Spring 2006
PHY 203: Introduction to Statistical Mechanics
Assignment #7
Due at the beginning of class on Wednesday, 3/29/06
Problems:
1. Deduce the virial expansion
∞
X
PV
=
al
nkT
l=1
λ3
v
!l−1
using the expressions:
1
P
= 3 g5/2 (z)
kT
λ
and
N − N0
1
= 3 g3/2 (z)
V
λ
with λ the thermal wavelength and g(z) the Bose Einstein functions. Verify the quoted
values of the virial coefficients – it is sufficient to find the first three terms, a1 , a2 , and
a3 . [30 points]
2. Consider an ideal Bose gas in the grand canonical ensemble and study fluctuations
in the total number of particles N and total energy E. Discuss, in particular, the
situation when the gas becomes highly degenrate. Use the following definitions:
∂hN i
∂µ
2
h(N − hN i) i = kT
2
h(E − hEi) i = kT
[20 points]
1
2
∂U
∂T
!
≡ (∆N )2
T,V
!
≡ (∆E)2
z,V