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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

Perseus MAST maxwelldemon-gamow.jpg Star cluster Quasiparticle


  Dr Paul Dellar, Prof Alexander Schekochihin, Prof James Binney
  TA: Glenn Wagner

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 2017

LECTURES
Fisher Room (DWB)
Monday 10:00-11:00 (weeks 1, 4-8)
Mondays 16:00-18:00 (weeks 1-8)
Tuesdays 14:00-16:00 (weeks 1, 3, 8)


CLASSES

  Fisher Room (DWB)
Tuesdays 14:00-16:00 (weeks 5, 7, 9)

EXAM

in week 0 of Hilary Term

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
Lectures 1-9 (Mon 9.10.17 - Mon 23.10.17) 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 (10:00-11:00; Mon 9.10.17)
Lectures 2-3 (16:00-18:00; Mon 9.10.17)
Lectures 4-5 (14:00-16:00; Tue 10.10.17)
Lectures 6-7 (16:00-18:00; Mon 16.10.17)
Lectures 8-9 (16:00-18:00; Mon 23.10.17)

Problem class 1
14:00-16:00 on Tue 7.11.17

Homework due 23:59 Sat 4.11.17
to Glenn Wagner (in TP or at Merton)

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
Lectures 10-19 (Tue 24.10.17 - Mon 13.11.17) 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
14:00-16:00 on Tue 21.11.17
Homework due 23:59 Sun 19.11.17
to Glenn Wagner (in TP or at Merton)

The latest version (19.11.17)
of the typed notes is available here.
Check back for upadtes!


Reading:


Lectures 10-11 (14:00-16:00; Tue 24.10.17) 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. Outline of the hierarchy of approximations: linear, quasilinear, weak turbulence, strong turbulence.

Lecture Notes sec 1.1-1.7, 1.9, 2

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

Lecture 12 (10:00-11:00; Mon 30.10.17) Linear theory: initial-value problem for the Vlasov-Poisson system, Laplace-transform solution, the dielectric function, Landau prescription for calculating velocity integrals.

Lecture Notes sec 3.1, 3.2

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)



Lectures 13-14 (16:00-18:00; Mon 30.10.17) Solving the plasma dispersion relation in the slow damping/growth limit. Langmuir waves. Landau damping and kinetic instabilities. Sound waves, their damping, ion-acoustic instability, ion-Langmuir oscillations.

You are ready to do Q1-3 of the Problem Set (+Q4 if you read the optional sec 5)

Lecture Notes sec 3.3-3.5, 3.7-3.10


Lecture 15 (10:00-11:00; Mon 6.11.17) Energy conservation. Heating. Entropy and free energy. 

Lecture Notes sec 4.1-4.2

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

Lectures 16-17 (16:00-18:00; Mon 6.11.17) Perturbed distribution function: ballistic response and phase mixing. Phase mixing: role of collisions; coarse-graining. Structure of the perturbed distribution near a resonance: the Case-van Kampen mode. Quasilinear theory: general scheme. QLT for bump-on-tail instability in 1D.

You are ready to do Q5-7 of the Problem Set

Lecture Notes sec 4.3-4.6, 7.1-7.3



Lecture 18 (10:00-11:00; Mon 13.11.17) QLT for bump-on-tail instability in 1D cont'd:  wave spectrum and the QL plateau, heating of the thermal bulk.

You are ready to do Q8-9 of the Problem Set

Lecture Notes sec 7.4-7.6

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


Lecture 19 (16:00-17:00; Mon 13.11.17) Quasiparticles.

Lecture Notes sec 7.9

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
Lectures 20-28 (Mon 13.11.17 - Tue 28.11.17) Prof James Binney
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 20 (17:00-18:00; Mon 13.11.17)
Lecture 21 (10:00-11:00; Mon 20.11.17)
Lectures 22-23 (16:00-18:00; Mon 20.11.17)
Lecture 24 (10:00-11:00; Mon 27.11.17)
Lectures 25-26 (16:00-18:00; Mon 27.11.17)
Lectures 27-28 (14:00-16:00; Tue 28.11.17)

Problem Set 3: you will find it
in the typed Lecture Notes

Problem class 3
14:00-16:00 on Tue 5.12.17
Homework due 23:59 on 3.12.17
to Prof Binney (in TP)

Lecture Notes

READING LIST

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