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Table of Contents
Intro; Supervisor's Foreword; Abstract; Publications Related to this Thesis; Contents; Abbreviations; Some Important Notations; Physical Constants (SI Units); 1 Introduction; 1.1 The Hierarchy of Plasma Models; 1.1.1 Single Particle Motion; 1.1.2 Kinetic Theory; 1.1.3 Quasineutrality; 1.1.4 Fluid Models; 1.2 Collisions in Plasmas; 1.2.1 Collisional Plasmas; 1.2.2 Collisionless Plasmas; 1.3 Collisionless Plasma Equilibria; 1.3.1 The `Forward' and `Inverse' Approaches; 1.3.2 Motivating Translationally Invariant Vlasov-Maxwell (VM) Equilibria; 1.3.3 Magnetic Reconnection
1.3.4 Forward Approach for One-Dimensional (1D) VM Equilibria1.3.5 Inverse Approach for 1D VM Equilibria; 1.3.6 Previous Work on VM Equilibria; 1.4 Thesis Motivation and Outline; 1.4.1 Outline of the Thesis; References; 2 The Use of Hermite Polynomials for the Inverse Problem in One-Dimensional Vlasov-Maxwell Equilibria; 2.1 Preamble; 2.2 Introduction; 2.2.1 Hermite Polynomials in Fluid Closure; 2.2.2 Hermite Polynomials in VM Plasma Theory; 2.2.3 Hermite Polynomials for Exact VM Equilibria; 2.3 Formal Solution by Hermite Polynomials; 2.3.1 Weierstrass Transform
2.3.2 Two Interpretations with Respect to Our Equations2.3.3 Formal Inversion of the Weierstrass Transform; 2.4 Mathematical Validity of the Method; 2.4.1 Convergence of the Hermite Expansion; 2.5 Non-negativity of the Hermite Expansion; 2.5.1 Possible Negativity of the Hermite Expansion; 2.5.2 Detailed Arguments; 2.5.3 Summary; 2.6 Illustrative Case of the Use of the Method: Correspondence with the Fourier …; 2.7 Summary; References; 3 One-Dimensional Nonlinear Force-Free Current Sheets; 3.1 Preamble; 3.2 Introduction; 3.2.1 Force-Free Equilibria and the Plasma Beta
4 One-Dimensional Asymmetric Current Sheets4.1 Preamble; 4.2 Introduction; 4.2.1 Asymmetric Current Sheets; 4.2.2 Modelling the Magnetopause Current Sheet; 4.3 Exact VM Equilibria for 1D Asymmetric Current Sheets; 4.3.1 Theoretical Obstacles; 4.3.2 Outline of Basic Method; 4.4 The Numerical/``tanh'' Equilibrium DF; 4.4.1 The Pressure Function; 4.4.2 Inverting the Weierstrass Transform; 4.5 The Analytical/``Exponential'' Equilibrium DF; 4.5.1 The Pressure Tensor; 4.5.2 The DF; 4.5.3 Plots of the DF; 4.6 Discussion; References; 5 Neutral and Non-neutral Flux Tube Equilibria; 5.1 Preamble
1.3.4 Forward Approach for One-Dimensional (1D) VM Equilibria1.3.5 Inverse Approach for 1D VM Equilibria; 1.3.6 Previous Work on VM Equilibria; 1.4 Thesis Motivation and Outline; 1.4.1 Outline of the Thesis; References; 2 The Use of Hermite Polynomials for the Inverse Problem in One-Dimensional Vlasov-Maxwell Equilibria; 2.1 Preamble; 2.2 Introduction; 2.2.1 Hermite Polynomials in Fluid Closure; 2.2.2 Hermite Polynomials in VM Plasma Theory; 2.2.3 Hermite Polynomials for Exact VM Equilibria; 2.3 Formal Solution by Hermite Polynomials; 2.3.1 Weierstrass Transform
2.3.2 Two Interpretations with Respect to Our Equations2.3.3 Formal Inversion of the Weierstrass Transform; 2.4 Mathematical Validity of the Method; 2.4.1 Convergence of the Hermite Expansion; 2.5 Non-negativity of the Hermite Expansion; 2.5.1 Possible Negativity of the Hermite Expansion; 2.5.2 Detailed Arguments; 2.5.3 Summary; 2.6 Illustrative Case of the Use of the Method: Correspondence with the Fourier …; 2.7 Summary; References; 3 One-Dimensional Nonlinear Force-Free Current Sheets; 3.1 Preamble; 3.2 Introduction; 3.2.1 Force-Free Equilibria and the Plasma Beta
4 One-Dimensional Asymmetric Current Sheets4.1 Preamble; 4.2 Introduction; 4.2.1 Asymmetric Current Sheets; 4.2.2 Modelling the Magnetopause Current Sheet; 4.3 Exact VM Equilibria for 1D Asymmetric Current Sheets; 4.3.1 Theoretical Obstacles; 4.3.2 Outline of Basic Method; 4.4 The Numerical/``tanh'' Equilibrium DF; 4.4.1 The Pressure Function; 4.4.2 Inverting the Weierstrass Transform; 4.5 The Analytical/``Exponential'' Equilibrium DF; 4.5.1 The Pressure Tensor; 4.5.2 The DF; 4.5.3 Plots of the DF; 4.6 Discussion; References; 5 Neutral and Non-neutral Flux Tube Equilibria; 5.1 Preamble