TEST BANK FOR Statistical Physics of Fields Solution Manual By Mehran Kardar

1. Collective Behavior, From Particles to Fields . . . . . . . . . . . . . . . . 1
2. Statistical Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3. Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4. The Scaling Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 55
5. Perturbative Renormalization Group . . . . . . . . . . . . . . . . . . . 63
6. Lattice Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7. Series Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8. Beyond Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Solutions to problems from chapter 1- Collective Behavior, From Particles to Fields
1. The binary alloy: A binary alloy (as in β brass) consists of NA atoms of type A, and
NB atoms of type B. The atoms form a simple cubic lattice, each interacting only with its
six nearest neighbors. Assume an attractive energy of −J (J > 0) between like neighbors
A − A and B − B, but a repulsive energy of +J for an A − B pair.
(a) What is the minimum energy configuration, or the state of the system at zero temperature?
• The minimum energy configuration has as little A-B bonds as possible. Thus, at zero
temperature atoms A and B phase separate, e.g. as indicated below.
A B
(b) Estimate the total interaction energy assuming that the atoms are randomly distributed
among the N sites; i.e. each site is occupied independently with probabilities pA = NA/N
and pB = NB/N.
• In a mixed state, the average energy is obtained from
E = (number of bonds) × (average bond energy)
= 3N ·
􀀀
−Jp2
A − Jp2
B + 2JpApB

= −3JN

NA − NB
N
2
.
(c) Estimate the mixing entropy of the alloy with the same approximation. Assume
NA,NB ≫ 1.
• From the number of ways of randomly mixing NA and NB particles, we obtain the
mixing entropy of
S = kB ln

N!
NA!NB!

.
Using Stirling’s approximation for large N (lnN! ≈ N lnN −N), the above expression can
be written as
S ≈ kB (N lnN − NA lnNA − NB lnNB) = −NkB (pA ln pA + pB ln pB) .
1
(d) Using the above, obtain a free energy function F(x), where x = (NA−NB)/N. Expand
F(x) to the fourth order in x, and show that the requirement of convexity of F breaks
down below a critical temperature Tc. For the remainder of this problem use the expansion
obtained in (d) in place of the full function F(x).
• In terms of x = pA − pB, the free energy can be written as
F = E − TS
= −3JNx2 + NkBT

1 + x
2

ln

1 + x
2

+

1 − x
2

ln

1 − x
2

.
Expanding about x = 0 to fourth order, gives
F ≃ −NkBT ln 2 + N

kBT
2 − 3J

x2 +
NkBT
12
x4.
Clearly, the second derivative of F,
∂2F
∂x2 = N (kBT − 6J) + NkBTx2,
becomes negative for T small enough. Upon decreasing the temperature, F becomes
concave first at x = 0, at a critical temperature Tc = 6J/kB.
(e) Sketch F(x) for T > Tc, T = Tc, and T < Tc. For T < Tc there is a range of
compositions x < |xsp(T)| where F(x) is not convex and hence the composition is locally
unstable. Find xsp(T).
• The function F (x) is concave if ∂2F/∂x2 < 0, i.e. if
x2 <

6J
kBT − 1

.
This occurs for T < Tc, at the spinodal line given by
xsp (T) =
r
6J
kBT − 1,
2
T>Tc
T=Tc
T<Tc
T=0
F(x)/NJ
x
-1 +1
xsp(T)
as indicated by the dashed line in the figure below.
(f) The alloy globally minimizes its free energy by separating into A rich and B rich phases
of compositions ±xeq(T), where xeq(T) minimizes the function F(x). Find xeq(T).
• Setting the first derivative of dF (x) /dx = Nx

(kBT − 6J) + kBTx2/3
    
, to zero yields
the equilibrium value of
xeq (T) =


±
√3
r
6J
kBT − 1 for T < Tc
0 for T > Tc
.
(g) In the (T, x) plane sketch the phase separation boundary ±xeq(T); and the so called
spinodal line ±xsp(T). (The spinodal line indicates onset of metastability and hysteresis
effects.)
• The spinodal and equilibrium curves are indicated in the figure above. In the interval
between the two curves, the system is locally stable, but globally unstable. The formation
of ordered regions in this regime requires nucleation, and is very slow. The dashed area is
locally unstable, and the system easily phase separates to regions rich in A and B.
********
2. The Ising model of magnetism: The local environment of an electron in a crystal
sometimes forces its spin to stay parallel or anti-parallel to a given lattice direction. As
a model of magnetism in such materials we denote the direction of the spin by a single
3
ÿ1 1 x
T
Tc
x