PHYSICS 140A : STATISTICAL PHYSICS
HW ASSIGNMENT #6
(1) Consider a monatomic ideal gas, represented within the grand canonical ensemble.
Show that the probability of finding the system to have N atoms is given by the Poisson
distribution,
1 −hN i
PN =
e
hN iN .
N!
(2) Derive the grand canonical distribution when there are several types of particles present.
(3) An ideal paramagnet is described by the model in §3.11 of the notes, i.e.
Ĥ = −µ0 H
N
X
σj ,
j=1
where each σj = ±1. Suppose the system starts off at a temperature T = 10 mK and a field
H = 20 T. The field is then lowered adiabatically to H = 1 T. What is the final temperature
of the system?
(4) Consider a nonrelativistic ideal gas. From dimensional analysis, we conclude that
|p|k = Ck (mkB T )k/2 .
Find the constants Ck . Use the OCE.
(5) Show that
CV = −kB β 2
∂2
βF
.
∂β 2
(6) Consider a three state system with energy levels at ε = 0, ε = ∆, and ε = W , with
0 ≤ ∆ ≤ W . Compute the free energy for such a system, f (T ). Derive an expression for
the heat capacity c(T ). You may find the results from problem (5) useful. Plot the specific
heat c(T ) versus kB T /∆ for W = ∆, W = 2∆, and W = 6∆.
1