Kinetic Theory - Material for the year 2019-2020

2019-2020
Lecturer(s): 
Prof. Paul Dellar
External Lecturer(s): 
Dr Jean-Baptiste Fouvry, Prof. Alex Schekochihin
Course Term: 
Michaelmas

Assessment type:

Course Overview: 

28 lectures.
Course weight: 1.75 units
Areas: CMT, Astro, foundational course.
Sequel: Advanced Fluid Dynamics (HT), Collisionless Plasma Physics (HT), Collisional Plasma Physics(TT),
Galactic and Planetary Dynamics (HT).

Please use this page to submit your Kinetic Theory homework via the 'assignments' tab.
Visiting and Short Option Students Only: Please e-mail your work to your TA/Tutor by the required deadline, copying in mmathphys@maths.ox.ac.uk
Problem sheets are hosted via the link above.

Course Synopsis: 

Part I (9 lectures). Kinetic theory of gases. 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.

Part II (10 lectures). Kinetic theory of plasmas and quasiparticles. 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 (non-examinable), 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. Elements of kinetic stability theory. Quasilinear theory: general scheme. QLT for bump-on-tail instability in 1D. Introduction to quasiparticle kinetics.

Part III (9 lectures). Kinetic theory of self gravitating systems.
Unshielded nature of gravity and implications for self-gravitating systems. Virial theorem, negative specific heat and impossibility of thermal equilibrium. Escape, impact of fluctuations. Mean-field approximation, angle-action variables, self-consistent potential, biorthonormal potential-density pairs. Relaxation driven by fluctuations in mean-field. Long-time response to initial perturbation. Fokker-Planck equation. Computation of the diffusion coeffcients in terms of resonant interactions. Application to a tepid disc.