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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

ADVANCED TOPICS IN PLASMA PHYSICS

Perseus MAST maxwelldemon-gamow.jpg Star cluster Quasiparticle


  Dr Paul Dellar, Prof Alexander Schekochihin, Dr Jean-Baptiste Fouvry
  TA: Toby Adkins

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 2020

LECTURES
(28 hours)
Lindemann LT
Monday 10:00-11:30 (weeks 2,3,4,7,8)
Monday 16:00-18:00 (weeks 2,3,7,8)
Monday 16:30-18:00 (weeks 4,5,6)
Tuesday 12:00-13:00 (weeks 2,3,4,7,8)
Tuesday 16:30-18:00 (weeks 3,4)
on ZOOM (links by email, not public)

Our virtual blackboard is here

CLASSES

  TBA

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 hours (Mon 19.10.20 - Tue 27.10.20) 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 kinetic description, one can construct fluid equations for a collisional system close to Maxwellian equilibrium.

Lecture 1+ (10:00-11:30; Mon 19.10.20)
Lectures 2-3 (16:00-18:00; Mon 19.10.20)
Lecture 4 (12:00-13:00; Tue 20.10.20)
Lecture 5+ (10:00-11:30; Mon 26.10.20)
Lectures 6-7 (16:00-18:00; Mon 26.10.20)
Lecture 8 (12:00-13:00; Tue 27.10.20)

Problem Class 1
TBA

Homework due TBA
to Toby Adkins

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 (Tue 27.10.20 - Mon 16.11.20) 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
TBA
Homework due TBA
to Toby Adkins

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








Reading:



Lecture 9+ (16:30-18:00; Tue 27.10.20) Kinetic description of a plasma: Debye shielding,  micro- vs. macroscopic fields, Vlasov-Landau-Maxwell equations. Basic properties of the Landau collision integral.

Lecture Notes sec 1.1-1.6, 1.8-1.9

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 10+ (10:00-11:30; Mon 2.11.20) Plasma frequency. Slow equilibrium and fast fluctuations. 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. Landau prescription for calculating velocity integrals. Solving the plasma dispersion relation in the slow damping/growth limit. Langmuir waves.

Lecture Notes sec 1.8-3.4

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)

Sec 4 of my Notes is
extracurricular material.
You can read it if you like
(after Lecture 10,
you will know all you need to know
to read it),
but I am planning to cover this
in my TT-2020 follow-on course.




Lecture 11+ (16:30-18:00; Mon 2.11.20) Linear theory cont'd: Landau damping and kinetic instabilities, sound waves, their damping, ion-acoustic instability.

Lecture Notes sec 3.5, 3.7-3.9



Lecture 12 (12:00-13:00; Tue 3.11.20) Linear theory cont'd: ion-Langmuir oscillations. Summary of longitudinal waves. Beam instabilities.

You are ready to do Q1, 2, 3/4 of the Problem Set

Lecture Notes sec 3.10-11




Lecture 13+ (16:30-18:00; Tue 3.11.20) Energy conservation. Heating. Entropy and free energy. Perturbed distribution function: ballistic response and phase mixing. Role of collisions; coarse-graining. Structure of the perturbed distribution near a resonance: the Case-van Kampen mode.

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

Lecture Notes sec 5.1-5.6

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


Lecture 14+ (16:30-18:00; Mon 9.11.20) Quasilinear theory: general scheme. QLT for bump-on-tail instability in 1D. Quasiparticles.

You are ready to do Q9, 10, or 11 of the Problem Set

Lecture Notes sec 6.1,6.3-6.6

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


Lecture 15+ (16:30-18:00; Mon 16.11.20) Quasiparticles.

Lecture Notes sec 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 23.11.20 - Tue 1.12.20) Dr Jean-Baptiste Fouvry
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 16+ (10:00-11:30; Mon 23.11.20)
Lectures 17-18 (16:00-18:00; Mon 23.11.20)
Lecture 19 (12:00-13:00; Tue 24.11.20)
Lecture 20+ (10:00-11:30; Mon 30.11.20)
Lectures 21-22 (16:00-18:00; Mon 30.11.20)
Lecture 23 (12:00-13:00; Tue 1.12.20)

Problem Set 3: you will find it
on J-B Fouvry's webpage

Problem Class 3
TBA
Homework due TBA
to Toby Adkins

J-B Fouvry's webpage
for this part of the course
,
including lecture notes
and problem set

J. Binney's 2018 Lecture Notes





clerics_inv
ADVANCED TOPICS
in
 PLASMA PHYSICS


Prof Alexander Schekochihin
  TA: Toby Adkins


Information below is for TT-2020,
to be updated for TT-2021 closer to the time

LECTURES
(8 hours)
email Alex Schekochihin to receive Zoom link
Tuesday and Friday 17:00-18:00 (weeks 1-4)

clerics_inv_reflected





KINETICS of
QUASIPARTICLES


Lecture 1 (17:00-18:00; Tue 28.04.20) Reminder: QLT in the language of quasiparticles. Weak turbulence: kinetic equations for weakly interacting waves --- 3-wave and 4-wave interactions.

Lecture Notes sec 7.1-7.2.2

Problem Set: see Lecture Notes
Due: Sunday week 5 to Toby Adkins


Lecture 2 (17:00-18:00; Fri 1.05.20) WT cont'd: kinetic equations for Langmuir-sound turbulence, induced scattering, and "real" collisions. Statatistical mechanics of Quasiparticles. Validity of WT.

Lecture Notes sec 7.2.3-7.4


LANGMUIR
TURBULENCE

Lecture 3 (17:00-18:00; Tue 5.05.20) Zakharov equations. Hamiltonian form thereof. Derivation of WT kinetic equations via perturbation and random-phase approximation.

Lecture Notes sec 8.1, 8.3-8.4.2


 

Lecture 4 (17:00-18:00; Fri 8.05.20) Solution of WT equations. Direct and inverse cascades. Break down of WT.

Lecture Notes sec 8.4.3-8.4.7

Topics I have no time to cover, but advise you to read about: modulational instability, Langmuir collapse, strong Langmuir turbulence.

Lecture Notes sec 8.5-8.6
NONLINEAR STABILITY
(THERMODYNAMICS
of COLLISIONLESS
PLASMA)

Lecture 5 (17:00-18:00; Tue 12.05.20) General scheme of the thermodynamic method. Gardner's theorem and Helander's minimum-energy thermodynamics. Kruskal and Oberman's thermodynamics of small perturbations. Fowler's thermodynamics of finite perturbations.


I will not, alas, have time to teach linear stability, but your can read what I hope is a coherent, and reasonably short, introduction to this topic in my notes.

COLLISIONLESS
RELAXATION
&
PHASE-SPACE
TURBULENCE

Lecture 6 (17:00-18:00; Fri 15.05.20) Lynden-Bell's statistical mechanics for collisionless plasma. QL derivation of a "collisionless collision integral".

Lecture Notes sec 10.1-10.2.1




Lecture 7 (17:00-18:00; Tue 19.05.20) Microgranulation ansatz. Kadomtsev-Pogutse, Lenard-Balescu and Landau's collision integrals. Interpretation of "collisionless collision integrals" and open questions. Intro to plasma echo.

Lecture Notes sec 10.2.2-11.1




Lecture 8 (17:00-18:00; Fri 22.05.20) Phase-space turbulence under the "short-sudden" approximation. Stochastic echo. Adkins' solution in 1D and the effect of collisions.Implications for collisionless relxation.

Lecture Notes sec 11.2-11.4



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