physicslogo

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
  & Dr Eugene Vasiliev

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 2016

LECTURES
Room L4 (A. Wiles Building)
Monday 10:00-11:00 (week 1)
Fisher Room (DWB)
Mondays 10:00-11:00 (weeks 4-8); 16:00-18:00 (weeks 1-8)
& Tuesdays 14:00-15:00 (weeks 1 & 3), 14:00-16:00 (week 8)


CLASSES

  Fisher Room (DWB)
Tuesdays 14:00-16:00 (weeks 4, 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 1-8 (Mon 10.10.16 - Mon 24.10.16) 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 collision operator. 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 10.10.16) in L4, A. Wiles Building!
Lectures 2-3 (16:00-18:00; Mon 10.10.16) in Fisher Room!
Lecture 4 (14:00-15:00; Tue 11.10.16)
Lectures 5-6 (16:00-18:00; Mon 17.10.16)
Lectures 7-8 (16:00-18:00; Mon 24.10.16)

Problem class 1
14:00-16:00 on Tue 1.11.15

Homework due 23:59 Sun 30.10.16
to Dr Vasiliev (in TP)

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
Lectures 9-17 (Tue 25.10.16 - Mon 14.11.16) 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 (“bump 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. Quasilinear theory: general scheme. QLT for bump-on-tail instability in 1D.

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 typed Lecture Notes

Problem class 2
14:00-16:00 on Tue 22.11.16
Homework due 23:59 Sun 20.11.16
to Dr Vasiliev

The latest version (15.11.16)
of the typed notes is available here.
Check back for upadtes!


Reading:


Lecture 9 (14:00-15:00; Tue 25.10.16) Kinetic description of a plasma: Debye shielding,  micro- vs. macroscopic fields, Vlasov-Landau-Maxwell equations.

Lecture Notes sec 1.1-1.6

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 10 (10:00-11:00; Mon 31.10.16) Vlasov-Landau-Maxwell equations cont'd. Basic properties of the Landau collision integral. Plasma frequency. Slow equilibrium and fast fluctuations.

Lecture Notes sec 1.7, 1.9, 2.1-2.3


Lectures 11-12 (16:00-18:00; Mon 31.10.16) Slow equilibrium and fast fluctuations cont'd. Outline of the hierarchy of approximations: linear, quasilinear, weak turbulence, strong turbulence. 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.

Lecture Notes sec 2.3, 2.4, 3.1-3.5

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)
Krall & Trivelpiece sec 9
(kinetic stability theory)


Lecture 13 (10:00-11:00; Mon 7.11.16) Linear theory completed: sound waves, their damping, ion-acoustic instability, ion-Langmuir oscillations.

You are ready to do Q1-3 of the Problem Set

Lecture Notes sec 3.7-3.10


Lectures 14-15 (16:00-18:00; Mon 7.11.16) Energy conservation. Heating. Entropy and free energy. 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.

You are ready to do Q4-6 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 16 (10:00-11:00; Mon 14.11.16) Quasilinear theory: general scheme. QLT for bump-on-tail instability in 1D.

Lecture Notes sec 6.1-6.5

Krall & Trivelpiece sec 10
Kadomtsev sec I.3
Sagdeev & Galeev sec II-2
(...and read on for more advanced topics)
Tsytovich sec 3, 5, 7
(on plasmon kinetics and beyond)

Lecture 17 (16:00-17:00; Mon 14.11.16) QLT for bump-on-tail instability in 1D cont'd: : wave spectrum and the QL plateau, heating of the thermal bulk. Kinetics of plasmons.

You are ready to do Q7-8 of the Problem Set

Lecture Notes sec 6.6-6.7





PART III: KINETIC THEORY
OF SELF-GRAVITATING
SYSTEMS
Lectures 18-26 (Mon 14.11.16 - Tue 29.11.16) Prof James Binney
Mean-field models and their evolution exemplified by star clusters. Isothermal sphere, escape velocity, evaporation. Virial theorem, negative specific heat, gravothermal catastrophe. Heggie's theorem, binaries as a heat source. Angle-action coordinates, Jeans' theorem. Equation for slow evolution of mean-field distribution function: computation of cross correlations of fluctuating quantities (non-examinable). Enhanced diffusion along resonances. Application to stellar discs: particle dressing and swing amplification making Poisson noise anomalously large. Ultimate destabilisation of disc at collisionless level.

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 bejaviour of a kinetic system that is not close to a Maxwellian equilibrium.

Lecture 18 (17:00-18:00; Mon 14.11.16) Dr Eugene Vasiliev
Lecture 19 (10:00-11:00; Mon 21.11.16) Dr Eugene Vasiliev
Lectures 20-21 (16:00-18:00; Mon 21.11.16) Dr Eugene Vasiliev
Lecture 22 (10:00-11:00; Mon 28.11.16) Prof James Binney
Lectures 23-24 (16:00-18:00; Mon 28.11.16) Prof James Binney
Lectures 25-26 (14:00-16:00; Tue 29.11.16) Prof James Binney

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

Problem class 3
14:00-16:00 on Tue 6.12.16
Homework due 23:49 on 4.12.16
to Prof Binney

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)
  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)
  14. V. E. Zakharov, V. S. Lvov & G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer 1992) (Amazon) (updated online version)
PART III:
  1. J. Binney & S. Tremaine, Galactic Dynamics (Princeton University Press 2008) (Amazon)