PHYSICS 140B : STATISTICAL PHYSICS
WINTER 2012 MIDTERM EXAM
(1) Consider the Planck equation of state,
p=−
RT
b
a
ln 1 −
− 2 ,
b
v
v
where a and b are constants, and R = NA kB is the gas constant.
(a) What are the dimensions of a and b?
(b) Recall the virial expansion of the equation of state,
p = nkB T 1 + B2 n + B3 n2 + . . . ,
where n = NA /v is the number density. Find all the virial coefficients for the Planck
equation. You should treat Bj=2 differently from Bj>2 . Recall also that for |ε| < 1,
ln(1 + ε) = ε − 21 ε2 + 31 ε3 − . . . =
∞
X
εk
(−1)k−1
.
k
k=1
(c) Find the values vc , Tc , and pc at the critical point.
(2) The Hamiltonian for the four state (Z4 ) clock model is written
Ĥ = −J
X
n̂i · n̂j ,
hiji
where each local unit vector n̂i can take one of four possible values: n̂i ∈ {x̂, ŷ, −x̂, −ŷ}.
(a) Consider the Z4 clock model on a lattice of coordination number z. Make the mean
field assumption hn̂i i = mx̂. Expanding the Hamiltonian to linear order in the fluctuations, derive the mean field Hamiltonian for this model ĤMF .
(b) Rescaling θ = kB T /zJ and f = F/N zJ, where N is the number of sites, find f (m, θ).
(c) Find the mean field equation and the critical value θc .
(d) Is the transition second order or first order?
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