University of Edinburgh
School of Physics and Astronomy
Statistical Mechanics
Courses notes for Thermal Physics (PHYS09061) Semester 2
and Statistical Mechanics (PHYS09019)
Contributors: Martin Evans, Philippe Monthoux, Alexander Morozov & Richard Blythe
MY STUDY OF PHYSICS HELPS ME TO BETTER UNDERSTAND HOW THINGS WORK
IN THE REAL WORLD.
Kinematics and dynamics.
Physics deals with matter, energy, and their interactions or in other words; it can be described as
the study of matter and motion. This makes it
Lab 5 - Univariate Statistics with R
Tom Booth (minor revisions by James Stone)
It is nally time to do some stats!
So in todays lab we are going to start working with some data that looks a little more real. What do I
mean by real, well, messy! Up until n
Lab 3 - Univariate Statistics with R
Tom Booth (revisions by James Stone)
Some notes and observations from the labs so far
Everyone has made great progress so far and it is great for us to see students engaging with R and hopefully
seeing the utility of u
USMR Lab 8
1 A tiny bit more R!
1.1 ifelse
Todays lab is primarily focused on extending our knowledge/experience of conducting and interpreting
linear models. But before that, a brief note on if-else statements as you may have noticed examples
of them in
USMR Lab 7
Three dierent activities this week! These may require a bit of thinking; weve tried to include
enough information to help you nd solutions, but our solutions arent the only ones, obviously.
1 A brief note on
with()
You may have noticed a really
USMR Lab 6
November 5, 2015
This week well be merging data collected from a number of sources before running a simple
correlation and a linear model. You should end up knowing a bit more about data manipulation, as
well as more about running statistical t
Lab 4 - Univariate Statistics with R
Tom Booth (revisions by James Stone)
R skills: Writing functions and useful functions
Welcome to Lab 4
The primary purpose of this lab is to teach you some more advanced R skills. So far we have mainly considered
how t
Univariate Statistics with R: Lab 2
Friday 9th & Wednesday 14th October, 2015
Introduction
In this lab, you are going to begin to become accustomed to randomly generating data of dierent types
in R, generating appropriate plots for dierent types of data,
USMR Lab 9
This lab is a little less guided, and a little more like the take-home exam. The main
thing to remember when carrying out an exercise like this is to make it clear why you made the
decisions you made, either as a writeup or as comments (beginni
Modern Physics: Relativity
Introducing Workshops and Continuously-assessed Hand-ins
Welcome to the Modern Physics Workshops. These workshops are designed to help
you in several ways, so please make the most of them! The timetab le for the Modern
Physics P
Modern Physics Problem Sheet 5
Hand in Question
Hand in deadline 09:00 Monday 2 November 2015
5.1 Electron scattering
Note: This is question B.2. from the Modern Physics Dec 2012 exam paper.
A beam of electrons with kinetic energy 50 eV is red at a 100 nm
Modern Physics: Relativity
HAND IN QUESTION
Hand in deadline: 09.00 Monday 26th October
1. Faster than light neutrinos? Independent research expected
On the 22nd September 2011, the OPERA experiment announced a measurement indicating that neutrinos had tr
SOLUTIONS TO SHEET 1
Introducing Solution Sheets to Problems
Use these solutions with care! For this course, fairly complete solutions to all the tutorial problems will
be provided. Later in your studies you will nd some courses that provide far less comp
Modern Physics: Relativity
Note: Remember, only the rst question is for handing-in. The remainder provide the
material to be worked on in the Workshops. I recommend you look at these before
the Workshops - if you do youll get more out of them. Also, there
Modern Physics Solutions Sheet 3
Hand in Question
1) Past Exam Question: December 2012 B1
Past exam questions are an extremely valuable source of revision material. You should
know how to access them and be using them regularly.
The hand in problem for Mo
MODERN PHYSICS: SOLUTIONS TO SHEET 2
1. Contractionand Dilation sounds like pregnancy!
a) (i) Viewed from the laboratory, the pions will be time dilated. At a speed of 0.95c, one can calculate
a Lorentz factor of 3.20. The new characteristic decay time wi
Physics 2
Modern Physics: Relativity
HAND IN QUESTION
Hand in deadline: 09.00 Monday 26th October
1. Faster than light neutrinos? Independent research expected
On the 22nd September 2011, the OPERA experiment announced a measurement indicating that neutri
Modern Physics Problem Sheet 5
Hand in Question
Hand in deadline 09:00 Monday 2 November 2015
5.1
Electron scattering
Note: This is question B.2. from the Modern Physics Dec 2012 exam paper.
A beam of electrons with kinetic energy 50 eV is red at a 100 nm
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 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 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
FoMP: Probability & Statistics
1
Introduction
Probability theory allows us to make logical decisions in uncertain situations. This course
explains why probabilistic tools are essential both for theoretical and experimental physics.
It will also touch on t
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
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
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
Chapter 1
Getting Started with Matlab
The computational examples and exercises in this book have been computed using Matlab. Matlab is an interactive system designed specifically for scientific computation, and is used widely in universities and industry.
STATISTICAL MECHANICS
Microstates, Macrostates and the Boltzmann Law
2.1
Solutions: Tutorial Sheet 2
Microstates and Macrostates [K]
For the model magnet with N = 4 dipoles, there are 5 possible values of the energy,
corresponding to having n = 0, 1, 2, 3
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