PHYSICS 210A : STATISTICAL PHYSICS
HW ASSIGNMENT #9 SOLUTIONS
(1) Consider a two-state Ising model, with an added quantum dash of avor. You are
invited to investigate the transverse Ising model, whose Hamiltonian is written
H = J
z
i ,
xx
i j H
i
ij
where
PHYSICS 210A : STATISTICAL PHYSICS
HW ASSIGNMENT #5 SOLUTIONS
(1) For a noninteracting quantum system with single particle density of states g() = A r
(with 0), nd the rst three virial coefcients for bosons and for fermions.
Solution :
We have
(1)j 1 Cj (
PHYSICS 210A : STATISTICAL PHYSICS
HW ASSIGNMENT #4 SOLUTIONS
(1) A strange material obeys the equation of state E (S, V, N ) = a S 7 /V 4 N 2 , where a is a
dimensionful constant.
(a) What are the SI dimensions of a?
(b) Find the equation of state relati
PHYSICS 210A : STATISTICAL PHYSICS
HW ASSIGNMENT #3 SOLUTIONS
(1) Consider an ultrarelativistic ideal gas in three space dimensions. The dispersion is
(p) = pc.
(a) Find T , p, and within the microcanonical ensemble (variables S , V , N ).
(b) Find F , S
PHYSICS 210A : STATISTICAL PHYSICS
HW ASSIGNMENT #1
(1) Consider a system with K possible states | i , with i cfw_1, . . . , K , where the transition
rate Wij between any two states is the same, with Wij = > 0.
(a) Find the matrix ij governing the master
PHYSICS 210A : STATISTICAL PHYSICS
HW ASSIGNMENT #2 SOLUTIONS
(1) Compute the density of states D(E, V, N ) for a three-dimensional gas of particles with
Hamiltonian H = N A |pi |4 , where A is a constant. Find the entropy S (E, V, N ), the
i=1
Helmholtz
PHYSICS 210A : STATISTICAL PHYSICS
HW ASSIGNMENT #6 SOLUTIONS
(1) In our derivation of the low temperature phase of an ideal Bose condensate, we split
off the lowest energy state 0 but treated the remainder as a continuum, taking = 0 in
all expressions re
PHYSICS 210A : STATISTICAL PHYSICS
HW ASSIGNMENT #7 SOLUTIONS
(1) For each of the two cluster diagrams in Fig. 1, nd the symmetry factor s and write
an expression for the cluster integral b (T ).
Figure 1: Mayer cluster expansion diagrams.
Solution :
The
3 Approach to Equilibrium : Summary
Distributions: Equilibrium statistical mechanics describes systems of particles in terms
of time-independent statistical distributions. Where do these distributions come from?
How does a system with a given set of init
4 Statistical Ensembles : Summary
Distributions: Let () be a normalized distribution on phase space. Then
f () = Tr () f () =
d () f () ,
where d = W () i di is the phase space measure. For a Hamiltonian system of N
identical indistinguishable point part
2 Thermodynamics : Summary
Extensive and intensive variables: The equilibrium state of a thermodynamic system is characterized by specifying a number of state variables which can be either extensive (scaling linearly with system size), or intensive (scal
1 Probability Distributions : Summary
Discrete distributions: Let n label the distinct possible outcomes of a discrete random
process, and let pn be the probability for outcome n. Let A be a quantity which takes
values which depend on n, with An being th
PHYSICS 210A : STATISTICAL PHYSICS
HW ASSIGNMENT #8 SOLUTIONS
(1) Consider a ferromagnetic spin-S Ising model on a lattice of coordination number z .
The Hamiltonian is
H = J
i j 0 H
i ,
i
ij
where cfw_S, S + 1, . . . , +S with 2S Z.
(a) Find the mean el
PHYSICS 210A : STATISTICAL PHYSICS
FINAL EXAMINATION SOLUTIONS
All parts are worth 5 points each
(1) [40 points total] Consider a noninteracting gas of bosons in d dimensions. Let the
single particle dispersion be (k) = A |k| , where > 0.
(a) Find the sin