Solutions Manual for Statistical Physics of Fields By Mehran Kardar
Solutions Manual for Statistical Physics of Fields By Mehran Kardar
,Solutions Manual for Statistical Physics of Fields By Mehran Kardar
Problems & Solutions
for
Statistical Physics of Fields
Updated July 2008
by
Mehran Kardar
Department of Physics
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139, USA
Solutions Manual for Statistical Physics of Fields By Mehran Kardar
,Solutions Manual for Statistical Physics of Fields By Mehran Kardar
Table of Contents
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 Manual for Statistical Physics of Fields By Mehran Kardar
, Solutions Manual for Statistical Physics of Fields By Mehran Kardar
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 temper-
ature?
• 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 · —JpA
2 — Jp2 + 2Jp p
B A B
2
NA — NB
= —3JN .
N
(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
N!
S = kB ln .
NA!NB!
Using Stirling’s approximation for large N (ln N ! ≈ N ln N — N ), the above expression can
be written as
S ≈ kB (N ln N — NA ln NA — NB ln NB) = —NkB (pA ln pA + pB ln pB) .
1
Solutions Manual for Statistical Physics of Fields By Mehran Kardar
,Solutions Manual for Statistical Physics of Fields By Mehran Kardar
Problems & Solutions
for
Statistical Physics of Fields
Updated July 2008
by
Mehran Kardar
Department of Physics
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139, USA
Solutions Manual for Statistical Physics of Fields By Mehran Kardar
,Solutions Manual for Statistical Physics of Fields By Mehran Kardar
Table of Contents
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 Manual for Statistical Physics of Fields By Mehran Kardar
, Solutions Manual for Statistical Physics of Fields By Mehran Kardar
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 temper-
ature?
• 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 · —JpA
2 — Jp2 + 2Jp p
B A B
2
NA — NB
= —3JN .
N
(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
N!
S = kB ln .
NA!NB!
Using Stirling’s approximation for large N (ln N ! ≈ N ln N — N ), the above expression can
be written as
S ≈ kB (N ln N — NA ln NA — NB ln NB) = —NkB (pA ln pA + pB ln pB) .
1
