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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

COLLISIONLESS PLASMA PHYSICS

Perseus MAST maxwelldemon-gamow.jpg Star cluster Quasiparticle


  Dr Paul Dellar, Prof Alexander Schekochihin, Dr Chris Hamilton
  TA: Robbie Ewart

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.
clericsMS293.jpg
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 2023

LECTURES
(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
CLASSES
See below

Course materials, reading suggestions, scheduling notices,
problem sets to appear below.
clericsMS293_reflected.jpg
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: KINETIC THEORY
OF GASES
9.5 hours (Mon 9.10.23 - Mon 23.10.23) 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 9.10.23)
Lectures 2-3 (15:00-17:00; Mon 9.10.23)
Lecture 4 (12:00-13:00; Tue 10.10.23)
Lectures 5-6 (15:00-17:00; Mon 16.10.23)
Lecture 7 (12:00-13:00; Tue 17.10.23)
Lectures 8-9 (15:00-17:00; Mon 23.10.23)

Problem Class 1
Monday 6.11.23 (week 5)
15:00-17:00 in Martin Wood LT
 Homework due by 2.11.23 @23:59
to Robbie Ewart via Canvas

Lecture Notes

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





PART II: KINETIC THEORY
OF PLASMAS
& QUASIPARTICLES
10 hours (Mon 23.10.23 - Tue 14.11.23) 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
Monday 20.11.23 (week 7)
17:00-19:00 in Fisher Room
Homework due by 17.11.23 @23:59
to Robbie Ewart via Canvas

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






Reading:


Lecture 10+ (10:00-11:30; Mon 23.10.23) 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
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)


Lecture 11 (12:00-13:00; Tue 24.10.23) Slow equilibrium and fast fluctuations. Outline of the hierarchy of approximations: linear, quasilinear, weak turbulence, strong turbulence.

Lecture Notes sec 2.2-2.4
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 30.10.23) 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. Langmuir waves.

Lecture Notes sec 3.1-3.4


Lecture 13 (12:00-13:00; Tue 31.10.23) Landau damping and kinetic instabilities. Sound waves, their damping, ion-acoustic instability. Summary of longitudinal waves.

Lecture Notes sec 3.5, 3.8-3.11
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 6.11.23) Hydrodynamical beam instability.

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 7.11.23) 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 13.11.23) Quasilinear theory: general scheme. QLT for bump-on-tail instability in 1D: plateau, spectrum of waves.

Lecture Notes sec 6.1, 6.3-6.4



Lecture 17 (12:00-13:00; Tue 14.11.23) QLT for bump-on-tail instability in 1D: energy of resonant particles, heating of the thermal bulk.

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

Quasiparticle kinetics. Reformulation of QLT in quasiparticle formalism.
Lecture Notes sec 6.5-6.6, 7.1
Tsytovich sec 3, 5, 7
(on plasmon kinetics and beyond)
Peierls's and Ziman's books
(on electrons and phonons in metals)





PART III: KINETIC THEORY
OF SELF-GRAVITATING
SYSTEMS
9 hours (Mon 20.11.23 - Tue 28.11.23) 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 20.11.23)
Lectures 19-20 (15:00-17:00; Mon 20.11.23)
Lecture 21 (12:00-13:00; Tue 21.11.23)
Lecture 22+ (10:00-11:30; Mon 27.11.23)
Lectures 23-24 (15:00-17:00; Mon 27.11.23)
Lecture 25 (12:00-13:00; Tue 28.11.23)

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

Problem Class 3
Monday 4.12.23 (week 9)
16:00-18:00 in Fisher Room
Homework due by 1.12.23 @23:59
to Robbie Ewart 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





clerics_inv
COLLISIONLESS PLASMA PHYSICS (Part II)

Prof Alexander Schekochihin
  TA: Will Clarke

[Part I of this course, taught by Dr Plamen Ivanov in HT24 and dedicated to waves in magnetised plasmas, is not covered on this website; for course info, see Canvas]

Trinity Term 2024
LECTURES
(10 hours)
  Monday, Thursday & Friday 17:00-18:00
(weeks 1-4, but no lectures on Friday week 3 and Monday week 4)
in Lindemann LT

Canvas page for this course is here
clerics_inv_reflected
PART I: KINETIC MHD
& DRIFT KINETICS
Lecture 1 (17:00-18:00; Mon 22.04.24) MHD-like equations in a magnetised, kinetic plasma.

Lecture Notes sec 19.1-19.4

Lecture 2 (17:00-18:00; Wed 24.04.24) Ideal Ohm's law and the magnetised limit. Gyrotropy. Reduction of kinetic content to pressure anisotropy.

Lecture Notes sec 19.5-19.7

Lecture 3 (17:00-18:00; Fri 26.04.24) Alfven waves and firehose instability. Gyroaveraged kinetic equation. KHMD completed: parallel electric field. Kinetic equation in variables involving the magnetic moment.

Lecture Notes sec 19.8-19.10

You are ready to do Ex 19.6 from the Lecture Notes

Lecture 4 (17:00-18:00; Mon 29.04.24) Relation of kinetic equations to particle motion. General perturbation theory for drift-kinetic equations.

Lecture Notes sec 20.1-20.2

Lecture 5 (17:00-18:00; Wed 1.05.24) Particle motion in a strong magnetic field. Recovery of high-flow drift kinetics.

Lecture Notes sec 20.3-20.4

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

Lecture 6 (17:00-18:00; Fri 3.05.24) Summary of low-flow drift kinetics. Mirror instability.

Lecture Notes sec 20.5, 21.1.1

Lecture 7 (17:00-18:00; Mon 6.05.24) Mirror instability cont'd. Barnes (transit-time) damping/magnetic pumping/betatron acceleration.

Lecture Notes sec 21.1

You are ready to do Q2.2 of the Problem Set and Ex 21.3 from the Lecture Notes
Lecture 8 (17:00-18:00; Wed 8.05.24) Origin of pressure anisotropy. CGL equations, double adiabaticity, longitudinal invariant. 

Lecture Notes sec 21.2

You are ready to do Ex 21.4 from the Lecture Notes
Lecture Notes: Felix Parra's
original notes for this course
 are available here.
My own notes are here (Part IV).

Problem Set: Solve Felix Parra's
Problem Set 1 and
Q2.1, 2.2 from his Problem Set 2.
Solve also Ex 19.6, 21.3, 21.4 from
the Lecture Notes
(those three questions are optional).

Homework due
by TBA
to Will Clarke

Problem Class: TBA
PART II: ELECTROSTATIC
DRIFT KINETICS
Lecture 9 (17:00-18:00; Wed 15.05.24) Electrostatic regime of drift kinetics. Electron response. Fluid ITG instability.

Lecture Notes sec 22-22.2

Lecture 10 (17:00-18:00; Fri 17.05.24) Kinetic ITG instability. [Nature of electrostatic limit.]

Lecture Notes sec 22.3

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

Lecture 11 (17:00-18:00; Mon 20.05.24) TBC.

On TG-driven instabilities
in a fully electromagnetic setting,
see this paper by Adkins et al. (2022).

On kinetic ITG with superimposed
flow shear, this paper (2012).

On turbulence and transport
in TG-unstable plasmas,
Ivanov et al. (2020) and (2022)
and Adkins et al. (2023).

(obviously, this topic has been studied
by many people outside Oxford as well,
for well over 30 years, in fact:
to find out more,
follow the paper trail back in time
from the references above)







There will be an optional sequel to this course, taught by Dr Plamen Ivanov and dedicated to fluid and kinetic models of plasma turbulence.


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
PART III:
  1. J. Binney & S. Tremaine, Galactic Dynamics (Princeton University Press 2008) (Amazon)
  2. J. Binney, Dynamics of secular evolution, arXiv:1202.3403