STATISTICAL MECHANICS
Ideal Bose Gas: blackbody radiation; Bose-Einstein condensation
9.1
Tutorial Sheet 9
The sun as a black body [S]
In lectures it was shown that u()d gives the energy density per unit volume of blackbody
radiation.
Find the maximum of
STATISTICAL MECHANICS
The Classical Limit
7.1
Tutorial Sheet 7
Maxwell velocity distribution [R,S]
Review the arguments leading to the Maxwell distribution for molecular velocities
M
P (vx , vy , vz ) =
2kT
3/2
2
2
2
eM (vx +vy +vz )/2kT
Deduce the molecu
STATISTICAL MECHANICS
Quantum statistical mechanics
8.1
Tutorial Sheet 8
Density in the isothermal atmosphere via the chemical potential [S]
Consider a thin layer of atmosphere at height z containing n(z) molecules and of volume
V . By treating the system
STATISTICAL MECHANICS
The Ideal Gas
6.1
Tutorial Sheet 6
Distinguishability and the role of the N! [S]
A container is divided by a partition into two equal regions of volume V , each holding N
identical molecules of gas, at temperature T . The partition i
STATISTICAL MECHANICS
The Boltzmann Distribution and Free Energy; Magnetism
4.1
Tutorial Sheet 4
How to divide and conquer [S]
Show that, if the energy of a system can be written as a sum of contributions associated with
dierent aspects (magnetic, vibrati
STATISTICAL MECHANICS
The Einstein Model of a Solid; Low density gases
5.1
Tutorial Sheet 5
Statistical mechanics of the 1d harmonic oscillator [R]
Review the argument leading to the result (for the mean energy of a 1d harmonic oscillator
of frequency )
1
STATISTICAL MECHANICS
Temperature and the Boltzmann Distribution
3.1
Tutorial Sheet 3
Equilibrium conditions [S]
Review the argument showing that two systems free to exchange energy must, in equilib1
S
rium, have a common value of T E . Extend the argumen
STATISTICAL MECHANICS
Microstates, Macrostates and the Boltzmann Law
2.1
Tutorial Sheet 2
Microstates and macrostates [K]
Consider the model magnet of N dipoles (see lecture 3) each of which may exist in either
of two states (orientations). For the case N
STATISTICAL MECHANICS
Revision of Probability and Probability Distributions
Tutorial Sheet 1
The questions that follow on this and succeeding sheets are an integral part of this course. Cross
references to the questions are given in the printed lecture no
Section 16: The Ideal Bose Gas (Baierlein Ch. 6, Mandl Ch. 10)
In this lecture we will be dealing with bosons for which the relevant distribution is the
Bose-Einstein distribution
n() = f () =
1
exp [( )] 1
Recall this gives the average number of bosons i
Section 17: Radiation Gas; Cosmic Background Radiation
Here we nish o our study of blackbody (cavity) radiation.
17. 1. Stefan-Boltzmann Law
Recall that the spatial energy density in the frequency range + d is given by
u() =
3
8h
c3 exp (h) 1
Consider the
Section 18: Ideal Bose Gas at low T
18. 1. Bose-Einstein Condensation (Baierlein 9.4)
Finally we return to the ideal Bose gas with non-zero (i.e. the (average) number of bosons
is xed and we have to choose to give the correct value for N). Recalling the B
Section 15: The Ideal Fermi Gas
In this lecture we will mainly be concerned with the Fermi-Dirac distribution from key point
17. First we recover classical results from the quantum gas distributions.
15. 1. Low density limit
Consider the limit
ni =
e/kT 1
Section 12: Classical Results; Review of Theory of Gases
In the last lecture we saw how the semi-classical approach to an ideal gas recovered the
ideal gas law and equipartition in the low density limit. Physically this corresponded to
dtyp > typ i.e.
1/3
Section 14: Quantum Gases
Last lecture the grand canonical distribution was obtained. It applies to a system in equilibrium with a reservoir of energy and particles. This kind of system is referred to as an open
system. The grand canonical distribution is
Section 13: Systems with Varying Particle Number
So far we have developed the Boltzmann distribution where the number of particles N is
xed and the energy is a free macroscopic variable. In this lecture we consider systems where
the particle number is als
Section 11: Ideal Gas in the Low Density Limit
In the last lecture the calculation of Z(1) for the ideal gas was begun. In this lecture we
nish the calculation and discuss when the semi-classical treatment is valid. The concept of
a density of states is a
Section 9: Einstein Model of a Crystalline Solid
In this section we apply the Boltzmann distribution to the vibrational energy of atoms in a
solid. In so doing we use a fundamental model of quantum physicsthe quantum harmonic
oscillator. The material of t
Section 10: Ideal Gas and Indistinguishability
So far we have applied the Boltzmann theory for weakly interacting systems to solids and
their magnetic and vibrational properties. We now turn to the other example of a weakly
interacting system mentioned ba
Section 2: Basic distributions (Reading: second Year notes)
2. 1. The binomial distribution
First let us review the concepts of permutations and combinations. If we have N distinguishable objectssay numbered balls, or a pack of cardsthen if we select n ob
Section 8: More on Magnetism; Review
In this lecture we tie up some loose ends and take stock.
8. 1. Paramagnetism and ferromagnetism
So far we have considered the model magnet without really explaining why such a simple
model is a good caricature of the
Section 7: Systems of Weakly Interacting Constituents
In the previous lectures we have developed the Boltzmann distribution (lecture 5). We have
seen in lecture 6 how for large N thermodynamic variables become sharply dened and can
be obtained from the pa
Section 5: The Boltzmann Distribution
(Note: Baierlein covers the material of this lecture in chapter 5, but he has already discussed
the density of states in his chapter 4 which we shall not deal with until a bit later. Mandl
2.5 covers the same ground a
Section 4: Entropy, Equilibrium & the Second Law
At the end of the last lecture we introduced the Boltzmann Law as a denition of entropy.
In this lecture we shall try and understand the signicance of the Boltzmann Law and how
it explains the second law of
Section 1: Introduction and Revision
1. 1. Overview (reading Mandl 1.1)
Thermal Physics encompasses all parts of physics where ideas of temperature and entropy
come into play. As we shall see this implies the properties of macroscopic systems with a
large
Section 6: The Boltzmann Distribution ctd: Free energy minimisation Mandl 2.6,
Baierlein chapter 5
Last lecture we derived the Boltzmann distribution for a system in equilibrium with a heat
bath at temperature T and we looked at a simple example of a one
Section 3: Microstates and Macrostates
In the next two lectures we will develop the basic ideas of statistical mechanics in the context
of an isolated system. This will enable us to understand why macroscopic properties take on
their equilibrium values, w
Statistical Mechanics
Lecturer: Dr P. Monthoux, 3.3803 CSEC, [email protected]
Time: Tuesdays and Fridays 9.00am
Location: JCMB Lecture Theatre C
Web page: http:/www.ph.ed.ac.uk/pmonthou/Statistical-Mechanics
This course provides an introduction to the