MMathPhys CM and Astro/Plasmas/Fluids Strands

STATISTICAL MECHANICS (24 lectures, MT)

Lecturer: ??

Syllabus [written by F. Essler, R. Golestanian]:

Advanced Statistical Mechanics (16 lectures, HT)

Lecturer: Ramin Golestanian

Syllabus [written by R. Golestanian]:

NOTE from A.A.S.: I have removed Kinetic Theory and Fluid Dynamics material into corresponding courses.
Perhaps what remains should be renamed "Stochastic Methods"?
[I have also removed the eponymous course from the Maths part as redundant]

1. Dynamics of Stochastic Processes (12 lectures)
•    Langevin equation and mean-squared displacement versus time, fundamentals of Molecular Dynamics and Stochastic Rotation Dynamics simulation methods
•    Probabilistic description of stochastic process, Fokker-Planck equation
•    Kramers rate theory, escape probability and first-passage time
•    Master equation, equilibrium and detailed-balance, fundamentals of Monte Carlo simulation method, chemical reactions, one-step processes (traffic models), fundamentals of Lattice Boltzmann simulation method
•    Diffusion-reaction processes and pattern formation
•    Heterogeneous catalysis and the Michaelis-Menten rule in enzymatic reactions
•    Rectification of stochastic motion and Brownian ratchets

2. Fluctuations and Response (4 lectures)
•    Equilibrium fluctuations, correlation functions
•    Density fluctuations, hydrodynamic fluctuations and the long-time tail
•    Linear response theory, response function, causality and Kramers-Kronig relations
•    Fluctuation-dissipation theorem near equilibrium
•    Small-system (stochastic) thermodynamics, Jarzynskii inequality
•    Generalised fluctuation-dissipation theorem in nonequilibrium systems

Complex Systems (16 lectures, HT)

Lecturer: Mason Porter

Syllabus [written by M. Porter]: Maths C6.2b (in future)
(Prerequisite: Statistical Mechanics or another course in the same, e.g., Maths C6.2a)

Introduction to Networks (5 lectures)
Percolation, fractals, self-organized criticality, and power laws (2 lectures)
Stochastics and generative models: random walks, preferential attachment, master equations (3 lectures)
Dynamical systems on networks (3 lectures): includes models of epidemics, social influence, voter models, etc. and how they are affected by network architecture
Agent-based models (1 lecture)
Numerical methods (2 lectures): Monte Carlo, simulated annealing, etc.

Critical Phenomena (?? lectures, TT)

Lecturer: ??

Syllabus [written by ??]:
(Prerequisites: Quantum Field Theory, Statistical Mechanics)

Phase transitions in simple systems:
    Landau theory
    Scaling Hypothesis
        1. Saddle Point Approximation and Mean Field Theory
        2. Continuous Symmetry Breaking and Goldstone Modes
        3. Discrete Symmetry Breaking
Real-Space RG
        1. Block-spin transformations.
Basic theory of the RG
        1. 1-loop beta function.
        2. Callan-Szymanzik equation.

Soft Condensed Matter (20?? lectures, TT)

Lecturer: Ramin Golestanian

Syllabus [written by R. Golestanian]:
(Prerequisites: Statistical Mechanics, Advanced Statistical Mechanics, Kinetic Theory, Fluid Dynamics)

NOTE from A.A.S.: Could this be massaged into either 16 or 24 lectures? (given the material in other courses)

1. Polymers (6 lectures)
•    Equilibrium properties: random walk on a lattice, distribution of end-end distances, polymer elasticity, bead-spring model, self-avoiding walk and the theta point, Flory’s calculation of scaling for a single polymer chain, Flory-Huggins theory for concentrated polymer solutions, polymer—polymer mixtures and microphase separation
•    Dynamics: Rouse model, Zimm theory, reptation, polymers pushed through a channel, an example of de Gennes scaling
•    Semiflexible polymers

2. Membranes (1 lecture)
•    Bending elasticity, Helfrich energy
•    Dynamics

3. Liquid Crystals (5 lectures)
•    Equilibrium properties: Landau–de Gennes theory (order parameters and the elastic free energy), surfaces, Fredrick’s transition, more general geometries, twisted nematic displays, topological defects
•    Hydrodynamics, equations of nematodynamics, Leslie angle, active nematics, permeation in cholesterics

4. Colloids (6 lectures)
•    Equilibrium, vdW-Lifshitz theory of dispersion interactions, DLVO potential
•    Electrostatics, Debye screening, double-layer, Poisson-Boltzmann equation, counterion condensation, like-charge attraction
•    Electrokinetics
•    Electrophoresis, Thermophoresis, Diffusiophoresis, etc
•    Sedimentation, hydrodynamic interaction
•    Smoluchowski flocculation

5. Wetting (2 lectures)
•    Laplace pressure, capillarity, shape of interfaces, meniscus
•    Contact line, Young equation, partial and complete wetting, patterned substrates
•    Hydrodynamics, viscous fingering, contact line dynamics (Landau-Levich)

Topology in Quantum Systems (10?? lectures, TT)

Lecturer: ??

Syllabus [written by ??]:
(Prerequisites: ???)

NOTE from A.A.S.: Could this be massaged into 16 lectures? (might just as well)

Fractional Quantum Hall Effect
Topological Quantum Field Theories
Topological Quantum Computation

Conformal Field Theory --- see PP Strand

KINETIC THEORY (24 lectures, MT)

Lecturer: could be a joint operation: gas kinetics someone from Maths (Paul Dellar?), plasma Alex Schekochihin or the new UL, self-gravitating James Binney or John Magorrian

Syllabus [written by J. Binney, R. Golestanian, J. Magorrian, A. Schekochihin]:

1-2. Liouville Theorem. BBGKY hierarchy and derivation of Boltzmann's equation.
3. H theorem, Maxwell's distribution.
4-6. Derivation of fluid equations. Transport: viscosity and thermal diffusvity.
7. Onsager symmetries.
[NEW IDEAS: BASICS OF KINETIC LANGUAGE]
8. Kinetics in an external field (TBD) OR Diffusion (TBD).
9. Plasma: charged particles + self-consistent EM fields. Debye screening.
10-13. BBGKY for plasma. Landau collision integral. Outline of the derivation of two-fluid (Braginskii) equations. MHD.
14-16. Collisionless plasma in electrostatic field. Dielectric permittivity, Landau damping, kinetic instabilities, waves.
[NEW IDEA: PARTICLES CAN INTERACT WITH FIELDS]
17. Outline of the quasilinear theory and onwards.
18. Self-gravitating kinetics and the resultant fluid equations.
19-20. Invariants of motion and the Jeans theorem. Non-Maxwellian (collisionless) equilibria.
21. Anisotropic distributions.
[NEW IDEA: NOT EVERYTHING IS MAXWELLIAN]
22-24. Kinetics of quasiparticles (phonons, TBD). UV catastrophe.
[NEW IDEA: QUASIPARTICLES]

FLUID DYNAMICS (24 lectures, HT)

Lecturer: could be a joint operation: someone from Soft Matter? Maths? Alex Schekochihin or new plasma UL can do MHD

Syllabus [written by R. Golestanian, A. Schekochihin]
(Prerequisite: Kinetic Theory)

Fluid equations. Stress tensor. Conservation laws. Incompressibility. The Navier-Stokes Equation.
Low Reynolds number hydrodynamics.
---> How much of standard fluid dynamics shall we do? cf. Maths B6a, B6b
    Boundary layers? Flows past things?
    Rotating fluids? Stratified? Sheared? (but see GFD)
    Waves? Shock waves? (but there will be compressible hydro in AFD)
    Turbulence?
MHD
    MHD equations
    Conservation laws (mass, momentum, energy, cross-helicity, helicity, flux)
    Equilibria
    Energy principle
    Instabilities
    Waves
    Elsasser solutions
    Magnetic reconnection (intro)
Mathematical similarities and differences between MHD and polymeric fluids
Non-Newtonian fluid dynamics

Nonlinear Systems (16 lectures, HT)

Lecturer: Mason Porter

Syllabus [written by M. Porter]: Maths B8b

Galactic and Planetary Dynamics - Celestial Mechanics for the 21st Century (24 lectures, HT)

Lecturer: James Binney and/or John Magorrian

Syllabus [written by J. Magorrian]:
(Prerequisite: Kinetic Theory)

NOTE from A.A.S.: any chance this can be done in 16 lectures?

Introduction to prototypical systems: Galactic disk, globular
clusters, protoplanetary disks. Characteristic length
and time scales.
[1 lecture]

Collisionless approximation. Derivation of Jeans and virial
equations. Simple applications: need for closure relations.
[1 lecture]

Collisionless spherical systems: orbits; Jeans' theorem. Equilibrium
models: choice of f. Inferring f, Phi from observations.
[2 lectures]

Collisional spherical systems. Negative specific heat and
gravothermal catastrophe. Fokker--Planck equation:
fluctuation--dissipation theorem, equipartition. Application to
globular clusters.
[3 lectures]

Orbits in flattened, non-rotating potentials: integrals of motion,
orbit families. Introduction to action-angle variables: tori. Jeans'
theorem revisited. Simple flattened galaxy models.
[5+ lectures]

Orbits in rotating potentials. Lagrange points.
[1-2 lectures]

Disc dynamics: winding problem, density waves, bars.
[4+ lectures]

Interactions between stellar systems. Dynamical friction. Tidal
shocks. Disk heating mechanisms.
[3 lectures]

TO DO: merge protoplanetary discs into previous 3 headings
[+N lectures]

Collisions in protoplanetary disks. Coagulation equation and runaway
growth. And there was Man.
[1 lecture]

Stellar Structure and Evolution (16 lectures, HT)

This should be joint with C1. P. Podzialowski's course?

Plasma Physics (16 lectures, TT)

Lecturer: New Plasma Theory UL in 2012 or 2013 (so we might not have this course in the first year)
OR someone from Culham (Steve Cowley would be ideal, but probably won't do it for lack of time)
OR Tony Bell if he is persuadable

Syllabus [written by A. Schekochihin]:
(Prerequisites: Kinetic Theory, Fluid Dynamics)

1-2. Orbit theory, adiabatic invariants (advanced level, Kruskal-style)
3. Basic plasma parameters, kinetic theory (BBGKY), collisions
4. Two-fluid theory, Braginskii equations (collisional transport), MHD
5-7. Resistive MHD: magnetic reconnection, tearing modes (also two-fluid?)
8. Landau damping; waves and instabilities in unmagnetised plasmas
---> Some quasilinear theory and then weak turbulence?
9-11. Waves in magnetised plasmas
12. Cold plasma limit and waves
13-16. Reduced theories
    Drift kinetics and double-adiabatic theory
    Reduced MHD
    Gyrokinetics

Astrophysical Fluid Dynamics

Lecturer: Julien Devriendt and/or Adrianne Slyz?

Syllabus [written by J. Devriendt, A. Slyz]:
(Prerequisite: Fluid Dynamics)

Geophysical Fluid Dynamics

Lecturer: David Marshall or a new UL in AOPP?

Syllabus [written by D. Marshall]:
(Prerequisite: Fluid Dynamics)

Turbulence (16 lectures, TT)

Lecturer: Alex Schekochihin, but this may be an overload, perhaps someone from AOPP?

Syllabus [written by A. Schekochihin]:
(Prerequisite: Fluid Dynamics)

1. Kolmgorov 1941 theory and general philosophy of turbulent cascades (Obukhov)
2. Turbulent diffusion, mixing of a scalar
3. General framework of mean-field theory, closures (basic idea, not detailed exposition)
4. Kinematics of turbulence: correlation functions
5-6. Exact laws (Kolmogorov's 4/5 and Yaglom's 4/3)
7-8. Intermittency: basic ideas; refined similarity (Kolmogorov 1962); She-Leveque theory
9-10. Turbulence in systems with waves: introduction to weak turbulence theory
11. MHD (Alfvenic) turbulence and the general idea of critically balanced turbulence in wave systems
12-13. Rotating and stratified turbulence
14-15. Turbulent dynamo
16. Plasma turbulence
[last two topics discardable in case of Zeitnot]

Some relevant notes: Turbulence, Dynamo, Critical Balance

GENERAL RELATIVITY --- see PP Strand

Advanced General Relativity --- see PP Strand

Cosmology --- see PP Strand

Astroparticle Physics --- see PP Strand