The kinetic theory of self-gravitating systems

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Table of contents

1 Introduction 
1.1 Context
1.2 Stellar discs
1.3 Hamiltonian Dynamics
1.4 Overview
2 Secular diffusion 
2.1 Introduction
2.2 Collisionless dynamics
2.2.1 Evolution equations
2.2.2 Matrix method
2.2.3 Diffusion coefficients and statistical average
2.3 Self-induced collisional dynamics
2.3.1 Evolution equations
2.3.2 Fast timescale amplification
2.3.3 Estimating the collision operator
2.3.4 The Balescu-Lenard equation
2.3.5 The bare case: the Landau equation
2.3.6 The multi-component case
2.3.7 H-theorem
2.4 Conclusion
2.4.1 Future works
Appendices
2.A Derivation of the BBGKY hierarchy
2.B Derivation of the Balescu-Lenard equation via the BBGKY hierarchy
2.B.1 Solving for the autocorrelation
2.B.2 Application to inhomogeneous systems
2.B.3 Rewriting the collision operator
2.C Functional approach to the Landau equation
2.C.1 Functional integral formalism
2.C.2 Application to inhomogeneous systems
2.C.3 Inverting the constraint
2.C.4 Recovering the Landau collision operator
3 Razor-thin discs 
3.1 Introduction
3.2 Angle-action coordinates and epicyclic approximation
3.3 The razor-thin WKB basis
3.4 WKB razor-thin amplification eigenvalues
3.5 WKB limit for the collisionless diffusion
3.6 WKB limit for the collisional diffusion
3.7 Application to radial diffusion
3.7.1 A razor-thin disc model
3.7.2 Shot noise driven radial diffusion
3.7.2.1 Collisionless forced radial diffusion
3.7.2.2 Collisional radial diffusion
3.7.3 Diffusion timescale
3.7.4 Interpretation
3.8 Conclusion
3.8.1 Future works
4 Razor-thin discs and swing amplification 
4.1 Introduction
4.2 Calculating the Balescu-Lenard diffusion flux
4.2.1 Calculating the actions
4.2.2 The basis elements
4.2.3 Computing the response matrix
4.2.4 Sub-region integration
4.2.5 Critical resonant lines
4.3 Application to self-induced radial diffusion
4.3.1 Initial diffusion flux
4.3.2 Diffusion timescale
4.3.3 Why swing amplification matters
4.3.3.1 Turning off collective effects
4.3.3.2 Turning off loosely wound contributions
4.4 Comparisons with N􀀀body simulations
4.4.1 A N􀀀body implementation
4.4.2 Scaling with N
4.4.3 Scaling with
4.4.4 Secular phase transitions
4.5 Conclusion
4.5.1 Future works
Appendices 
4.A Kalnajs 2D basis
4.B Calculation of @
4.C Recovering unstable modes
4.C.1 The response matrix validation
4.C.2 The N􀀀body code validation
4.D The case of self-gravitating spheres
4.D.1 The 3D calculation
4.D.2 An exemple of application: the cusp-core problem
5 Thickened discs 
5.1 Introduction
5.2 Angle-action coordinates and epicyclic approximation
5.3 The thickened WKB basis
5.4 WKB thick amplification eigenvalues
5.4.1 WKB response matrix
5.4.2 A thickened Q factor
5.5 WKB limit for the collisionless diffusion
5.6 WKB limit for the collisional diffusion
5.7 Application to disc thickening
5.7.1 A thickened disc model
5.7.2 Shot noise driven resonant disc thickening
5.7.2.1 Collisionless forced thickening
5.7.2.2 Collisional thickening
5.7.2.3 Vertical kinetic heating
5.7.3 Diffusion timescale
5.7.4 Radial migration
5.7.5 Thickening induced by bars
5.7.6 GMCs triggered thickening
5.8 Conclusion
5.8.1 Future works
Appendices
5.A Antisymmetric basis
5.B A diagonal response matrix
5.C From thick to thin
5.C.1 The collisionless case
5.C.2 The collisional case
6 Quasi-Keplerian systems 
6.1 Introduction
6.2 The associated BBGKY hierarchy
6.3 Degenerate angle-action coordinates
6.4 Averaging the evolution equations
6.5 The degenerate Balescu-Lenard equation
6.5.1 The one-component Balescu-Lenard equation
6.5.2 The multi-component Balescu-Lenard equation
6.6 Applications
6.6.1 Razor-thin axisymmetric discs
6.6.2 Spherical clusters
6.6.3 Relativistic barrier crossing
6.7 Conclusion
6.7.1 Future works
Appendices
6.A Relativistic precessions
6.B Multi-component BBGKY hierarchy
6.C From Fokker-Planck to Langevin
7 Conclusion 
7.1 Overview
7.2 Outlook and future works