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 
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) ESSENTIAL DOWNLOADS: 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). 

PART I: BASIC THERMODYNAMICS 
Lectures
17 will be given by Prof
Julien Devriendt A. Boothroyd's handouts (2014) for this
part of the course:
Handout 1 Handout 2 Handout 3 Handout 4 Handout 5 Handout 6 Handout 7 Lecture Plan for Lectures 17 A. Boothroyd's Lecture Notes (2014) for Lectures 17 

You are ready to do Problem Sets 1 and 2  
PART II: KINETIC THEORY 
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 Maxwell'sdistribution) 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 10Key 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 timedependent solutions given periodic boundary conditions (real frequencies, complex wavenumbers), (iii) how to find timedependent 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 boundaryvalue 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 & LifshitzKinetics Chapter 1 (hardcore Russian treatment); Chapman & Cowling Chapters 3, 6, 7, 9, 10 (hardcore 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 proofread!), 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 ESSENTIAL DOWNLOADS: Reading List
LECTURES 

PART III: FOUNDATIONS OF STATISTICAL MECHANICS 
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 12 Jaynes Sec 11.4 (assigning probabilities fairly) Blundell Sec 14.8, 15.115.2 (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 afterlecture 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 GibbsShannon ("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. 
RECOMMENDED: 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 11.13 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 Complexity (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 23 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 VVI 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 67 (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 23 Blundell 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 "GibbsShannon 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 21.121.5 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 statisticalmechanical,
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 12 Blundell Sec 4.44.6 (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 PS5 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 6.36.4 Blundell Sec 15.4 Landau & Lifshitz Sec 56 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 Entropies," 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) 

PART IV: STATISTICAL MECHANICS OF SIMPLE SYSTEMS 
Lectures 810 will be given by Prof
Julien Devriendt 

PART V: OPEN SYSTEMS 
Lecture 11 (12:00 Thu 9.02.17) Grand
canonical
ensemble. Chemical potential. Thermodynamics of open systems. Particle equilibrium. Reading: Blundell Sec 22.14; Kittel Sec 14; Kardar Sec 4.9; Landau & Lifshitz Sec 3536 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 isolatedsystem 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 22.56;
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 101102, 104105 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 ionisationrecombination 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. Finitetemperature
corrections and heat capacity. Reading: Blundell Sec 30.2; Kittel Sec 1920; Kardar Sec 7.5; Landau & Lifshitz Sec 57,58; Schroedinger Sec 8(a) 
If you
want to learn more about the thermodynamics of highdensity, highmass 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. BoseEinstein condensation. Reading: Blundell Sec 30.34; Kittel Sec 21; Kardar Sec 7.6; Landau & Lifshitz Sec 62; Schroedinger Sec 8(b) 
2001 Nobel Prize ("for the achievement of BoseEinstein condensation...") 

Lectures 1819 (on thermal
radiation) will be given by Prof
Julien Devriendt A. Boothroyd's handouts (2015):


You are ready to do Problem Set 7  
PART VII: THERMODYNAMICS OF REAL GASES 
Lectures 2024 will be given by Prof
Julien Devriendt A. Boothroyd's handouts (2015):
Handout
12
Handout 13 Handout 14 Handout 15 Additional handout: Van der Vaals Gas pVT surface of a real gas Extracurricular handout: The Third Law 

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

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 23 (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): 2013Q6, Q10, 2012Q10, 2010Q4, 2006Q7, 2005Q7, 2003Q8 There will (hopefully) be a Q&A session in the 2nd hour of the lecture. Revision Lecture Notes Notes on some past questions 
