PHYSICS 140B : STATISTICAL PHYSICS
HW ASSIGNMENT #8 SOLUTIONS
(1) Consider a monatomic ideal gas in the presence of a temperature gradient T . Answer
the following questions within the framework of the relaxation time approximation to the
Boltzmann equati
PHYSICS 140B : STATISTICAL PHYSICS
HW ASSIGNMENT #2 SOLUTIONS
(1) For each of the cluster diagrams in Fig. 1, nd the symmetry factor s and write an
expression for the cluster integral b .
(a)
(b)
(c)
(d)
Figure 1: Cluster diagrams for problem 1.
Solution
PHYSICS 140B : STATISTICAL PHYSICS
HW ASSIGNMENT #3 SOLUTIONS
(1) How would you dene the pair distribution function gab (r ) for a molecular liquid? You
might wish to dene xa as the position of the ith atom of species a in molecule I and then
Ii
dene an a
PHYSICS 140B : STATISTICAL PHYSICS
HW ASSIGNMENT #1
(1) Consider a spin-1 Ising chain with Hamiltonian
H = J
Sn Sn+1 ,
n
where each Sn takes possible values cfw_1, 0, 1.
(a) Find the transfer matrix for the this model.
(b) Find an expression for the free
PHYSICS 140B : STATISTICAL PHYSICS
WINTER 2012 FINAL EXAM
(1) Consider an Ising model on a square lattice with Hamiltonian
H = J
S i j ,
iA j B
where the sum is over all nearest-neighbor pairs, such that i is on the A sublattice and j is
on the B sublatti
PHYSICS 140B : STATISTICAL PHYSICS FINAL EXAMINATION 100 POINTS TOTAL
(1) Consider a spin-2 Ising model with Hamiltonian
H = 1 2
i,j
Jij Si Sj H
i
Si
where Si cfw_2, 1, 0, 1, 2. The system is on a simple cubic lattice, with nearest neighbor coupling J1 /
PHYSICS 140B : STATISTICAL PHYSICS
WINTER 2012 FINAL EXAM SOLUTIONS
(1) Consider an Ising model on a square lattice with Hamiltonian
H = J
S i j ,
iA j B
where the sum is over all nearest-neighbor pairs, such that i is on the A sublattice and j is
on the
PHYSICS 140B : STATISTICAL PHYSICS
HW ASSIGNMENT #4 SOLUTIONS
(1) The Dieterici equation of state is
p (v b) = RT exp
a
vRT
.
(a) Find the critical point (pc , vc , Tc ) for this equation of state
(b) Writing p = p/pc , v = v/vc , and T = T /Tc , rewrite
PHYSICS 140B : STATISTICAL PHYSICS
HW ASSIGNMENT #5 SOLUTIONS
(1) Consider the S = 1 Ising Hamiltonian
H = J
zz
Si Sj H
z
Si +
i
ij
z
Si
2
,
i
z
where Si cfw_1, 0, +1.
(a) Making the mean eld approximation in the rst term, but treating the other terms
ex
PHYSICS 140B : STATISTICAL PHYSICS
WINTER 2012 MIDTERM EXAM SOLUTIONS
(1) Consider the Planck equation of state,
p=
RT
b
ln 1
b
v
a
,
v2
where a and b are constants, and R = NA kB is the gas constant.
(a) What are the dimensions of a and b?
Clearly [a] =
PHYSICS 140B : STATISTICAL PHYSICS
WINTER 2012 MIDTERM EXAM
(1) Consider the Planck equation of state,
p=
RT
b
ln 1
b
v
a
,
v2
where a and b are constants, and R = NA kB is the gas constant.
(a) What are the dimensions of a and b?
(b) Recall the virial e
PHYSICS 140B : STATISTICAL PHYSICS
MIDTERM EXAM
9:00 am 9:50 am / Friday Feb. 13 / WLH 2008
Consider a four-state ferromagnetic Ising model with the Hamiltonian
H = J
Si Sj H
Si ,
i
ij
where the rst sum is over all links of a lattice of coordination numbe
PHYSICS 140B : STATISTICAL PHYSICS
MIDTERM EXAM SOLUTIONS
Consider a four-state ferromagnetic Ising model with the Hamiltonian
H = J
Si Sj H
Si ,
i
ij
where the rst sum is over all links of a lattice of coordination number z . The spin variables
Si take v
PHYSICS 140B : STATISTICAL PHYSICS
HW ASSIGNMENT #6 SOLUTIONS
(1) Consider a q -state Potts model on the body-centered cubic (BCC) lattice. The Hamiltonian is given by
H = J
i , j ,
ij
where i cfw_1, . . . , q on each site.
(a) Following the mean eld tre
PHYSICS 140B : STATISTICAL PHYSICS
HW ASSIGNMENT #7 SOLUTIONS
(1) The Hamiltonian for the three state (Z3 ) clock model is written
H = J
ij
ni nj ,
where each local unit vector ni can take one of three possible values:
n=x
n = 1x +
2
,
3
2y
,
1
n = 2x
3
PHYSICS 140B : STATISTICAL PHYSICS
FINAL EXAMINATION SOLUTIONS
100 POINTS TOTAL
(1) Consider a spin-2 Ising model with Hamiltonian
1
H = 2
i,j
Jij Si Sj H
Si
i
where Si cfw_2, 1, 0, 1, 2. The system is on a simple cubic lattice, with nearest neighbor
coup