Maxwell's demon
(drawing by
G. Gamow)

To play the good family doctor who warns about reading something prematurely, simply because it would be premature for him his whole life long — I’m not the man for that. And I find nothing more tactless and brutal than constantly trying to nail talented youth down to its ‘immaturity’, with every other sentence a ‘that's nothing for you yet’. Let him be the judge of that! Let him keep an eye out for how he manages.

Thomas Mann,
Doctor Faustus

Prof Alexander Schekochihin and Prof Julien Devriendt

What you see below will initially be lecture content, notes and downloads from the 2015-16 course.
These will be updated as the course progresses and in the case of essential downloads, the updates will be marked in RED.

So do please visit this website regularly if you wish to stay abreast of the latest developments.

The current version (23.02.2017) of the pdf file containing the typed version is linked here.
This file is being edited/corrected/amended as we proceed.
I will be very grateful for your feedback: comments, error corrections, views etc.

A sketch of students (or, perhaps, fellows) in a manuscript
of William of Ockham's commentary on Aristotle's
Physics (MS293 from the Merton College library,
image courtesy of J. Walwarth)

Michaelmas Term 2016
(skip to HT)
(skip to TT Revision Lectures)


Reading List
(notes, reading material and various other info are likely to appear here)

NB: The reading suggestions are only suggestions --- the way material is presented in those sources is not always identical to our exposition. See Reading List (above) for full book titles and publication info.

A sketch of students (or, perhaps, fellows) in a manuscript
of William of Ockham's commentary on Aristotle's
Physics (MS293 from the Merton College library,
image courtesy of J. Walwarth)
Lectures 1-7 will be given by Prof Julien Devriendt

You are ready to do Problem Sets 1 and 2

Lecture 1 (12:00 Wed 16.11.16) Statistical description of a gas. Energy. Thermodynamic limit. Kinetic calculation of pressure.

Reading: Blundell Sec 3 (a primer on probability), 6.1 (pressure)
Kardar Chapter 2 (a more advanced primer on probability)
Pauli Sec 24 (pressure)
Chapman & Cowling Chapters 2, 4 (much more advanced treatment)
Note: Please make sure you have a practical command of the notions from probability theory that I am using: random variables, their averages (means), probability density function, joint distribution, indeoendent random variables, change of variables for pdfs. This may require reading a book! (for example Sinai's book suggested in the right column) It is better to sort this out in your mind early, otherwise initial confusion will fester and undermine you ability to follow the rest of the course.

Related mathematics:
Probability and Statistics by S. Biller

A superb
(mathematical but accessible)
course on probability theory:
Ya. G. Sinai, Probability Theory:
An Introductory Course

An unorthodox but fascinating book
on probability:
E. T. Jaynes, Probability Theory:
The Logic of Science (CUP 2004)

A philosophical treatment of
what it all means by a thinker
now very much back in fashion:
J. M. Keynes, A Treatise on
Probability (1920, reprinted 2008)

Lecture 2 (12:00 Thu 17.11.16) Kinetic calculation of pressure cont'd. Particle distribution functions. Pressure vs. energy. Isotropic distributions. Classical ideal gas. Maxwell's distribution.

 Reading: Blundell Sec 5
Pauli Sec 25
Chapman & Cowling Chapters 2, 4 (much more advanced treatment)

Lecture 3 (12:00 Fri 18.11.16) Equation of state and temperature; heat capacity of monatomic ideal gas. Effusion. Collisions.
Reading: Blundell Sec 7 (effusion), 8 (collisions);
Pauli Sec 28 (effusion), 26 (collisions);
Chapman & Cowling Chapter 5 (much more advanced treatment)

A treatment of collisions
(and derivation of
from the original source:
L. Boltzmann,
Lectures on
Gas Theory
(Dover 1995)

You are ready to do Problem Set 3

Lecture 4 (12:00 Wed 23.11.16) Local Maxwellian equilibrium. Conservation laws: internal energy.

Reading: Blundell Sec 10

Lecture 5 (12:00 Thu 24.11.16) Conservation laws: momentum. Thermal conductivity and viscosity. Relaxation to global equilibrium. Dimensional estimates of the thermal conductivity and viscosity. Separation of scales between local and global relaxation.
Reading: Blundell Sec 10
Key thought #1: All diffusion processes are similar (diffusion of particles, momentum, energy...).
Key thought #2: Relaxation to equilibrium occurs in two stages: first to local equilibrium (on collision time scale, so very quickly), then to global equilibrium (diffusively, so slowly) --- both processes are collisional, but different in speed and in the nature of the physics involved.
Key thought #3: Transport equations are expressions of conservation laws. We use kinetic theory to calculate the fluxes of the conserved quantities (particle number, momentum, energy).

Note: Methods to solve the heat diffusion equation were covered in Mathematical Methods. There are some questions on this in Problem Set. Do go back and brush up your understanding of (i) how to find steady solutions given appropriate boundary conditions (temperature or heat flux on two boundaries), (ii) how to find time-dependent solutions given periodic boundary conditions (real frequencies, complex wavenumbers), (iii) how to find time-dependent solutions given initial conditions (real wavenumbers, complex frequencies) [see my typed notes or Blundell Sec 10 and Appendix C.12.].

Here are some notes on dimensional
analysis (from a summer school
in Oxford), with an elementary
treatment of a fun example
of the role of viscosity: how
do bubbles rise in a fluid?

On diffusion, learn from the master:
A. Einstein, Investigations on the Theory of
the Brownian Movement
(Dover 1956)
[the theory of Brownian motion will
be taught in the 3rd year, paper BI,
but there's nothing there that you can't
understand now;
see also Blundell Sec 33.1]

You are ready to start working on Problem Set 4 (vacation work)

Lecture 6 (12:00 Fri 25.11.16) Transport equations with sources and sinks; steady solutions for boundary-value problems. Kinetic derivation of the transport equations and transport coefficients (thermal conductivity and viscosity): the simplified "dodgy" derivation and its critique.

Reading: Blundell Sec 10
Blundell Sec 9 (the simplified derivation);
Pauli Sec 27 (another version of this);
Kardar Chapter 3 (more advanced than my treatment,

includes general derivation of Boltzmann's equation and various attendant matters,
in particular H theorem; treatment of fluid equations similar to mine);
Kittel Sec 40, 43;
Landau & Lifshitz-Kinetics Chapter 1 (hard-core Russian treatment);
Chapman & Cowling  Chapters 3, 6, 7, 9, 10 (hard-core Cambridge treatment)

Lecture 7 (12:00 Wed 30.11.16) Kinetic expressions for fluxes. Kinetic equation. Local conservation laws rederived: continuity and momentum equations.
Reading: See Lecture 6

Lecture 8 (12:00 Thu 1.11.16) Local conservation laws cont'd: energy equation. Collision operator. Solution of the kinetic equation.
Reading: See Lecture 6

Lecture 9 (12:00 Fri 2.12.16) Solution of the kinetic equation cont'd.  Calculation of fluxes.
Reading: See Lecture 6
What I have shown you is a very simplified version of this calculation. Do read more advanced texts and find out how it is really done!

I have posted an update to my lecture notes (4.12.16, not yet properly proof-read!), with some typos corrected and also with a new extracurricucular section on the kinetics of "Brownian praticles" --- a neat illustration of how to set up kinetic theory for another example of a particle system (slightly different from ideal gas). In this example, I will actually show how to derive an explicit collision operator and what to do with it.

If you liked Kinetic Theory
and can't wait till MMathPhys
to learn more, click here.

Hilary Term 2017


Reading List


Lecture 1 (12:00 Wed 18.01.17) Our programme: from microphysics to macrophysics (what we need to do to construct the thermodynamics of a given system). Micorostates. SM definition of pressure. Principle of maximum entropy.

Reading: Binney Notes Sec 1
Schroedinger Chapter 1-2
Jaynes Sec 11.4 (assigning probabilities fairly)
Blundell Sec 14.8, 15.1-15.2 (entropy, probability, information),
App C.3 (Stirling's formula)
My account of the principle of maximum entropy and subsequent developments (continued in the next lecture) owes most to J. Binney's notes, Schroedinger's book and Jaynes' book and papers. This is not necessarily the standard approach and I invite you to read other books on the subject and form your own view of what makes sense. There will be more discussion of the conceptual issues underpinning this approach in later lectures (see Lecture Notes 12).
    To avoid one particular confusion arising (which I detected in some after-lecture questions today), let me reiterate what I said in the lecture about my attitude to the equal a priori probabilities principle. The fact that it has this grand name might have suggested that it would be the foundation of what follows. In fact, I only mentioned it as a particularly simple example of a fair assignment of probabilities. In practice, we will have no use for completely isolated systems, about which we are 100% ignorant, but instead focus on systems for which at least some mean quantities can be known (measured). The statistical inferences we will make about such systems will be based on the principle of maximum entropy. This approach belongs to the Gibbs-Shannon ("canonical") family of treatments, rather than the Boltzmann one ("microcanonical"). A logic chart of both is on p. 126 of Lecture Notes 12, where I will discuss in great detail how various constructions of Statistical Mechanics compare.

James Binney's 2002 lecture notes
on Statistical Mechanics

E. T. Jaynes, Probability Theory:
The Logic of Science (CUP 2004)

C. Shannon's original paper:
Bell System Tech. J. 27, 379 (1948)
(the first in the linked issue).

Also you might enjoy reading this:
E. T. Jaynes, "Information Theory and
Statistical Mechanics,"
Phys. Rev. 106, 620 (1957)

On adiabatic processes (and pressure),
see Binney & Skinner,
Quantum Mechanics, sec 11.1-3

A very nice and readable book:
I. Ford, Statistical Physics:
An Entropic Approach

(Wiley 2013)
...although he prefers starting with
the Boltzmannite microcanonical
formulation, like most texts
--- so I continue to recommend
Schroedinger's lectures
as primary reading

Another very good read is
J. P. Sethna, Statistical Mechanics:
Entropy, Order Parameters, and

(OUP 2006),
where you will find a great number
of cool and modern examples

Lecture 2 (12:00 Thu 19.01.17) Principle of maximum entropy cont'd. Method of Lagrange multipliers.
Reading: Binney Notes Sec 3.0
Schroedinger Chapter 2-3
Blundell App C.13 (Lagrange multipliers)
Binney Notes Sec 2 (Shannon's theorem)
Shannon's paper Sec 6 and App 2
Jaynes Sec 11.3 (Shannon's theorem)

Note that in order for the results derived from the maximum entropy principle to be useful, the maximum must be quite sharp and the resulting distributions should not have too much variance (fluctuations around the mean values should be small). All this can be quite rigorously demonstrated for large systems and is basically related to the quality of the "thermodynamic limit." You will find these discussions in Schroedinger Chapters V-VI and some further relevant maths in Jaynes Sec 11.7.
    A perceptive student might wonder whether entropy is a human construct. Basically, yes, it is a measure of uncertainty that helps us make the best (unbiased) statistical inference about the state of a largely unknown system. I will discuss this issue a lot more in Lectures 6-7 (but no one stops you thinking about it until then.)

Lecture 3 (12:00 Fri 20.01.17) Canonical ensemble: Gibbs distribution. Construction of thermodynamics.

Reading: Schroedinger Chapter 2-3
Example 14.7 (Gibbs),
Sec 20 (from partition function to thermodynamics)

On composite systems in QM,
see Binney & Skinner,
Quantum Mechanics, sec 6.1

If you want some intellectual
stimulation of an abstract kind,
see what mathematicians can do
to thermodynamics:
axiomatic thermodynamics
by E. Lieb & J. Yngvason,
Physics Reports 310, 1 (1999)

(open at your own risk;
you have to be over the age of consent!)

An older account of Caratheodory's
axiomatic thermodynamics
is in Pauli Sec 11

Lecture 4 (12:00 Wed 25.01.17) Additivity of entropy. Thermal equilibrium and the validity of the SM definition of temperature. Heat bath and the physical interpretation of the canonical ensemble. Mechanical and dynamical equilibria.

Reading: Binney Notes Sec 3.0.3 (composite systems)
Landau & Lifshitz Sec 10 (dynamical equlibria), 12 (pressure equilbrium)
Best way to understand the stability arguments is to think about cases when the conclusions don't apply. For example, Problem Set 5 contains some examples of negative temperatures: why are negative tempeartures possible there despite my arguments to the contrary in today's lecture?

Here's an interesting recent take
on the issue of negative temperatures:
J. Dunkel & S. Hilbert,
Nature Phys. 10, 67 (2014)
and a "popular" recap of their article:
I. M. Sokolov, Nature Phys. 10, 7 (2014)
(note that what these people mean by
"Gibbs entropy" is not the same thing as
our "Gibbs-Shannon entropy")

Lecture 5 (12:00 Thu 26.01.17) Thermodynamic stability, positivity of temperature and pressure. SM of classical monatomic ideal gas.

Reading: Blundell Sec 21.1-21.5
One of you asked today how the result that heat capacity must be positive in order to guarantee thermal stability squares with the fact that the heat capacity of black holes is negative (their temperature, as black bodies emitting Hawking radiation, is inversely proportional to mass, while their entropy is proportional to the area of the horizon, hence to its radius squared, hence to mass squared; thus, decreasing T is accompanied by increasig S). Does this mean they are not in equilibrium? Is there thermal instability? The answer appears to be yes. Indeed, we know black holes evaporate and, considering a population of black holes, we would expect this to be unstable: they would merge into ever bigger, but also "colder", black holes. Note that the question of what are the microstates of a black hole is an active research topic. String theorists claim to have the answer for some cases and are able to calculate black hole entropy from it.

You are ready to start working on Problem Set 5

Lecture 6 (12:00 Fri 27.01.17) SM of classical monatomic ideal gas cont'd. Thermodynamics of ideal gas.
Reading: see above
We have now encountered three sets of definitions of temperature and pressure, viz., thermodynamical, kinetic and statistical-mechanical, and proved they are all equivalent. Ponder how this worked and see if you really understand the logic involved. This is a good way to revise.

Lecture 7 (12:00 Wed 1.02.17) Boltzmann entropy. Microcanonical vs. canonical ensemble/Boltzmann vs. Gibbs. Meaning of probabilities. Second law and the loss of information.

Reading: Schroedinger Sec 1-2
Blundell Sec 4.4-4.6 (from microcanonical to canonical)
Kardar Sec 4.2, 4.6 (from microcanonical to canonical)
Binney Notes Sec 5.0
Jaynes Sec 11.8 (meaning of probabilities, extra constraints etc.)
Landau & Lifshitz Sec 3, 4, 7, 8 (microcanonical approach)

I mentioned the intriguing concept of the "thermal death of the Universe" possibly without sufficient explanation. The idea is that if the entropy of the Universe keeps increasing, we are moving towards an eventual state where it is globally maximal and so there are no gradients of any kind --- it all ends up in the ultimate boring state of deadly homogeneity --- which means, sadly for us, that no structures and so no life can survive. This realisation, the (possibly apocryphal) story goes, caused Boltzmann great distress and ultimately led to his suicide. In contrast, Schroedinger, whose understanding of statistical mechanics was (as evidenced by his book) deeper and more compelling than Boltzmann's, was quite a positive fellow, worried less about death, and indeed, among other things, wrote a book "What is Life?" which I highly recommend. He was also a Fellow of Magdalen here in Oxford until his insistence on having two wives became untenable and he moved to catholic Ireland, which, interestingly, proved more tolerant.

Subjectivity of the a priori probabilities: if two observers have different information, will they obtain different statistical mechanics and so different predictions? Does the heat capacity of a box of gas depend on who is looking?! I have provided some comments on this in my Lecture Notes 12.3 --- or you can read the discussion in Jaynes Sec 11.8. There is a question in PS-5 which provides an example of what happens when a superfluous constraint is imposed.

There is (alas!) no time to cover the density matrix and the way information is lost from the quantum mechanical viewpoint (which may be how it's "really" lost). You will find a treatment of this in Lecture Notes 13 and recommended readings below.

Reading: Binney & Skinner Sec 6.3-6.4
Blundell Sec 15.4
Landau & Lifshitz Sec 5-6
Kardar Sec 6.5

If you are interested in the history of
ideas, here's a good read about
Einstein's struggles with entropy,
probability and the second law:
A. Pais, "Subtle is the Lord..."
(OUP 1982) Ch. 4

Possibly the first paper that
made the connection between
entropy and information
(and sucessfully exorcised
Maxwell's demon):
Leo Szilard, Z. Physik 53, 840 (1929)
--- English translation in
Behavioral Sci. 9, 301 (1964)

Second Law:
E. T. Jaynes, "Gibbs vs Boltzmann
Am. J. Phys. 33, 391 (1965)

Entropy and thermodynamics
from the QM perspective:
E. T. Jaynes, "Information Theory and
Statistical Mechanics II,"
Phys. Rev. 108, 171 (1957)
Lectures 8-10 will be given by Prof Julien Devriendt
A. Boothroyd's handouts (2015):
Handout 8
Handout 9
Handout 10

Lecture 11 (12:00 Thu 9.02.17) Grand canonical ensemble. Chemical potential. Thermodynamics of open systems. Particle equilibrium.

Reading: Blundell Sec 22.1-4;
Kittel Sec 14;
Kardar Sec 4.9;
Landau & Lifshitz Sec 35-36
Note that, like temperature, the chemical potential can be introduced purely thermodynamically (as energy cost of adding particles to the system) and then shown to be the same as the parameter that appears in the grand canonical distribution. Also, instead of maximising entropy, as I have done, one can derive the grand canonical distribution from the microcanonical isolated-system set up by considering a small subsystem of the world exchanging energy and particles with its surroundings. This is how it is done in most textbooks. 

I have added a fun exercise (Ex 14.7)
to the Lecture Notes, on a kind of
grand canonical approach to the
statistical mechanics of black holes.
It is based on the paper
by G. Gour,
Phys Rev. D 61, 021501(R)

I have also included this exercise
as a new optional vacation work
question (R.7) in the
updated Revision Problem Set,
along with an opportunity to
be creative about elastic chains (R.6).

Note by the way an intriguing
connexion between gravity
and elasticity:
E. Verlinde JHEP04(2011)029

Lecture 12 (12:00 Fri 10.02.17) Chemical potential of a classical ideal gas. Equilibria of systems in external fields. Chemical potential and the Gibbs function.
Reading: Blundell Sec 22.5-6;
Kittel Sec 14, 15;
Kardar Sec 4.9;
Landau & Lifshitz Sec 24, 25

Lecture 13 (12:00 Wed 15.02.17) Multispecies (multicomponent) systems: generalisation of the grand canonical ensemble. Gibbs phase rule. Chemical equilibrium. Law of mass action.

Reading: Blundell Sec 28.5, 22.8;
Kittel Sec 16;
Landau & Lifshitz Sec 101-102, 104-105
Another interesting topic that involves the use of chemical potential but that I have not covered is the thermodynamics of solutions. Read Landau & Lifshitz Chapter IX if you want to find out about that (also see Blundell & Blundell Sec 22.9 about osmotic pressure).

The ionisation-recombination equilibrium
(see Problem Set 6) has interesting applications to Early Universe
("the recombination epoch").
Here and here are some lecture notes
(from the US) on this subject.

You are ready to do Problem Set 6

PART VI: QUANTUM GASES Lecture 14 (12:00 Thu 16.02.17) Quantum gases. Pauli exclusion principle. Partition function for fermions and bosons. Occupation number statistics. Preview of interesting limits.

Reading: Blundell Chapter 29;
Kardar Sec 7.1, 7.3;
Landau & Lifshitz Sec 53, 54;
Schroedinger Sec 8

Lecture 15 (12:00 Fri 17.02.17) Calculations in the continuous limit. Classical limit. Degeneration.
Reading: Blundell Sec 30.1;
Kardar Sec 7.4;
Landau & Lifshitz Sec 56;
Schroedinger Sec 8

Lecture 16 (12:00 Wed 22.02.17) Degenerate Fermi gas. Fermi energy. Thermodynamics of Fermi gas at T=0. Finite-temperature corrections and heat capacity.

Reading: Blundell Sec 30.2;
Kittel Sec 19-20;
Kardar Sec 7.5;
Landau & Lifshitz Sec 57,58;
Schroedinger Sec 8(a)

If you want to learn more about
the thermodynamics of high-density,
high-mass systems (including
stability of neutron stars,
which you will encounter
in Problem Set 7),
see Landau Chapter XI.

Chandrasekhar got the
1983 Nobel Prize for his theory
of the structure and evolurion of stars.
Here is his Nobel lecture
on the subject.

Lecture 17 (12:00 Thu 23.02.17) Degenerate Bose gas. Bose-Einstein condensation.

Reading: Blundell Sec 30.3-4;
Kittel Sec 21;
Kardar Sec 7.6;
Landau & Lifshitz Sec 62;
Schroedinger Sec 8(b)

2001 Nobel Prize
("for the achievement of
Bose-Einstein condensation...")

Lectures 18-19 (on thermal radiation) will be given by Prof Julien Devriendt
A. Boothroyd's handouts (2015):
Handout 11
Additional handout: Greenhouse Effect

You are ready to do Problem Set 7

Lectures 20-24 will be given by Prof Julien Devriendt
A. Boothroyd's handouts (2015):

You are ready to do Problem Set 8 (vacation work)

clericsMS293.jpgclerics_inv_reflectedclerics_inv_reflectedclericsMS293.jpg Trinity Term 2017: Revision Lectures

Revision Lecture 1 (TBA, Prof Devriendt)
A. Boothroyd's revision handouts (2015):
Revision questions
Solutions to revision questions
Strategy: Thermo
Strategy: Stat Mech

Revision Lectures 2-3 (TBA, Prof Schekochihin)
I will discuss some topics from Kinetic Theory and Quantum Gases.
In anticipation of the lecture, you might wish to look at the following questions from past papers (all June):
2013-Q6, Q10, 2012-Q10, 2010-Q4, 2006-Q7, 2005-Q7, 2003-Q8
There will (hopefully) be a Q&A session in the 2nd hour of the lecture.
Revision Lecture Notes
Notes on some past questions