Description
1. Find the second virial coefficient for a three-dimensional, non-relativistic, non-interacting
Fermi gas. [Hint: remember for an ideal Bose gas we have PV/T = Vv−1
Q
g5/2(e
βµ).]
2. Given a system with the two-particle (or density-density) correlation function
g(R) = 1 + Ae−R/`
, (1)
where A, and ` are constants, find the number fluctuations of the system,
h∆N2
i
hNi
=
hN2 − hNi
2
i
hNi
.
What is h∆N2
i/N for a Boltzmann gas? What is h∆N2
i/N for a non-interacting Fermi
gas? [Calculate these directly from the grand canonical distribution function] Qualitatively, what does this tell us about spatial correlations in a Fermi gas?
3. Consider a van Der Waals gas with equation of state
P =
T
v − b
−
a
v
2
,
where a and b are constants and v = V/N. We want to think about the properties of the
liquid-gas phase transition in the low temperature limit when the gas phase behaves
like an ideal gas and the specific volume of the high-density phase approaches the
density b so that we can consider only the first order correction in T. The first four
parts of the problem derive some general properties of the van der Waals gas and the
liquid-gas phase transition it undergoes, while the last three consider the properties of
this transition in the low temperature limit.
(a) Derive the Maxwell relation
∂P
∂T
N,V
=
∂S
∂V
T,N
which also implies
T
∂P
∂T
N,V
= P +
∂E
∂V
N,T
.
(b) Find the difference in energies per particle at fixed temperature between specific
volumes v1 and v2
e(T, v1) − e(T, v2),
using the result from part (a).
(c) Find the difference in entropy per particle at fixed temperature between specific
volumes v1 and v2
s(T, v1) − s(T, v2),
using the result from part (a).
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(d) Find the conditions for liquid-gas phase coexistence. Stated another way, find the
pressure of the both phases in terms of T, the specific volume in the gas phase,
vg, and the specific volume in the liquid phase, vl
.
(e) At low temperature at the liquid-gas phase transition, the gas phase is at very
low density and the liquid phase is nearly at the maximum density, 1/b. Stated
in terms of the specific volumes, vg vl ∼ b and vg
√
a. In this limit, pressure
equality of the two phases gives the liquid specific volume to first order in T as
vl = b + Tb2/a. In this limit, show that the specific volume of the gas phase is
approximately given by
vg ≈
Tb2
a
exp a
Tb
(f) What is the latent heat across the phase transition at low temperature?
(g) What is dP/dT along the phase co-existence line in the low temperature limit?
4. Show that in the mean field approximation the magnetic susceptibility of the Ising
model is given by
χ =
∂M
∂B
T
= Nµ
2
B
1 − hσi
2
T − (1 − hσi
2)Tc
(2)
5. Consider the Ising ferromagnet in zero field, in the case where the spin can take three
values σ = −1, 0, 1.
(a) Find the equation for the mean field free energy.
(b) Find an implicit equation for the mean field magnetization.
(c) Find the critical temperature, is it lower or higher than the the σ = ±1 case?
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