The following lecture topics were covered in class. Notes have been posted where available.

Introduction and Basic Transport Concepts

- Form of Transport Equations
- Random Walk Picture – Guiding Centers
- Coulomb Cross Section and Estimate
- Fusion Numbers: (a) Banana Diffusion, (b) Bohm and Gyro-Bohm Diffusion
- Transport Matrix Structure: (a) Onsager Symmetry

Diffusion Equation Solutions and Scaling

- Initial Value Problem
- Steady State Heating Problem (temperature) w/ Power Source
- Density Behavior: (a) Include Pinch Effect
- Magnetic Field Diffusion
- Velocity Space Diffusion: (a) Relaxation Behavior w/o Friction, (b) Need for Friction in Equilibration

Coulomb Collision Operator Derivation

- Written Notes for these Lectures (2 sets):

(1) Collisions and Transport Theory I (PDF)

(2) Collisions and Transport Theory II: Electrical Conductivity - Spitzer Problem (PDF) - Fokker-Planck Equation Derivation

Coulomb Collision Operator Derivation II

- Calculation of Fokker-Planck Coefficients
- Debye Cutoff: (a) Balescu-Lenard form and (b) Completely Convergent Form
- Collision Operator Properties: (a) Conservation Laws, (b) Positivity, (c) H-Theorem

Coulomb Collision Operator Derivation III

- Electron-ion Lorentz Operator
- Energy Equilibration Terms
- Electrical Conductivity - The Spitzer-Harm Problem: (a) Example of Transport Theory Calculation
- Runaway Electrons

Classical (collisional) Transport in Magnetized Plasma

- Moment Equations
- Expansion About Local Thermal Equilibrium (Electron Transport)
- Linear Force/Flux Relations
- Transport Coefficients: Dissipative and Non-dissipative Terms
- Physical Picture of Non-dissipative Terms: (a) “Diamagnetic” Flow Terminology and Physics from Pressure Balance and Show that B
**in**< B**out**, (b) “Magnetization” Flow Terminology from FLR, J=Curl M - Physical Picture of Dissipative Flows: (a) Guiding Center Scattering, (b) Random Walk

Classical Transport in Guiding Center Picture

- Alternate formulation displays microscopic physics more clearly (needs Gyrofrequency » Collision Frequency)
- Follows Hierarchy of Relaxation Processes - “Collisionless Relaxation”
- Transformation to Guiding Center Variables: (a) Physical Interpretation
- Gyro-averaged Kinetic Equation IS Drift Kinetic Equation
- Gyro-averaged Collision Operator: Spatial KINETIC Diffusion of Guiding Center
- Transport Theory Ordering

Classical Transport in Guiding Center Picture II

- Expansion of Distribution Function and Kinetic Equation: (a) Maximal Ordering (Math and Physics)
- Zero Order Distribution - Local Maxwellian
- 1st order - Generalized Spitzer problem: (a) Inversion of (Velocity Space) (b) Collision Operator, (c)Integrability Conditions and Identification of Thermodynamic Forces
- 2nd order - Transport Equations: (a) Integrability Conditions Yield Transport Equations, (c) Complete Specification of Zero Order f
- Transport Coefficient Evaluations: (a) Equivalence to Prior Results
- Physical Picture of Flows: (a) Guiding Center Flows and “Magnetization” Flows

Random (Stochastic) Processes, Fluctuation, etc. (Intro.)

- Probability and Random Variables
- Ensemble Averages
- Stochastic Processes: (a) Fluctuating Electric Fields, (b) Correlation Functions, (c) Stationary Random Process
- Integrated Stochastic Process - Diffusion: (a) Example of Integral of Electric Field Fluctuations giving Velocity Diffusion, (b) Integrated Diffusion Process

Distribution Function of Fluctuations

- Central Limit Theorem
- “Normal Process” Definition: (a) Cumulant Expansion Mentioned, (b) Example of Guiding Center Diffusion Coefficient

Fluctuation Spectra – Representation of Fields

- Fourier Representation of Random Variable: (a) Mapping of “All Curves” to Set of All Fourier Coefficients, (b) Fourier Spectral Properties for Stationary Process, (c) Equivalence of “Random Phase Approximation”
- Physical Interpretation in Terms of Waves
- Definition of Spectrum as FT of Correlation Function
- Generalize to Space & Time Dependent Fields: (a) Statistical “Homogeneity”
- Continuum Limit Rules

Diffusion Coefficient from Fluctuation Spectrum

- Stochastic Process Evaluation of Particle Velocity Diffusion Coefficient from Homogeneous, Stationary Electric Field Fluctuation Spectrum
- Physical Interpretation via Resonant Waves
- Superposition of Dressed Test Particles - Field Fluctuations
- Diffusion (Tensor) from Discreteness Fluctuations - Collision Operator
- Correlation Time Estimates

Turbulent Transport – Drift Waves

- Space Diffusion of Guiding Center from Potential Fluctuations and ExB Drift
- Estimates and Scalings from Drift Wave Characteristics: (a) Bohm scaling, (b) Gyro-Bohm Scaling from Realistic Saturated Turbulence Level

Coulomb Collision Operator Properties

- Correct Details of Electron-ion Operator Expansion Including Small v Behavior
- Energy Scattering

Full Classical Transport in Magnetized Plasma Cylinder

- Includes Ion and Impurity Transport
- Estimates and Orderings for Electron and Ion Processes
- Ambipolarity and Two “Mantra” of Classical Transport: (a) “Like Particle Collisions Produce no Particle Flux”, (b) “Collisional Transport is Intrinsically Ambipolar”, (c) Microscopic Proof of Mantra for Binary Collisions
- Moment Equation Expressions for Perpendicular Flows: (a) Flux-Friction Relations, (b) Leading Order Approximations
- Particle Flux Relations
- Non-Ambipolar Fluxes, Viscosity, Plasma Rotation: (a) Limits to Mantra, Calculation of Ambipolar Field, (b) Impurity Transport, and Steady State Profiles

Fast ion Collisions, Alpha Slowing Down and Fusion Alpha Distribution

Like-Particle Collisional Transport

- Ion Thermal Conduction Calculation
- Guiding Center Picture Calculation
- Heat Flux - Heat Friction Relation

Neoclassical Transport

- Introductory concepts: (a) Particle orbits and Magnetic Geometry, (b) Particle Mean Flux Surface, Moments, Flows and Currents
- Tokamak Orbit Properties: (a) Trapped Particle Fraction, (b) Bounce Time (Circulation Time)
- Bounce Averages
- Tokamak Moments and Flux-Surface averages: (a) Constant of Motion variables, (b) Moments @ Fixed Space Position, (c) Flux-Surface Averaged Moments, (d) Bootstrap Current (Magnetization Piece)
- Moment Relations and Definitions
- Bounce Average Kinetic Equation Derivation
- Perturbation Theory for The “Banana” Regime
- Banana Regime Transport Theory: (a) Particle Moment, (b) Energy Moment, (c) Toroidal Current, (d) Transport Coefficient Formalism
- Structure of the Transport Matrix: (a) Onsager Symmetry
- Evaluation of Neoclassical Transport
- Analytic Details of Thermal Conduction Calculation Including Complete Expression

Ware Pinch Effect

Magnetization Bootstrap Current

Simplified Implicit Transport Coefficient

Diagonal Transport Coefficients

Onsager Symmetry of Transport Coefficients