Razor-thin discs and swing amplification 

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Secular diffusion The work presented in this chapter is based on Fouvry et al. (2015d,b, 2016a,b).


The previous chapter described the typical fate of self-gravitating systems which can be briefly summed up as follows. As a result of both phase mixing (see figure 1.3.4) and violent relaxation (see figure 1.3.5), self-gravitating systems very efficiently reach quasi-stationary states for the collisionless mean field dy-namics. The systems are virialised and the mean potentials do not strongly fluctuate anymore. Stars follow their orbit set up by the mean field potential and are typically uniformly distributed in phase along each of them. Yet, as gravity is a long-range interaction, self-gravitating systems have the ability to amplify and dress perturbations (see, e.g., figure 1.3.6). These collective effects have two main con-sequences. They may first lead to the spontaneous growth of dynamical instabilities if ever the system was dynamically unstable. Moreover, even for genuinely stable systems, these effects can also lead to polarisation, i.e. a dressing of perturbations and therefore a boost in amplitude of the fluctuations in the system. This self-gravitating amplification is especially important for cold dynamical systems, i.e. within which most of the gravitational support comes from centrifugal forces and for which the velocity dispersion is low. This makes the system strongly responsive. This is for example important for stellar discs, where new stars, born on the cold orbits of the gas, are constantly being supplied to the system. Once the system has reached a quasi-stationary state through these various mixing processes, the mean collisionless dynamics maintains stationarity and such a quiescent system can now only slowly evolve on long timescales.1 This is the timescale for secular evolution, which will be our main interest here. At this stage, only additional fluctuations can drive the system’s evolution. Such considerations fall within the general framework of the fluctuation-dissipation theorem, for which fluctuations occur-ring in the system lead to its dissipation and diffusion. Let us now introduce an important dichotomy on which the two upcoming sections rely. There are two main channels to induce fluctuations in a system. Fluctuations of the first type are induced by external stochastic perturbations, whose non-stationary con-tributions will be felt by the system and will lead therein to slow orbital distortions. As will be discussed in detail in the next section, the efficiency of such secular dynamics is dictated in particular by the match between the temporal frequencies of these perturbations and the system’s natural intrinsic frequencies. We call this framework the collisionless framework. Another source of fluctuations is also present in any system made of a finite number N of particles: these are finiteN effects, also called Poisson shot noise. This graininess can not only be triggered by the finite number of constituents in the system, but can also originate from the variety of its components, e.g., the existence of a mass spectrum of compo-nents. As a direct consequence of the finite number of particles, the system’s self-induced potential is not perfectly smooth, and therefore fluctuates around its mean quasi-stationary value. These unavoid-able and non-vanishing fluctuations may then act as the source of a secular irreversible evolution. We call this framework the collisional framework, in the sense that is relies on encounters between the finite number of particles. Let us finally note that whatever the source of the perturbations, these fluctuations are dressed by collective effects. A proper accounting of the importance of the gravitational polarisation is at the heart of the upcoming derivations.
This dichotomy is essential for all the upcoming sections. It allows us to distinguish secular evolution induced by the system’s environment from secular evolution induced by the system’s internal proper-ties. It is therefore an useful tool to disentangle the respective contributions from nurture and nature in driving the evolution of a self-gravitating system. The aim of the present chapter is to detail the rel-evant formalisms allowing for the description of long-term evolutions induced by (internal or external) potential fluctuations. The following chapters will illustrate applications of this formalism to various astrophysical systems. Let us first focus in section 2.2 on the collisionless framework, where the dynam-ics is driven by external perturbations. Then, in section 2.3, we will consider the collisional framework of diffusion, sourced by the discreteness of these self-gravitating systems.

Collisionless dynamics

Let us first describe the collisionless diffusion that external potential fluctuations may induce. Such externally driven secular evolution can be addressed via the so-called dressed secular collisionless dif-fusion equation, where the source of evolution is taken to be potential fluctuations from an external bath. It has already been a theme of active research, as we now briefly review. Binney & Lacey (1988) com-puted the first- and second-order diffusion coefficients in action space describing the orbital diffusion occurring in a system because of fluctuations in the gravitational potential. This first approach however did not account for collective effects, i.e. the ability of the system to dress and amplify perturbations. Weinberg (1993) emphasised the importance of self-gravity for the non-local and collective relaxation of stellar systems. Weinberg (2001a,b) considered similar secular evolutions while accounting for the self-gravitating amplification of perturbations, and studied the impacts of the properties of the noise processes. Ma & Bertschinger (2004) relied on a quasilinear approach to investigate the diffusion of dark matter induced by cosmological fluctuations. Pichon & Aubert (2006) sketched a time-decoupling approach to solve the collisionless Boltzmann equation in the presence of external perturbations and applied it to a statistical study of the effect of dynamical flows through dark matter haloes on secu-lar timescales. The approach developed therein is close to the one presented in Fouvry et al. (2015d). Chavanis (2012a) considered the evolution of homogeneous collisionless systems when forced by an ex-ternal perturbation, while Nardini et al. (2012) investigated similarly the effects of stochastic forces on the long-term evolution of long-range interacting systems.
In the upcoming section, let us follow Fouvry et al. (2015d) and present a derivation of the appropri-ate secular resonant collisionless dressed diffusion equation. This derivation is based on a quasilinear timescale decoupling of the collisionless Boltzmann equation. This yields two evolution equations, one for the fast dynamical evolution and amplification of perturbations within the system, and one for the secular evolution of the system’s mean DF.

Evolution equations

Let us consider a collisionless self-gravitating quasi-stationary system undergoing external stochastic perturbations. The mean system being quasi-stationary, we introduce its quasi-stationary Hamiltonian H0, associated with the mean potential 0. We assume that throughout its evolution, the system remains integrable, so that one can always define an angle-action mapping (x; v)7!( ; J) appropriate for the Hamiltonian H0. Thanks to Jeans theorem (Jeans, 1915), the mean DF of the system, F , depends only on the actions, so that F =F (J; t). We suppose that an external source is perturbing the system, and we expand the system’s total DF and Hamiltonian as
F tot(J; ; t) = F (J; t)+ F (J; ; t) ; (2.1)
Htot(J; ; t) = H0(J; t)+ e(J; ; t)+ s(J; ; t) :
In the decompositions from equation (2.1), one should pay attention to the presence of two types of potential perturbations. Here, e corresponds to an external stochastic perturbation, while s corre-sponds to the self-response of the system induced by its self-gravity (Weinberg, 2001a). This additional perturbation is crucial to capture the system’s gravitational susceptibility, i.e. its ability to amplify per-turbations. We place ourselves in the limit of small perturbations, so that F F , and e; s 0. Assuming that the system evolves in a collisionless fashion, its dynamics is fully described by the colli-sionless Boltzmann equation (1.9)
With this convention, F tot corresponds to the direction along which individual particles diffuse. Equa-tion (2.31) is the main result of this section.
Let us now briefly discuss the physical content of equation (2.31). First, because it is written as the divergence of a flux, the total number of stars is conserved during the diffusion. One can also note that the diffusion coefficients Dm(J) from equation (2.32) capture the joint and coupled contributions from the external perturbations (via the autocorrelation matrix Cb) and from the self-gravitating suscep-tibility of the system (via the response matrix Mc). The total diffusion coefficients appear therefore as a collaboration between the strength of the external pertubations and the local strength of the system’s amplification. As equation (2.31) describes a resonant diffusion, the external perturbing power spec-trum and the system’s susceptibility have to be evaluated at the local intrinsic frequency ! =m . In this sense, this diffusion equation is appropriate to capture the nature of a collisionless system, via its natural frequencies and susceptibility, as well as its nurture, via the structure of the power spectrum of the external perturbations.
In addition, one can also note that the diffusion equation (2.31) takes the form of a strongly anisotropic diffusion equation in action space. It is anisotropic not only because the diffusion coef-ficients Dm(J) depend on the position in action space, but also because the diffusion associated with one resonance vector m correponds to a diffusion in the preferential direction of the vector m. For a given resonance m, the diffusion is maximum along m and vanishes in the orthogonal directions. A qualitative illustration of the properties of equation (2.31) is given in figure 2.2.1. Finally, note that equation (2.31) is indeed an illustration of the fluctuation-dissipation theorem. The autocorrelation of the fluctuating potential drives the diffusion of the system’s orbital structure.

Self-induced collisional dynamics

In the previous section, we considered the collisionless case where a secular diffusion is induced by external perturbations. However, a given self-gravitating system, even when isolated, may also undergo a secular evolution as a result of its own intrisinc graininess. This is a collisional evolution sourced by finiteN effects.
The dynamics and thermodynamics of systems with long-range interactions has recently been a subject of active research (Campa et al., 2009; Campa et al., 2014), which led to a much better under-standing of the equilibrium properties of these systems, their specificities such as negative specific heats (Antonov, 1962; Lynden-Bell & Wood, 1968; Lynden-Bell, 1999), as well as various kinds of phase transitions and ensemble inequivalences. However, the precise description of their dynamical evolution remains to be improved to offer explicit predictions. We refer the reader to Chavanis (2010, 2013a,b) for a historical account of the development of kinetic theories of plasmas, stellar systems, and other systems with long-range interactions, but let us briefly recall here the main milestones.
The first kinetic theory focusing on the statistical description of the evolution of a large number of particles was considered by Boltzmann in the case of dilute neutral gases (Boltzmann, 1872). For such systems, particles do not interact except during strong local collisions. The gas is assumed to be spa-tially homogeneous and Boltzmann equation describes the evolution of the system’s velocity distribu-tion f(v; t) as a result of strong collisions. This kinetic equation satisfies a H-theorem, associated with an increase of Boltzmann’s entropy.
Boltzmann’s approach was extended to charged gases (plasmas) by Landau (Landau, 1936). For plasmas, particles interact via long-range Coulombian forces, but because of electroneutrality and Debye shielding (Debye & Hückel, 1923a,b), these interactions are screened on a lengthscale of the order of the Debye length, and collisions become essentially local. Neutral plasmas are spatially homogeneous, so that the kinetic equation describes again the evolution of the velocity DF f(v; t), driven by close electrostatic encounters. Because the encounters are weak, one can expand the Boltzmann equation in the limit of small deflections and perform a linear trajectory approximation. In the weak coupling approximation, this leads to the so-called Landau equation. The Landau equation exhibits two formal divergences: one at small scales due to the neglect of strong collisions and one logarithmic divergence at large scales due to the neglect of collective effects, i.e. the dressing of particles by their polarisation cloud (a particle of a given charge has the tendency to be surrounded by a cloud of particles of opposite charges). Landau regularised these divergences by introducing a lower cut-off at the impact parameter producing a deflection of 90 (this is the Landau length) as well as an upper cut-off at the Debye length.
Collective effects were later rigourously taken into account in Balescu (1960) and Lenard (1960), lead-ing to the Balescu-Lenard equation for plasmas. The Balescu-Lenard equation is similar to the Landau equation, except that it includes the square of the dielectric function in the denominator of the potential of interaction in Fourier space. This dieletric function first appeared as a probe of the dynamical stability of plasmas based on the linearised Vlasov equation (Vlasov, 1938, 1945). In the Balescu-Lenard equation, the dielectric function accounts for Debye shielding and removes the large scale logarithmic divergence present in the Landau equation. The Landau equation is recovered from the Balescu-Lenard equation by replacing the dressed potential of interaction by its bare expression, i.e. by replacing the dielectric function by unity. In addition, the Balescu-Lenard equation, as given originally by Balescu and Lenard, exhibits a local resonance condition, encapsulated in a Dirac Dfunction. For such systems, resonant contributions are the drivers of the secular evolution. Integrating over this resonance condition leads to the original form of the kinetic equation given by Lindau.
In parallel to the developments of kinetic equations for plasmas, the secular evolution of self-gravitating systems was also investigated. Self-gravitating systems are spatially inhomogeneous, but the first kinetic theories (Jeans, 1929; Chandrasekhar, 1942, 1943a,b) were all based on the assumption that collisions (i.e. close encounters) between stars can be treated with a local approximation, as if the system were infinite and homogeneous. Relying on the idea that a given star undergoes a large number of weak deflections, Chandrasekhar (1949) developed an analogy with Brownian motion. He started from a Fokker-Planck writing of the diffusion equation and computed the diffusion and friction coef-ficients relying on a binary collision theory. This led to a kinetic equation, often called Fokker-Planck equation in astrophysics, which is the gravitational equivalent of the Landau equation from plasmas. This equation exhibits similarly two divergences: one at small scales due to the mishandling of strong collisions, and one at large scales due to the local approximation, i.e. the assumption that the system is infinite and homogeneous. In the treatment of Chandrasekhar, strong collisions are taken into account without having to introduce a cut-off, so that the small scale divergence is regularised at the gravita-tional Landau length. The large scale divergence is usually regularised by introducing a cut-off at the Jeans length, which is the gravitational equivalent of the Debye length. This gravitational Landau equa-tion is often considered to be relevant to describe the collisional dynamics of spherical systems such as globular clusters. Let us however note that the associated treatment based on the local approximation remains unsatisfactory, in particular because of the unavoidable appearance of a logarithmic divergence at large scales. In addition, within this framework, one cannot account for collective effects, i.e. the dressing of stars by their polarisation cloud, i.e. the fact that the gravitational force being attractive, a given star has the tendency to be surrounded by a cloud of stars. This increases its effective gravitational mass and reduces the collisional relaxation time.
In order to fully account for these properties, the kinetic theory of self-gravitating systems was re-cently generalised to fully inhomogeneous systems, either when collective effects are neglected (Cha-vanis, 2010, 2013b) leading to the inhomogeneous Landau equation, or when they are accounted for leading to the inhomogeneous Balescu-Lenard equation (Heyvaerts, 2010; Chavanis, 2012b). These ki-netic equations, presented and discussed in detail in the upcoming section, are valid at order 1=N, where N is the number of stars in the system. Having accounted for the finite extension of the system, these equations no longer present divergence at large scales. In order to deal with the system’s inhomogeneity, they are written in angle-action coordinates (see section 1.3), which allow for the description of stars’ in-tricate dynamics in spatially inhomogeneous and multi-periodic systems. These equations involve sim-ilarly a resonance condition encapsulated in a Dirac Dfunction (see figure 2.3.2), which generalises the one present in the homogeneous Balescu-Lenard equation. Finally, in order to capture collective effects, the inhomogeneous Balescu-Lenard equation also involves the system’s response matrix (see equation (2.17)) expressed in angle-action variables. This generalises the dielectric function appearing in the homogeneous Balescu-Lenard equation for plasmas. This dressing accounts for anti-shielding, i.e. the fact that the gravitational mass of a star is enhanced by its polarisation, leading to a reduction of the relaxation time. The upcoming chapters will emphasise how these powerful and predictive kinetic equations may be used in the astrophysical context to probe complex secular regimes.
There are two standard methods to derive kinetic equations for a Nbody system with long-range pairwise interactions. The first approach is based on Liouville’s equation for the Nbody distribu-tion function of the system. One has to write the first two equations of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. The hierarchy is then closed by considering only contributions of order 1=N. One may then solve the second equation of the BBGKY hierarchy to express the 2body cor-relation function in terms of the system’s 1 body DF. One finally substitutes this expression in the first equation of the BBGKY hierarchy to obtain the closed self-consistent kinetic equation satisfied by the 1body DF. The same results can also be obtained thanks to projection operator techniques. The second method relies on the Klimontovich equation (Klimontovich, 1967), which describes the dynamics of the system’s DF written as a sum of D functions. This exact DF is then decomposed in two parts, a smooth component and fluctuations. One can then write two evolution equations, one for the smooth mean component, and one for the fluctuations. This coupled system is then closed by neglecting non-linear terms in the evolution of the fluctuations (quasilinear approximation). The final step in this approach is to solve the equation for the fluctuations to express their properties as a function of the underlying smooth component. Injecting this result in the first evolution equation for the smooth part, one obtains a self-consistent kinetic equation. These two methods are physically equivalent, while technically dif-ferent. Finally, we recently presented in Fouvry et al. (2016a,b) a third approach based on a functional rewriting of the evolution equations. This approach starts from the first two equations of the BBGKY hierarchy truncated at order 1=N. Introducing auxiliary fields, the evolution of the two coupled dynam-ical quantities, 1body DF and 2body autocorrelation, can then be rewritten as a traditional functional integral. By functionally integrating over the 2body autocorrelation, one obtains a new contraint con-necting the 1body DF and the auxiliary fields. When inverted, this constraint finally allows for the derivation of the closed non-linear kinetic equation satisfied by the 1body DF.
In the upcoming sections, we will follow Chavanis (2012b) and present a derivation of the inhomo-geneous Balescu-Lenard equation based on the resolution of the Klimontovich equation. We decided to present this derivation in the main text, in order to emphasise the various similarities it shares with the previous collisionless diffusion equation. In Appendix 2.A, we present the derivation of the BBGKY hi-erarchy. This allows us to revisit in Appendix 2.B the derivation of the inhomogeneous Balescu-Lenard equation first presented by Heyvaerts (2010) and based on the direct resolution of the BBGKY hierarchy. Finally, in Appendix 2.C, we consider the third approach to the derivation of kinetic equations based on a functional integral rewriting.

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In this chapter, we presented two important sources of diffusion to induce secular evolution in self-gravitating systems. The first source, presented in section 2.2, considers the case of a collisionless sys-tem undergoing external perturbations. The second source, presented in section 2.3, is captured by the Balescu-Lenard equation, which describes the long-term effects of finiteN fluctuations on isolated dis-crete self-gravitating systems. In our two derivations, we emphasised the strong similarities existing be-tween the two approaches, as can be seen in particular in their similar decoupled evolution equations. Let us finally underline that both equations (2.31) and (2.67) share the properties that they describe strongly anisotropic diffusion in action space (see figure 2.2.1), account for the system’s internal sus-ceptibility (via the response matrix from equation (2.17) and the associated gravitational polarisation, see figure 1.3.6). Because they are sourced by different fluctuations, either external or internal, these two orbital diffusion processes provide the ideal frameworks in which to study the secular evolution of self-gravitating systems.
The rest of the thesis is focused on illustrating for various astrophysical systems how these for-malisms allow for a detailed description of their secular dynamics. In chapter 3, we will consider the case of razor-thin stellar discs. In order to obtain simple quadratures for the diffusion fluxes, we will develop a razor-thin WKB formalism (i.e. restriction to radially tightly wound perturbations) provid-ing a straightforward understanding of the regions of maximum amplification within the disc. We will illustrate how the functional form of the diffusion coefficients explains the self-induced formation of resonant ridges in the disc’s DF, as observed in numerical simulations. In chapter 4, we will resort to the same razor-thin stellar discs, but will devote our efforts to correctly account for the disc’s self-gravity and the associated strong amplification. This will be shown to significantly hasten the diffusion in the disc. In addition, in Appendix 4.D, we will illustrate how the same method may also be applied to study the long-term dynamics of 3D spherical systems such as dark matter haloes. This framework provides a promising way to investigate the secular transformation of dark matter haloes’ cusps into cores. In chap-ter 5, we will extend our WKB approximation to apply it to thickened stellar discs. We will investigate various possible mechanisms of thickening such as the disc’s internal Poisson shot noise, a series of cen-tral decaying bars, or the joint evolution of giant molecular clouds within the disc. Finally, in chapter 6, we will consider the case of quasi-Keplerian systems, such as galactic centres, for which the presence of a dominating central body imposes a degenerate Keplerian dynamics. Once tailored for such systems, we will detail in particular how the Balescu-Lenard formalism recovers the process of « resonant relaxation » specific to these systems.

Future works

The previous formalisms could be generalised in various ways.
In Appendix 2.C, we presented a new method based on a functional approach to derive the inho-mogeneous Landau equation. Because of the simplicity of the required calculations, this throws new light on the complex dynamical processes at play. One could hope to generalise this calculation to ac-count for collective effects and recover the inhomogeneous Balescu-Lenard equation. Such a calculation is expected to be more demanding, as it will involve a Fredholm type equation, such as equation (2.125). Similarly, we showed in Fouvry et al. (2016b) how the same functional approach could also be transposed to the kinetic theory of two-dimensional point vortices (Chavanis, 2012d,c). One should investigate other physical systems for which this approach could also be successful. Finally, it would be of particular in-terest to apply this method to derive a closed kinetic equation when higher order correlation terms are accounted for. This could for example allow us to describe the dynamics of 1D homogeneous systems, for which the 1=N Balescu-Lenard collision term vanishes by symmetry (Eldridge & Feix, 1963; Kadomt-sev & Pogutse, 1970). This is also the case for the Hamiltonian Mean Field model (HMF) (Chavanis et al., 2005; Bouchet & Dauxois, 2005).
Inspired by Pichon & Aubert (2006), the previous approaches could also be extended and developed for open systems, by accounting for possible sources and sinks of particles. Similarly, it could also prove interesting to investigate the Balescu-Lenard equation in a context where the system’s number of par-ticles gets to evolve during the secular evolution, to describe for example the progessive dissolution of overdensities, etc. Similarly, as can be seen in the proposed derivations, all these formalisms rely on the fundamental assumption of integrability, i.e. on the existence of angle-action coordinates. It would be of interest to investigate as well how such approaches could be tailored to deal with chaotic behaviours and their associated diffusions. Finally, one could also investigate within these frameworks the role that gas may play on the dynamical properties of the system. Indeed, one crucial property of gas is that it cannot shell-cross, it shocks. This typically means that the gas component is dynamically much colder than its stellar counterpart, which alters the system’s dynamical susceptibility.

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
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 Collisionless forced radial diffusion 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 Turning off collective effects 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
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 Collisionless forced thickening Collisional thickening 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
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
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


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