PHYSICS 210A : STATISTICAL PHYSICS
HW ASSIGNMENT #7
(1) Study carefully example problem 1.8. Then consider the integral
1
F (λ) = √
2π
Z∞
1 2 λ 6
dx exp − x − x .
2
6!
−∞
(a) Find the coefficients in the perturbation expansion,
∞
X
F (λ) =
Cn λn .
n=0
(b) Compute the remainder after N terms, RN (λ), defined as
RN (λ) = F (λ) −
N
X
Cn λn .
n=0
Using Mathematica or some other numerical method, compute F (λ), RN (λ), and
plot the ratio SN (λ) = RN (λ)/F (λ) for N <
∼ 50 and for various values of λ. (See Fig.
1.3 from the example problems.)
(c) Discuss the diagrammatic interpretation of the perturbation expansion. Sketch some
low order diagrams and identify their symmetry factors.
(2) In HW problem 2.4, you considered the thermodynamic properties associated with the
grand partition function Ξ(V, z) = (1 + z)V /v0 1 + z αV /v0 . Consider now the following
partition function:
(
αV /Kv0 )
K
Y
z
Ξ(V, z) = (1 + z)V /v0
.
1+
σj
j=1
Consider the thermodynamic limit where α is a number on the order of unity, V /v0 → ∞,
and K → ∞ but with Kv0 /V → 0. For example, we might have K ∝ (V /v0 )1/2 .
(a) Show that the number density is
α
1 z
+
n(T, z) =
v0 1 + z v0
Z|z|
dσ g(σ) ,
0
where
K
1 X
g(σ) =
δ(σ − σj ) .
K
j=1
1
(b) Derive the corresponding expression for p(T, z).
(c) In the thermodynamic limit, the spacing between consecutive σj values becomes
infinitesimal. In this case, g(σ) approaches a continuous distribution. Consider the
flat distribution,
(
w−1 if r < σ < r + w
1
g(σ) = Θ(σ − r) Θ(r + w − σ) =
w
0
otherwise.
The model now involves three dimensionless parameters1 : α, r, and w. Solve for
z(v). You will have to take cases, and you should find there are three regimes to
consider2 .
(d) Plot pv0 /kB T versus v/v0 for the case α =
1
4
and r = w = 1.
(e) Comment on the critical properties (i.e. the singularities) of the equation of state.
(3) For the van der Waals interaction,
( 6 )
σ
σ 12
−
,
v(r) = 4ǫ
r
r
compute the second virial coefficient B2 (T ), expressed as a power series in 4ǫ/kB T . Start
d
by writing d3r = 4π dr r 2 with r 2 = 31 dr
(r 3 ), and then integrate by parts, applying the
6
d
in a Taylor
to the Mayer function. Then expand exp k4ǫT σr
differential operator dr
B
series, and make an appropriate change of variables so you can do the integrals. Evaluate
the first ten expansion coefficients in B2 .
(4) Find an expression for the screened potential of a test charge Q in a two-dimensional
system using an appropriate generalization of Debye-H
ückel theory. The unscreened in
terparticle potential is v(r, r ′ ) = −2qq ′ ln |r − r ′ |/a , where a is a constant. Assume two
species of charge, with q = ±e, for the plasma.
1
2
The quantity v0 has dimensions of volume and disappears from the problem if one defines ṽ = v/v0 .
You should find that a fourth regime, v < (1 + r −1 )v0 , is not permitted.
2