvalue
equation for R factorizes, so you only have to solve a quadratic equation.
(2) Consider a two-state Ising model on a one-dimensional ring, where each site supports a
spin σn = ±1. The Hamiltonian is
Ĥ = −J
X 1 + σn−1 σn 1 + σn σn+1 2
n
(a) Show explicitly that Pσσ0 =
1
2
2
.
1 + σσ 0 is zero if σ 6= σ 0 and is unity if σ = σ 0 .
(b) Show that Ĥ is proportional to the number of consecutive triples of sites (n−1 , n , n+
1) for which σn−1 = σn = σn+1 .
(c) Calculate the partition function for a ring of N = 4 sites. There are 24 = 16 states of
the ring, so you can attack the problem by direct enumeration of all the possibilities.
Remember that the system is a ring.
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(3) For the Mayer cluster expansion, write down all possible unlabeled connected subgraphs
γ which contain four vertices. For your favorite of these animals, identify its symmetry factor
sγ , and write down the corresponding expression for the cluster integral bγ . For example,
for the diagram with four vertices the symmetry factor is s = 8 and the cluster integral
is
Z
Z
Z
Z
1
d
d
d
b =
d r1 d r2 d r3 ddr4 f (r12 ) f (r23 ) f (r34 ) f (r14 )
8V
Z
Z
Z
1
d
d
=
d r1 d r2 ddr3 f (r12 ) f (r23 ) f (r1 ) f (r3 ) .
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(You’ll have to choose a favorite other than .) If you’re really energetic, compute sγ and
bγ for all of the animals with four vertices.
(4) Explain physically, using pictures and/or drawing analogies to other common knowledge
(of upper division physics majors) why the pair distribution function g(r) should have the
features it does in fig. 5.7 of the notes.
(5) An ionic solution of dielectric constant and mean ionic density n fills a grounded
conducting sphere of radius R. A charge Q lies at the center of the sphere. Calculate the
ionic charge density as a function of the radial coordinate r, assuming Q/r kB T .
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