Oxford Master Course in Mathematical and Theoretical Physics
Centre for Postgraduate Training in Plasma Physics and High Energy Density Science

followed  by


Perseus MAST maxwelldemon-gamow.jpg Star
                    cluster Quasiparticle

  Dr Paul Dellar, Prof Alexander Schekochihin, Dr Chris Hamilton
  TA: Georgia Acton

This is a core MMathPhys course which we expect to be of interest to graduate students specialising in the physics (or applied mathematics) of gases and plasmas, astrophysics, and condensed matter.

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 2022

(28.5 hours)
Monday 10:00-11:30 (weeks 1,3-8)
Monday 15:00-17:00 (weeks 1-3,7-8)

Tuesday 12:00-13:00 (weeks 1-8)
in Lindemann LT


See below

Course materials, reading suggestions, scheduling notices,
problem sets to appear below.

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)

9.5 hours (Mon 10.10.22 - Mon 24.10.22) Dr Paul Dellar
Timescales and length scales. Hamiltonian mechanics of N particles. Liouville's Theorem. Reduced distributions. BBGKY hierarchy. Boltzmann-Grad limit and truncation of BBGKY equation for the 2-particle distribution assuming a short-range potential. Boltzmann's collision operator and its conservation properties. Boltzmann's entropy and the H-theorem. Maxwell-Boltzmann distribution. Linearised collision operator. Model collision operators: the BGK operator, Fokker-Planck operator. Derivation of hydrodynamics via Chapman-Enskog expansion. Viscosity and thermal conductivity.

The objective of this part of the course is to introduce the basic language of kinetic theory and show how, starting from a system of interacting particles, we can derive first a kinetic description for a single-particle distribution function, and second fluid equations to describe a collisional system close to Maxwellian equilibrium.

Lecture 1+ (10:00-11:30; Mon 10.10.22)
Lectures 2-3 (15:00-17:00; Mon 10.10.22)
Lecture 4 (12:00-13:00; Tue 11.10.22)
Lectures 5-6 (15:00-17:00; Mon 17.10.22)
Lecture 7 (12:00-13:00; Tue 18.10.22)
Lectures 8-9 (15:00-17:00; Mon 24.10.22)

Problem Class 1
Monday 7.11.22 @15:00-17:00
in Lindemann LT
Homework due by 3.11.22 @12:00
to Georgia Acton via Canvas

Lecture Notes

Paul Dellar's webpage
for this part of the course
including lecture notes,
problem set,
and reading suggestions


10 hours (Mon 24.10.22 - Fri 15.11.22) Prof Alexander Schekochihin
Kinetic description of a plasma: Debye shielding, micro- vs. macroscopic fields, Vlasov-Maxwell equations. Klimontovich's version of BBGKY (non-examinable). Plasma frequency. Partition of the dynamics into equilibrium and fluctuations. Linear theory: initial-value problem for the Vlasov-Poisson system, Laplace-tranform solution, the dielectric function, Landau prescription for calculating velocity integrals, Langmuir waves, Landau damping and kinetic instabilities (driven by beams, streams and bumps on tail), Weibel instability, sound waves, their damping, ion-acoustic instability, ion-Langmuir oscillations. Energy conservation. Heating. Entropy and free energy. Ballistic response and phase mixing. Role of collisions; coarse-graining. Elements of kinetic stability theory. Quasilinear theory: general scheme. QLT for bump-on-tail instability in 1D. Introduction to quasiparticle kinetics.

The main new set of concepts in this part of the course, compared to Part I, is about the behaviour of a system that has not only particles but also fields: how do they exchange energy? how do they interact?

Problem Set 2: you will find it
in the Lecture Notes

Problem Class 2
Friday 25.11.22 @17:00-19:00
in Fisher Room

Homework due by 21.11.22 @12:00
to Georgia Acton via Canvas

The latest version
of the typed notes is available here.
Check back for upadtes!


Lecture 10+ (10:00-11:30; Mon 24.10.22) Kinetic description of a plasma: Debye shielding,  micro- vs. macroscopic fields, Vlasov-Landau-Maxwell equations. Basic properties of the Landau collision integral. Plasma frequency. Slow equilibrium and fast fluctuations.

Lecture Notes sec 1.1-1.6, 1.8-2.3

Klimontovich sec 4, 5, 11
(his version of BBGKY etc.)
Helander sec 3 (coll. operator)
Helander sec 4 (fluid eqns)
(Chapman-Enskog for plasma, original derivation)

Lecture 11 (12:00-13:00; Tue 25.10.22) Outline of the hierarchy of approximations: linear, quasilinear, weak turbulence, strong turbulence.

Linear theory: initial-value problem for the Vlasov-Poisson system, Laplace-transform solution, the dielectric function, plasma dispersion relation.

Lecture Notes sec 2.3-3.1

Zakharov et al.; Nazarenko
(general scheme of weak turbulence theory)
Kadomtsev; Sagdeev & Galeev
(from linear to QL to WT for plasma)

Landau's paper (original derivation)
Hazeltine & Waelbroeck sec 6.3, 6.4
Alexandrov et al. sec 2, 4
(all the waves catalogued, with an emphasis on plasma as a dielectric)

Lecture 12+ (10:00-11:30; Mon 31.10.22) Linear theory cont'd:  Landau prescription for calculating velocity integrals. Solving the plasma dispersion relation in the slow damping/growth limit. Langmuir waves. Landau damping and kinetic instabilities.

Lecture Notes sec 3.2-3.5

Lecture 13 (12:00-13:00; Tue 1.11.22) Hydrodynamic beam instability. Sound waves, their damping, ion-acoustic instability.

Lecture Notes sec 3.7-3.9

Sec 4 of my Notes is
extracurricular material.
You can read it if you like
(after Lecture 13,
you know all you need to know
to read it)

Lecture 14+ (10:00-10:30; Mon 7.11.22) Summary of longitudinal waves. Ion-Langmuir oscillations.
Lecture Notes sec 3.10-3.11

You are ready to do Q1-5 of the Problem Set

Energy conservation. Heating. Entropy and free energy.

Lecture Notes sec 5.1-5.2

Hazeltine & Waelbroeck sec 6.2
(Landau damping and phase mixing
without Laplace transforms)

Lecture 15 (12:00-13:00; Tue 8.11.22) Perturbed distribution function: ballistic response and phase mixing. Role of collisions; coarse-graining. Structure of the perturbed distribution near a resonance.

You are ready to do Q6-8 of the Problem Set

Lecture Notes sec 5.3-5.6

Krall & Trivelpiece sec 10
Kadomtsev sec I.3
Sagdeev & Galeev sec II-2
(...and read on for more advanced topics)

Lecture 16+ (10:00-11:30; Mon 14.11.22) Quasilinear theory: general scheme. QLT for bump-on-tail instability in 1D: plateau, spectrum of waves.

You are ready to do Q9-12 of the Problem Set

Lecture Notes sec 6.1, 6.3-6.6, 7.1

Lecture 17 (12:00-13:00; Tue 15.11.22) Quasiparticle kinetics. Reformulation of QLT in quasiparticle formalism..

Tsytovich sec 3, 5, 7
(on plasmon kinetics and beyond)
Peierls's and Ziman's books
(on electrons and phonons in metals)


9 hours (Mon 21.11.22 - Tue 29.11.22) Dr Chris Hamilton
Unshielded nature of gravity and implications for self-gravitating systems. Mean-field approximation with simple examples. Negative specific heat and impossibility of thermal equilibrium. Relaxation driven by fluctuations in mean field. Evaporation. Angle-action variables. Potential-density pairs. Long-time response to initial perturbation. Fokker-Planck equation. Computation of the diffusion coefficients in terms of resonant interactions. Application to a tepid disc.

Here again there are particles (stars) and fields (gravitational). The key feature is that there are no collisions at all and one must understand the behaviour of a kinetic system that is not close to a Maxwellian equilibrium.

Lecture 18+ (10:00-11:30; Mon 21.11.22)
Lectures 19-20 (15:00-17:00; Mon 21.11.22)
Lecture 21 (12:00-13:00; Tue 22.11.22)
Lecture 22+ (10:00-11:30; Mon 28.11.22)
Lectures 23-24 (15:00-17:00; Mon 28.11.22)
Lecture 25 (12:00-13:00; Tue 29.11.22)

Problem Set 3: you will find it
on C. Hamilton's webage

Problem Class 3
Wednesday 7.12.22 @14:00-16:00
in DWB Seminar Room

Homework due by 2.12.22 @12:00
to Georgia Acton via Canvas

C. Hamilton's webpage
for this part of the course
including lecture notes
and problem set

J. Binney's 2018 Lecture Notes
J.-B. Fouvry's 2021 Lecture Notes



Prof Alexander Schekochihin
  TA: Michael Nastac

[Part I of this course, taught by Dr Plamen Ivanov in HT21 and dedicated to waves in magnetised plasmas, is not covered on this website; for course info, see Canvas]

Trinity Term 2022

(10 hours)
  Monday, Wednesday & Friday 17:00-18:00
(week 1-4, but no lectures on Friday week 1 and Monday week 3)
in Lindemann LT

Canvas page for this course is here

NOTE: The lecture plan below may change as we progress


Lecture 1 (17:00-18:00; Mon 25.04.22) MHD equations in a strongly magnetised, kinetic plasma. Gyrotropy. Reduction of kinetic content to pressure anisotropy.

Lecture 2 (17:00-18:00; Wed 27.04.22) Alfven waves and firehose instability. Gyroaveraged kinetic equation.

Lecture 3 (17:00-18:00; Mon 2.05.22) KHMD completed: parallel electric field. Kinetic equation in variables involving the magnetic moment. Relation of kinetic equations to particle motion. Particle motion in a strong magnetic field.

Lecture 4 (17:00-18:00; Wed 4.05.22) Particle motion in strong magnetic field cont'd. Rederivation of high-flow drift kinetics. 

You are ready to do Q1.1-1.5 of the Problem Set

Lecture 5 (17:00-18:00; Fri 6.05.22) Derivation of low-flow drift kinetics.

Lecture 6 (17:00-18:00; Wed 11.05.22) Mirror instability: formal derivation.

Lecture Notes sec 9
Parra Notes IV sec 3

You are ready to do Q2.2 of the Problem Set

Lecture 7 (17:00-18:00; Fri 13.05.22) Mirror instability cont'd: physics. Barnes (transit-time) damping/magnetic pumping/betatron acceleration. Origin of pressure anisotropy. CGL equations. 

Lecture Notes sec 9-10

Lecture 8 (17:00-18:00; Mon 16.05.22) CGL equations cont'd: double adiabaticity, longitudinal invariant. 

Lecture Notes: I shall use
Lecture Notes by Prof Felix Parra,
(see also his reading list)
+ my own notes,
which will be posted here.
FP's notes are also available here.

My lectures will present the subject
in a somewhat differed order than FP's did,
but if you read
his notes
ahead of my lectures,
you may find that
comparing the two expositions
will help you achieve greater enlightenment
(and do the problem set).

Problem Set: Solve Prof Parra's
Problem Set 1 and
Q2.1, 2.2 from his Problem Set 2

Homework due by midnight 31.05.22
to Michael Nastac via Canvas

Problem Class: 16:00-18:00,
Monday 6.06.22
in DWB 501

Electrostatic regime of drift kinetics. Electron response.

Lecture 9 (17:00-18:00; Wed 18.05.22) Fluid ITG instability.

Lecture 10 (17:00-18:00; Fri 20.05.22) Kinetic ITG instability. [Nature of electrostatic limit.]

You are ready to do Q1.6, 2.1 of the Problem Set

READING LIST for the Kinetic Theory Course

PART I: see reading suggestions on Paul Dellar's course webpage

PART II (including "further reading"):
  1. A. F. Alexandrov, L. S. Bogdankevich & A. A. Rukhadze, Principles of Plasma Electrodynamics (Springer 1984) (Amazon)
  2. S. I. Braginskii, "Transport processes in a plasma," Rev. Plasma. Phys. 1, 205 (1965) (pdf)
  3. S. C. Cowley, Lecture notes on plasma physics (UCLA 2003-07)
  4. R. D. Hazeltine & F. L. Waelbroeck, The Framework Of Plasma Physics (Perseus Books 1998) (Amazon)
  5. P. Helander & D. J. Sigmar, Collisional Transport in Magnetized Plasmas (CUP 2005) (Amazon)
  6. B. B. Kadomtsev, Plasma Turbulence (Academic Press 1965) (pdf)
  7. Yu. L. Klimontovich, The Statistical Theory of Non-Equilibrium Processes in a Plasma (Pergamon 1967) (Amazon)
  8. N. A. Krall & A. W. Trivelpiece, Principles of Plasma Physics (McGraw-Hill 1973) --- available in TP Discussion Room Library
  9. L. Landau, "On the vibrations of the electronic plasma," J. Phys. USSR 10, 25 (1946)  (pdf)
  10. E. M. Lifshitz & L. P. Pitaevskii, Physical Kinetics (Volume 10 of L. D. Landau and E. M. Lifshitz's Course of Theoretical Physics) (Elsevier 1976) (Amazon)
  11. S. Nazarenko, Wave Turbulence (Springer 2011) (Amazon)
  12. R. Z. Sagdeev & A. A. Galeev, Nonlinear Plasma Theory (W. A. Benjamin 1969) (pdf)
  13. V. N. Tsytovich, Lectures on Nonlinear Plasma Kinetics (Springer 1995) (Amazon) --- available in TP Discussion Room Library
  14. V. E. Zakharov, V. S. Lvov & G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer 1992) (Amazon) (updated online version)
  15. J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids (OUP 2001) (Amazon) --- available in TP Discussion Room Library
  1. J. Binney & S. Tremaine, Galactic Dynamics (Princeton University Press 2008) (Amazon)
  2. J. Binney, Dynamics of secular evolution, arXiv:1202.3403