E XERCISES STATISTICAL MECHANICS , EXTRA
AUTUMN 2007
Consider a classical system of N identical particles, that is described by a Hamiltonian H . Assume that
we found an approximated Hamiltonian H0 , from which we can determine all statistical quantities.
E XERCISES STATISTICAL MECHANICS , WEEK 10
AUTUMN 2007
1. Exercise 3.14 (Alben model)
on the extra page.
2. Phase transitions in a magnetic system
A magnetic system with magnetization m can be subject to a homogeneous magnetic eld H .
For small values of
E XERCISES STATISTICAL MECHANICS , WEEK 11
AUTUMN 2007
1. Landau-Ginzburg theory
The Landau-Ginzburg Hamiltonian is given by
H =
r
K
r
[m(r)]2 + m2 (r) + um4 (r) .
2
2
For system of spins on a square lattice with lattice constant a, the gradient must be
VOLUME 84, NUMBER 13
PHYSICAL REVIEW LETTERS
27 MARCH 2000
High-Field Electrical Transport in Single-Wall Carbon Nanotubes
Zhen Yao,1 Charles L. Kane,2 and Cees Dekker1
1
Department of Applied Physics and DIMES, Delft University of Technology, Lorentzweg
E XERCISES STATISTICAL MECHANICS , WEEK 13
AUTUMN 2007
1. Bolzmann Equation
A particle with mass m, that can only move in a straight line 0 q L , experiences a gravitational force with potential V (q) = mgq; de collissions at q = 0 and q = L are elastic.
E XERCISES STATISTICAL MECHANICS ,
AUTUMN 2007
WEEK
8
1. Exercise 4.6.1 (High temperature expansion and Kramers-Wannier duality)
on the extra pages.
2. Relation between scattering cross section and structure factor
In this exercise we analyse the relation
E XERCISES STATISTICAL MECHANICS , WEEK 6
AUTUMN 2007
1. Tonks Gas
Consider a one dimensional gas of particles of length a conned to a strip of length L. The
particles interact through the potential
U (xi x j ) =
0
for |xi x j | < a and
for |xi x j | > a.
E XERCISES STATISTICAL MECHANICS , WEEK 7
AUTUMN 2007
1. Van der Waals equation of state
Consider a gas whose equation of state is given by
P+
a
(v b) = kT
v2
where v is the specic volume, v = V /N . This is the equation of state of a van der Waals gas.
I
E XERCISES STATISTICAL MECHANICS , WEEK 8
AUTUMN 2007
1. Exercise 5.7.7 (Superuidity for Hardcore Bosons) on the extra pages.
2. Perturbation Theory for Liquids
In this exercise we will investigate the ideas on which the modern perturbation theories of
li