KINETIC THEORY
Oxford Master Course in Mathematical and Theoretical Physics
("MMathPhys")
&
Centre for Postgraduate Training in Plasma Physics and High Energy Density Science
followed by
COLLISIONAL PLASMA PHYSICS
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Dr Paul Dellar, Prof Alexander Schekochihin, Dr Robert Ewart
TA: Lucas McConnell
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 2025 LECTURES (28 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 CLASSES See below Course materials, reading suggestions, scheduling notices, problem sets to appear below. Canvas page for this course is here |
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). |
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| PART I: KINETIC THEORY OF GASES |
9
hours (Mon 13.10.25 - Mon 27.10.25)
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 13.10.25) Lectures 2-3 (15:00-17:00; Mon 13.10.25) Lecture 4 (12:00-13:00; Tue 14.10.25) Lectures 5-6 (15:00-17:00; Mon 20.10.25) Lecture 7 (12:00-13:00; Tue 21.10.25) Lectures 8+ (15:00-16:30; Mon 27.10.25) |
Problem Class 1 by Dr Paul Dellar Monday 10.11.25 (week 5) @15:00-17:00 in Lindemann LT Homework due by 17:00 on 6.11.25 to Lucas McConnell Lecture Notes Paul Dellar's webpage for this part of the course, including lecture notes, problem set, and reading suggestions |
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| PART II: KINETIC THEORY OF PLASMAS & QUASIPARTICLES |
10
hours (Mon
27.10.25 - Tue 18.11.25) 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 by Lucas McConnell Friday 28.11.25 (week 7) @15:30-17:30 in Lindemann LT Homework due by 17:00 on 24.11.25 to Lucas McConnell The latest version of the typed notes is available here. Check back for upadtes! Reading: |
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| Lecture 9+ (10:00-11:30; Mon
27.10.25) Kinetic description of a plasma:
Debye shielding, micro- vs. macroscopic fields,
Vlasov-Landau-Maxwell equations. Basic properties of the
Landau collision integral. Plasma frequency. Lecture Notes sec
1.1-1.6, 1.8, 2.1
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Klimontovich
sec 4, 5, 11 (his version of BBGKY etc.) Helander sec 3 (coll. operator) Helander sec 4 (fluid eqns) Braginskii (Chapman-Enskog for plasma, original derivation) |
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Lecture 10 (12:00-13:00; Tue 28.10.25)
Slow equilibrium and fast fluctuations. Outline of the
hierarchy of approximations: linear, quasilinear, weak
turbulence, strong turbulence.
Lecture Notes sec
2.2-2.4
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Zakharov
et
al.; Nazarenko (general scheme of weak turbulence theory) Kadomtsev; Sagdeev & Galeev (from linear to QL to WT for plasma) |
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Lecture 11+ (10:00-11:30; Mon 3.11.25)
Linear theory: initial-value problem for the
Vlasov-Poisson system, Laplace-transform solution, the
dielectric function, plasma dispersion relation.
Landau prescription for calculating velocity
integrals. Solving the plasma dispersion relation in
the slow damping/growth limit.
Lecture Notes sec
3.1-3.3
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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) Here is a handy primer on complex analysis. |
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| Lecture 12 (12:00-13:00; Tue 4.11.25)
Langmuir waves, Landau damping and kinetic
instabilities. |
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) |
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| Lecture 13+ (10:00-11:30; Mon 10.11.25)
Hydrodynamical beam instability. Sound waves, their damping,
ion-acoustic instability. Summary of longitudinal waves.
You are ready to do Q1-5 of the Problem Set Lecture Notes sec
3.7-3.11
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Hazeltine & Waelbroeck sec 6.2 (Landau damping and phase mixing without Laplace transforms) |
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Lecture 14 (12:00-13:00; Tue 11.11.25) Energy
conservation. Heating. Entropy and free energy.
Perturbed distribution function: ballistic response
and phase mixing.
You are ready to do Q6-8 of the Problem Set Lecture Notes sec
5.1-5.4
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Krall & Trivelpiece
sec 10 Kadomtsev sec I.3 Sagdeev & Galeev sec II-2 (...and read on for more advanced topics) |
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| Lecture
15+ (10:00-11:30;
Mon 17.11.25) Role of collisions,
irreversibility of Landau damping. Quasilinear theory:
general scheme. QLT
for bump-on-tail instability in 1D: plateau, spectrum of
waves, energy of resonant particles, heating of the
thermal bulk. You are ready to do Q9-13 of the Problem Set |
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| Lecture
16 (12:00-13:00;
Tue 18.11.25) Quasiparticle
kinetics. Reformulation of QLT in quasiparticle
formalism. Lecture Notes sec 7.1
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Tsytovich sec 3, 5, 7 (on plasmon kinetics and beyond) Peierls's and Ziman's books (on electrons and phonons in metals) |
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| PART III: KINETIC THEORY OF SELF-GRAVITATING SYSTEMS |
9
hours (Mon 24.11.25 - Tue 2.12.25) Dr Robert Ewart 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 17+ (10:00-11:30; Mon 24.11.25) Lectures 18-19 (15:00-17:00; Mon 24.11.25) Lecture 20 (12:00-13:00; Tue 25.11.25) Lecture 21+ (10:00-11:30; Mon 1.12.25) Lectures 22-23 (15:00-17:00; Mon 1.12.25) Lecture 24 (12:00-13:00; Tue 2.12.25) |
Problem Set 3
is here (new in
2025) Problem Class 3 by Dr Robert Ewart Thursday 11.12.25 (week 9) @15:30-17:30 in Lindemann LT Homework due by 17:00 on 8.12.25 to Lucas McConnell J. Binney's 2018 Lecture Notes J.-B. Fouvry's 2021 Lecture Notes C. Hamilton's 2024 Lecture Notes |
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Materials for the Collision Plasma Physics are being updated in real time |
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COLLISIONAL PLASMA PHYSICS Prof Alexander Schekochihin TA: Richard Nies Trinity Term 2026 LECTURES (16 hours) Weeks 1,2: Wed, Thu, Fri 11:00-12:00 Week 4: Thu, Fri 11:00-12:00 Week 5, 6, 7: Fri 11:00-12:00 & 12:00-13:00 Fisher Room, DWB Weeks 6, 7: Thu 11:00-12:00 501 (Seminar Room), DWB Canvas page for this course is here |
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| PART I: COLLISIONS & FLUCTUATIONS |
Lecture
1 (11:00-12:00;
Wed 29.04.26) Intro to collisions.
Klimontovich description of a plasma. General form of
collision integrals, relation to correlations and
fluctuations.
My Lecture Notes 1-2Lecture 2 (11:00-12:00; Thu 30.04.26) Quasilinear collision integrals. Lecture 3 (11:00-12:00; Fri 1.05.26) Conservation laws. Stosszahlansatz and Lenard-Balescu collision integral. Lecture 4 (11:00-12:00; Wed 6.05.26) H-theorem and thermal equilibrium. Landau collision integral. Meaning of divergences, Coulomb integral, collision rate. Parra Notes I Kunz Notes I-II, IV-VII |
Lecture Notes: ---Felix Parra's original notes for this course are available here or here. ---You can also read Matt Kunz's notes for the Princeton course "Irreversible Processes in Plasmas", available here. ---My own (handwritten) notes will be posted here; I might type them up in real time, in which case they will appear in the regular updates of my KT Notes (the new Part II+updates elsewhere). Problem Sets: Coming soon Homework due by TBA to Richard Nies Problem Class I: TBA TBA Problem Class II: TBA TBA |
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| PART II: COLLISIONAL RELAXATION & RESISTIVE MHD |
Lecture 5 (11:00-12:00; Thu 7.05.26)
Inter-species collisions: collision rates.
Lecture 6 (11:00-12:00; Fri 8.05.26) TBA Lecture 7 (11:00-12:00; Thu 21.05.26) TBA Lecture 8 (11:00-12:00; Fri 22.04.26) TBA Lectures 9-10 (11:00-13:00; Fri 29.05.26) TBA |
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| PART III: COLLISIONAL TRANSPORT & BRAGINSKII MHD |
Lecture
11 (11:00-12:00;
Thu 4.06.26) TBA Lecture 12-13 (11:00-13:00; Fri 5.06.26) TBA Lecture 14 (11:00-12:00; Thu 11.06.26) TBA Lecture 15-16 (11:00-13:00; Fri 12.04.26) TBA |
READING LIST
KINETIC
THEORY PART I: see reading suggestions on Paul
Dellar's
course webpage
KINETIC THEORY PART II & COLLISIONAL PLASMA PHYSICS:
KINETIC THEORY PART II & COLLISIONAL PLASMA PHYSICS:
- A. F. Alexandrov, L. S. Bogdankevich & A. A.
Rukhadze, Principles
of Plasma Electrodynamics (Springer 1984) (Amazon)
- S. I. Braginskii, "Transport processes in a plasma," Rev. Plasma. Phys. 1, 205 (1965) (pdf)
- S. C. Cowley, Lecture
notes
on plasma physics (UCLA 2003-07)
- R. D. Hazeltine & F.
L. Waelbroeck, The
Framework Of Plasma Physics (Perseus
Books 1998) (Amazon)
- P. Helander & D. J. Sigmar, Collisional Transport in
Magnetized Plasmas (CUP 2005) (Amazon)
- B. B. Kadomtsev, Plasma Turbulence (Academic Press 1965) (pdf)
- Yu. L. Klimontovich, The Statistical Theory of Non-Equilibrium Processes in a Plasma (Pergamon 1967) (Amazon)
- N. A. Krall & A. W. Trivelpiece, Principles of Plasma Physics (McGraw-Hill 1973)
- L. Landau, "On the vibrations of the electronic plasma," J. Phys. USSR 10, 25 (1946) (pdf)
- 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)
- S. Nazarenko, Wave
Turbulence (Springer 2011) (Amazon)
- R. Z. Sagdeev & A. A. Galeev, Nonlinear Plasma Theory (W. A. Benjamin 1969) (pdf)
- V. N. Tsytovich, Lectures on Nonlinear Plasma Kinetics (Springer 1995) (Amazon)
- N. G. van Kampen & B. U. Felderhof, Theoretical
Methods in Plasma Physics (North-Holland 1967) (pdf)
- V. E. Zakharov, V. S. Lvov & G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer 1992) (Amazon) (updated online version)
- J. M. Ziman, Electrons
and Phonons: The Theory of Transport Phenomena in
Solids (OUP 2001) (Amazon)
- J. Binney & S. Tremaine, Galactic Dynamics (Princeton University Press 2008) (Amazon)
- C. Hamilton & J.-B. Fouvry, "Kinetic Theory of
Stellar Systems: A Tutorial," Phys.
Plasmas 31, 120901 (2024)
- J. Binney, Stellar Dynamics for Physicists (2026)
(Amazon)