Appendix
A Relativistic Precessions
In this Appendix, let us briefly detail the content of the averaged relativistic corrections encompassed by the potential \(\overline{\Phi }_{\mathrm {r}}\) present in Eqs. (6.45) and (6.46). As we aim for explicit expressions of these corrections, let us use the 3D Delaunay variables from Eq. (6.19). In addition, we assume for simplicity that the spin of the BH is aligned with the \({z-}\)direction and introduce its spin parameter \({ 0 \!\le \! s \!\le \!1 }\). We follow Merritt (2015) in order to recover explicit expressions for these precession frequencies.
The first relativistic correction is associated with a 1PN effect (i.e. a correction of order \({ 1/c^{2} }\)), called the Schwarzschild precession. Equation (5.103) in Merritt (2015) gives us that during one Keplerian orbit of duration \({ T_\mathrm{Kep} \!=\! 2 \pi / \Omega _\mathrm{Kep} \!=\! 2 \pi I^{3} / (G M_{\bullet })^{2} }\), the slow angle g is modified by an amount
$$\begin{aligned} \Delta g_\mathrm{rel}^\mathrm{1PN} = g (T_\mathrm{Kep}) \!-\! g (0) = \frac{6 \pi G M_{\bullet }}{c^{2} a (1 \!-\! e^{2})} \, . \end{aligned}$$
(6.97)
This precession corresponds to a precession of the orbit’s pericentre, while the orbit remains in its orbital plane. To the change from Eq. (6.97), one can straightforwardly associate an averaged precession frequency \({ \dot{g}_\mathrm{rel}^\mathrm{1PN} \!=\! \Delta g_\mathrm{rel}^\mathrm{1PN} / T_\mathrm{Kep} }\) reading
$$\begin{aligned} \dot{g}_\mathrm{rel}^\mathrm{1PN} = \frac{3 (G M_{\bullet })^{4}}{c^{2} I^{3} L^{2}} = \frac{\partial H_\mathrm{rel}^\mathrm{1PN}}{\partial L} \, , \end{aligned}$$
(6.98)
where the semi-major axis a and the eccentricity e respectively satisfy \({ a \!=\! I^{2} / (G M_{\bullet }) }\) and \({ 1 \!-\! e^{2} \!=\! (L / I)^{2} }\). We also introduced the Hamiltonian \(H_\mathrm{rel}^\mathrm{1PN}\) as
$$\begin{aligned} H_\mathrm{rel}^\mathrm{1PN} (I , L) = - \frac{3 (G M_{\bullet })^{4}}{c^{2}} \frac{1}{I^{3} L} \, . \end{aligned}$$
(6.99)
The next order relativistic corrections are associated with a 1.5PN effect (i.e. a correction of order \({1/c^{3}}\)) called the Lense-Thirring precession. Following Eq. (5.118) of Merritt (2015), during one Keplerian orbit this effect leads to a precession of the slow angle g given by
$$\begin{aligned} \Delta g_\mathrm{rel}^\mathrm{1.5PN} = g (T_\mathrm{Kep}) \!-\! g (0) = - \frac{12 \pi s}{c^{3}} \bigg [ \frac{G M_{\bullet }}{(1 \!-\! e^{2}) a} \bigg ]^{3/2} \cos (i) \, , \end{aligned}$$
(6.100)
where we recall that we assume that the BH’s spin is aligned with the \({z-}\)direction. We also introduced the orbit’s inclination i such that \({ L_{z} \!=\! L \cos (i) }\). One can straightforwardly associate a precession frequency \({ \dot{g}_\mathrm{rel}^\mathrm{1.5PN} \!=\! \Delta g_\mathrm{rel}^\mathrm{1.5PN} / T_\mathrm{Kep}}\) to this change, so that
$$\begin{aligned} \dot{g}_\mathrm{rel}^\mathrm{1.5PN} = - \frac{6s}{c^{3}} \frac{(G M_{\bullet })^{5} L_{z}}{I^{3} L^{4}} = \frac{\partial H_\mathrm{rel}^\mathrm{1.5PN}}{\partial L} \, . \end{aligned}$$
(6.101)
In Eq. (6.101), we introduced the Hamiltonian \(H_\mathrm{rel}^\mathrm{1.5PN}\), which captures the corrections associated with the BH’s spin as
$$\begin{aligned} H_\mathrm{rel}^\mathrm{1.5PN} (I , L , L_{z}) = \frac{2 s (G M_{\bullet })^{5}}{c^{3}} \frac{L_{z}}{I^{3} L^{3}} \, . \end{aligned}$$
(6.102)
Such a Hamiltonian also induces relativistic precessions w.r.t. the second slow angle h associated with the slow action \(L_{z}\). We do not detail here how these precessions are indeed correctly described by the Hamiltonian \(H_\mathrm{rel}^\mathrm{1.5PN}\).
Let us finally write the explicit expression of the averaged potential corrections \(\overline{\Phi }_{\mathrm {r}}\) appearing in Eqs. (6.45) and (6.46). One has to pay a careful attention to the normalisation conventions introduced in Eqs. (6.2), (6.14), and (6.38). One gets
$$\begin{aligned} \overline{\Phi }_{\mathrm {r}} (I , L , L_{z}) = \frac{1}{(2 \pi )^{d - k}} \frac{M_{\bullet }}{M_{\star }} \bigg [ H_\mathrm{rel}^\mathrm{1PN} (I , L) + H_\mathrm{rel}^\mathrm{1.5PN} (I , L , L_{z}) \bigg ] \, . \end{aligned}$$
(6.103)
Following Eq. (6.50), this relativistic potential correction immediately leads to the associated precession frequencies \(\varvec{\Omega }^{\mathrm {s}}_\mathrm{rel}\) w.r.t. the slow angles \(\varvec{\theta }^{\mathrm {s}}\), which read
$$\begin{aligned} \varvec{\Omega }_\mathrm{rel}^{\mathrm {s}} = \frac{\partial \overline{\Phi }_{\mathrm {r}}}{\partial \varvec{J}^{\mathrm {s}}} = \frac{(G M_{\bullet })^{4}}{(2 \pi )^{d - k} c^{2}} \frac{M_{\bullet }}{M_{\star }} \frac{\partial }{\partial \varvec{J}^{\mathrm {s}}} \bigg [ - \frac{3}{I^{3} L} + \frac{2 G M_{\bullet }}{c} \frac{s L_{z}}{I^{3} L^{3}} \bigg ] \, . \end{aligned}$$
(6.104)
Let us finally note that gravitational waves and the associated dissipations (Hopman and Alexander 2006) are not accounted for in Eq. (6.104), hence the possibility to obtain a Hamiltonian formulation for these precessions.
B Multi-component BBGKY Hierarchy
In this Appendix, let us detail how one can adapt the formalism presented in Sect. 6.2 to the case where the system is composed of multiple components. The different components are indexed by the letters “\(\mathrm {a}\)”, “\(\mathrm {b}\)”, etc. We assume that the component “\(\mathrm {a}\)” is made of \(N_{\mathrm {a}}\) particles of individual mass \(\mu _{\mathrm {a}}\). The total mass of the component “\(\mathrm {a}\)” is written as \(M_{\star }^{\mathrm {a}}\). When accounting for multiple components and placing ourselves within the democratic heliocentric coordinates from Eq. (6.3), the system’s total Hamiltonian from Eq. (6.7) becomes
$$\begin{aligned} H =&\, \sum _{\mathrm {a}} \sum _{i = 1}^{N_{\mathrm {a}}} \frac{\mu _{\mathrm {a}}}{2} ( \varvec{v}_{i}^{\mathrm {a}} )^{2} + \sum _{\mathrm {a}} \mu _{\mathrm {a}} M_{\bullet } \sum _{i = 1}^{N_{\mathrm {a}}} U (|\varvec{x}_{i}^{\mathrm {a}}|) + \sum _{\mathrm {a}} \mu _{\mathrm {a}} M_{\star } \sum _{i = 1}^{N_{\mathrm {a}}} \Phi _{\mathrm {r}} (\varvec{x}_{i}^{\mathrm {a}}) \nonumber \\&\, + \sum _{\mathrm {a}} \mu _{\mathrm {a}}^{2} \sum _{i< j}^{N_{\mathrm {a}}} U (|\varvec{x}_{i}^{\mathrm {a}} \!-\! \varvec{x}_{j}^{\mathrm {a}}|) + \sum _{{\mathrm {a}} < {\mathrm {b}}} \sum _{i = 1}^{N_{\mathrm {a}}} \sum _{j = 1}^{N_{\mathrm {b}}} \mu _{\mathrm {a}} \mu _{\mathrm {b}} U (|\varvec{x}_{i}^{\mathrm {a}} \!-\! \varvec{x}_{i}^{\mathrm {b}}|) + \frac{1}{2 M_{\bullet }} \bigg [ \sum _{\mathrm {a}} \mu _{\mathrm {a}} \sum _{i = 1}^{N_{\mathrm {a}}} \varvec{v}_{i}^{\mathrm {a}} \bigg ]^{2} \, , \end{aligned}$$
(6.105)
where we noted as \({ \Gamma _{i}^{\mathrm {a}} \!=\! (\varvec{x}_{i}^{\mathrm {a}} , \varvec{v}_{i}^{\mathrm {a}}) }\) the position and velocity of the \(i^\mathrm{th}\) particle of the component “\(\mathrm {a}\)”. In Eq. (6.105), the various terms are respectively the kinetic energy of the particles, the Keplerian potential due to the central BH, the relativistic potential corrections \(\Phi _{\mathrm {r}}\), the self-gravity among a given component, the interactions between particles of different components, and finally the additional kinetic terms introduced by the change of coordinates from Eq. (6.3). One should also pay attention to the normalisation of the relativistic component \(\Phi _{\mathrm {r}}\), as we wrote this potential as \({ \mu _{\mathrm {a}} M_{\star } \Phi _{\mathrm {r}} }\), where we introduced the system’s total active mass as \({ M_{\star } \!=\! \sum _{\mathrm {a}} M_{\star }^{\mathrm {a}} }\) to have a writing similar to Eq. (6.7). Let us now introduce the system total PDF \({ P_\mathrm{tot} (\Gamma _{1}^{\mathrm {a}} , \ldots , \Gamma _{N_{\mathrm {a}}}^{\mathrm {a}}, \Gamma _{1}^{\mathrm {b}} ,\ldots , \Gamma _{N_{\mathrm {b}}}^{\mathrm {b}} , \ldots ) }\) which gives the probability of finding at time t the particle 1 of the component “\(\mathrm {a}\)” at position \(\varvec{x}_{1}^{\mathrm {a}}\) and velocity \(\varvec{v}_{1}^{\mathrm {a}}\), etc. We normalise \(P_\mathrm{tot}\) following the convention from Eq. (2.94). Similarly to Eq. (6.8), \(P_\mathrm{tot}\) evolves according to Liouville’s equation which reads
$$\begin{aligned} \frac{\partial P_\mathrm{tot}}{\partial t} + \sum _{\mathrm {a}} \sum _{i = 1}^{N_{\mathrm {a}}} \bigg [ \dot{\varvec{x}}_{i}^{\mathrm {a}} \!\cdot \! \frac{\partial P_\mathrm{tot}}{\partial \varvec{x}_{i}^{\mathrm {a}}} + \dot{\varvec{v}}_{i}^{\mathrm {a}} \!\cdot \! \frac{\partial P_\mathrm{tot}}{\partial \varvec{v}_{i}^{\mathrm {a}}} \bigg ] = 0 \, . \end{aligned}$$
(6.106)
Following Eq. (2.97), we define the system’s reduced PDFs \(P_{n}^{\mathrm {a}_{1} , \ldots , \mathrm {a}_{n} }\) by integrating \(P_\mathrm{tot}\) over all particles except n particles belonging respectively to the components \(\mathrm {a}_{1}\), ..., \(\mathrm {a}_{n}\). Our aim is now to write the two first equations of the associated BBGKY hierarchy. In order to clarify the upcoming calculations, let us from now on neglect any contributions associated with the last kinetic terms from Eq. (6.105). Indeed, in the single-component case, we justified in Eq. (6.41) that these terms, once averaged over the fast Keplerian motion, do not contribute to the system’s dynamics at the order considered in our kinetic developments. To get the evolution equation for \(P_{1}^{\mathrm {a}}\), one integrates Eq. (6.106) over all phase space coordinates except \(\Gamma _{1}^{\mathrm {a}}\) and relies on the symmetry of \(P_\mathrm{tot}\) w.r.t. interchanges of particles of the same component. One gets
$$\begin{aligned}&\, \frac{\partial P_{1}^{\mathrm {a}}}{\partial t} + \varvec{v}_{1}^{\mathrm {a}} \!\cdot \! \frac{\partial P_{1}^{\mathrm {a}}}{\partial \varvec{x}_{1}^{\mathrm {a}}} + \bigg [ M_{\bullet } \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} 0} + M_{\star } \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} \mathrm {r}} \bigg ] \!\cdot \! \frac{\partial P_{1}^{\mathrm {a}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} \nonumber \\&\, + (N \!-\! 1) \, \mu _{\mathrm {a}} \!\! \int \!\! \mathrm {d}\Gamma _{2}^{\mathrm {a}} \, \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} 2^{\mathrm {a}}} \!\cdot \! \frac{\partial P_{2}^{\mathrm {a}\mathrm {a}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \sum _{\mathrm {b}\ne \mathrm {a}} N_{\mathrm {b}} \, \mu _{\mathrm {b}} \!\! \int \!\! \mathrm {d}\Gamma _{2}^{\mathrm {b}} \, \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} 2^{\mathrm {b}}} \!\cdot \! \frac{\partial P_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} = 0 \, . \end{aligned}$$
(6.107)
In Eq. (6.107), we used the same notations as in Eq. (6.9), and introduced as \(\varvec{\mathcal {F}}_{1^{\mathrm {a}} 0}\) the force exerted by the BH on particle \(1^{\mathrm {a}}\), \(\varvec{\mathcal {F}}_{1^{\mathrm {a}} \mathrm {r}}\) as the force acting on particle \(1^{\mathrm {a}}\) due to the relativistic corrections, and finally \(\varvec{\mathcal {F}}_{ij}\) as the force between two particles. In order to get the second equation of the BBGKY hierarchy, one should proceed similarly and integrate Eq. (6.106) w.r.t. all particles except two. At this stage, two different cases should be investigated, depending on whether one considers \(P_{2}^{\mathrm {a}\mathrm {a}}\) or \(P_{2}^{\mathrm {a}\mathrm {b}}\) (with \({ \mathrm {a}\!\ne \! \mathrm {b}}\)). Let us first consider the diffusion equation satisfied by \(P_{2}^{\mathrm {a}\mathrm {a}}\), which ensues from Eq. (6.106) by integrating it w.r.t. all phase space coordinates except \(\Gamma _{1}^{\mathrm {a}}\) and \(\Gamma _{2}^{\mathrm {a}}\). It reads
$$\begin{aligned}&\, \frac{\partial P_{2}^{\mathrm {a}\mathrm {a}}}{\partial t} + \varvec{v}_{1}^{\mathrm {a}} \!\cdot \! \frac{\partial P_{2}^{\mathrm {a}\mathrm {a}}}{\partial \varvec{x}_{1}^{\mathrm {a}}} + \varvec{v}_{2}^{\mathrm {a}} \!\cdot \! \frac{\partial P_{2}^{\mathrm {a}\mathrm {a}}}{\partial \varvec{x}_{2}^{\mathrm {a}}} + \mu _{\mathrm {a}} \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} 2^{\mathrm {a}}} \!\cdot \! \frac{\partial P_{2}^{\mathrm {a}\mathrm {a}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \mu _{\mathrm {a}} \varvec{\mathcal {F}}_{\! 2^{\mathrm {a}} 1^{\mathrm {a}}} \!\cdot \! \frac{\partial P_{2}^{\mathrm {a}\mathrm {a}}}{\partial \varvec{v}_{2}^{\mathrm {a}}} \nonumber \\&\, + \bigg [ M_{\bullet } \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} 0} + M_{\star } \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} \mathrm {r}} \bigg ] \!\cdot \! \frac{\partial P_{2}^{\mathrm {a}\mathrm {a}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \bigg [ M_{\bullet } \varvec{\mathcal {F}}_{\! 2^{\mathrm {a}} 0} + M_{\star } \varvec{\mathcal {F}}_{\! 2^{\mathrm {a}} \mathrm {r}} \bigg ] \!\cdot \! \frac{\partial P_{2}^{\mathrm {a}\mathrm {a}}}{\partial \varvec{v}_{2}^{\mathrm {a}}} \nonumber \\&\, + (N_{\mathrm {a}} \!-\! 2) \, \mu _{\mathrm {a}} \!\! \int \!\! \mathrm {d}\Gamma _{3}^{\mathrm {a}} \, \bigg [ \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} 3^{\mathrm {a}}} \!\cdot \! \frac{\partial P_{3}^{\mathrm {a}\mathrm {a}\mathrm {a}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \varvec{\mathcal {F}}_{\! 2^{\mathrm {a}} 3^{\mathrm {a}}} \!\cdot \! \frac{\partial P_{3}^{\mathrm {a}\mathrm {a}\mathrm {a}}}{\partial \varvec{v}_{2}^{\mathrm {a}}} \bigg ] \nonumber \\&\, + \sum _{\mathrm {b}\ne \mathrm {a}} N_{\mathrm {b}} \, \mu _{\mathrm {b}} \!\! \int \!\! \mathrm {d}\Gamma _{3}^{\mathrm {b}} \, \bigg [ \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} 3^{\mathrm {b}}} \!\cdot \! \frac{\partial P_{3}^{\mathrm {a}\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \varvec{\mathcal {F}}_{\! 2^{\mathrm {a}} 3^{\mathrm {b}}} \!\cdot \! \frac{\partial P_{3}^{\mathrm {a}\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{2}^{\mathrm {a}}} \bigg ] = 0 \, . \end{aligned}$$
(6.108)
When the two particles do not belong to the same component, the second equation of the hierarchy becomes
$$\begin{aligned}&\, \frac{\partial P_{2}^{\mathrm {a}\mathrm {b}}}{\partial t} + \varvec{v}_{1}^{\mathrm {a}} \!\cdot \! \frac{\partial P_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{x}_{1}^{\mathrm {a}}} + \varvec{v}_{2}^{\mathrm {b}} \!\cdot \! \frac{\partial P_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{x}_{2}^{\mathrm {b}}} + \mu _{\mathrm {b}} \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} 2^{\mathrm {b}}} \!\cdot \! \frac{\partial P_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \mu _{\mathrm {a}} \varvec{\mathcal {F}}_{\! 2^{\mathrm {b}} 1^{\mathrm {a}}} \!\cdot \! \frac{\partial P_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{2}^{\mathrm {b}}} \nonumber \\&\, + \bigg [ M_{\bullet } \varvec{\mathcal {F}}_{\!1^{\mathrm {a}} 0} + M_{\star } \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} \mathrm {r}} \bigg ] \!\cdot \! \frac{\partial P_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \bigg [ M_{\bullet } \varvec{\mathcal {F}}_{\! 2^{\mathrm {b}} 0} + M_{\star } \varvec{\mathcal {F}}_{\! 2^{\mathrm {b}} \mathrm {r}} \bigg ] \!\cdot \! \frac{\partial P_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{2}^{\mathrm {b}}} \nonumber \\&\, + (N_{\mathrm {a}} \!-\! 1) \, \mu _{\mathrm {a}} \!\! \int \!\! \mathrm {d}\Gamma _{3}^{\mathrm {a}} \, \bigg [ \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} 3^{\mathrm {a}}} \!\cdot \! \frac{\partial P_{3}^{\mathrm {a}\mathrm {b}\mathrm {a}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \varvec{\mathcal {F}}_{\! 2^{\mathrm {b}} 3^{\mathrm {a}}} \!\cdot \! \frac{\partial P_{3}^{\mathrm {a}\mathrm {b}\mathrm {a}}}{\partial \varvec{v}_{2}^{\mathrm {b}}} \bigg ] + (N_{\mathrm {b}} \!-\! 1 ) \, \mu _{\mathrm {b}} \!\! \int \!\! \mathrm {d}\Gamma _{3}^{\mathrm {b}} \, \bigg [ \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} 3^{\mathrm {b}}} \!\cdot \! \frac{\partial P_{3}^{\mathrm {a}\mathrm {b}\mathrm {b}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \varvec{\mathcal {F}}_{\! 2^{\mathrm {b}} 3^{\mathrm {b}}} \!\cdot \! \frac{\partial P_{3}^{\mathrm {a}\mathrm {b}\mathrm {b}}}{\partial \varvec{v}_{2}^{\mathrm {b}}} \bigg ] \nonumber \\&\, + \sum _{\mathrm {c}\ne \mathrm {a}, \mathrm {b}} N_{\mathrm {c}} \, \mu _{\mathrm {c}} \!\! \int \!\! \mathrm {d}\Gamma _{3}^{\mathrm {c}} \, \bigg [ \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} 3^{\mathrm {c}}} \!\cdot \! \frac{\partial P_{3}^{\mathrm {a}\mathrm {b}\mathrm {c}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \varvec{\mathcal {F}}_{\! 2^{\mathrm {b}} 3^{\mathrm {c}}} \!\cdot \! \frac{\partial P_{3}^{\mathrm {a}\mathrm {b}\mathrm {c}}}{\partial \varvec{v}_{2}^{\mathrm {b}}} \bigg ] = 0 \, . \end{aligned}$$
(6.109)
Let us now adapt the definition of the reduced DFs from Eq. (2.99) to the multi-component case. We therefore introduce the system’s renormalised DFs \(f_{1}^{\mathrm {a}}\), \(f_{2}^{\mathrm {a}\mathrm {b}}\), and \(f_{3}^{\mathrm {a}\mathrm {b}\mathrm {c}}\) as
$$\begin{aligned}&\, f_{1}^{\mathrm {a}} \!=\! \mu _{\mathrm {a}} N_{\mathrm {a}} P_{1}^{\mathrm {a}} \, ; \nonumber \\&\, f_{2}^{\mathrm {a}\mathrm {a}} \!=\! \mu _{\mathrm {a}}^{2} N_{\mathrm {a}} (N_{\mathrm {a}} \!-\! 1) P_{2}^{\mathrm {a}\mathrm {a}} \; ; \; f_{2}^{\mathrm {a}\mathrm {b}} \!=\! \mu _{\mathrm {a}} \mu _{\mathrm {b}} N_{\mathrm {a}} N_{\mathrm {b}} P_{2}^{\mathrm {a}\mathrm {b}} \, ; \\&\, f_{3}^{\mathrm {a}\mathrm {a}\mathrm {a}} \!=\! \mu _{\mathrm {a}}^{3} N_{\mathrm {a}} (N_{\mathrm {a}} \!-\! 1) (N_{\mathrm {a}} \!-\! 2) P_{3}^{\mathrm {a}\mathrm {a}\mathrm {a}} \; ; \; f_{3}^{\mathrm {a}\mathrm {a}\mathrm {b}} \!=\! \mu _{\mathrm {a}}^{2} \mu _{\mathrm {b}} N_{\mathrm {a}} (N_{\mathrm {a}} \!-\! 1) N_{\mathrm {b}} P_{3}^{\mathrm {a}\mathrm {a}\mathrm {b}} \; ; \; f_{3}^{\mathrm {a}\mathrm {b}\mathrm {c}} \!=\! \mu _{\mathrm {a}} \mu _{\mathrm {b}} \mu _{\mathrm {c}} N_{\mathrm {a}} N_{\mathrm {b}} N_{\mathrm {c}} P_{3}^{\mathrm {a}\mathrm {b}\mathrm {c}} \, , \nonumber \end{aligned}$$
(6.110)
where “\(\mathrm {a}\)”, “\(\mathrm {b}\)”, and “\(\mathrm {c}\)” are associated with different components. These detailed normalisations allow us to rewrite Eq. (6.107) under the general form
$$\begin{aligned} \frac{\partial f_{1}^{\mathrm {a}}}{\partial t} + \varvec{v}_{1}^{\mathrm {a}} \!\cdot \! \frac{\partial f_{1}^{\mathrm {a}}}{\partial \varvec{x}_{1}^{\mathrm {a}}} + \bigg [ M_{\bullet } \varvec{\mathcal {F}}_{\!1^{\mathrm {a}} 0} + M_{\star } \varvec{\mathcal {F}}_{\!1^{\mathrm {a}} \mathrm {r}} \bigg ] \!\cdot \! \frac{\partial f_{1}^{\mathrm {a}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \sum _{\mathrm {b}} \!\! \int \!\! \mathrm {d}\Gamma _{2}^{\mathrm {b}} \, \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} 2^{\mathrm {b}}} \!\cdot \! \frac{\partial f_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} = 0 \, , \end{aligned}$$
(6.111)
where one should note that the sum over “\(\mathrm {b}\)” runs for all components. Similarly, Eqs. (6.108) and (6.109) can both be cast under the same generic form reading
$$\begin{aligned}&\, \frac{\partial f_{2}^{\mathrm {a}\mathrm {b}}}{\partial t} + \varvec{v}_{1}^{\mathrm {a}} \!\cdot \! \frac{\partial f_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{x}_{1}^{\mathrm {a}}} + \varvec{v}_{2}^{\mathrm {b}} \!\cdot \! \frac{\partial f_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{x}_{2}^{\mathrm {b}}} + \mu _{\mathrm {b}} \varvec{\mathcal {F}}_{\!1^{\mathrm {a}}2^{\mathrm {b}}} \!\cdot \! \frac{\partial f_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \mu _{\mathrm {a}} \varvec{\mathcal {F}}_{\!2^{\mathrm {b}} 1^{\mathrm {a}}} \!\cdot \! \frac{\partial f_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{2}^{\mathrm {b}}} \nonumber \\&\, + \bigg [ M_{\bullet } \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} 0} + M_{\star } \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} \mathrm {r}} \bigg ] \!\cdot \! \frac{\partial f_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \bigg [ M_{\bullet } \varvec{\mathcal {F}}_{\! 2^{\mathrm {b}}0} + M_{\star } \varvec{\mathcal {F}}_{\! 2^{\mathrm {b}} \mathrm {r}} \bigg ] \!\cdot \! \frac{\partial f_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{2}^{\mathrm {b}}} \nonumber \\&\, + \sum _{\mathrm {c}} \!\! \int \!\! \mathrm {d}\Gamma _{3}^{\mathrm {c}} \, \bigg [ \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} 3^{\mathrm {c}}} \!\cdot \! \frac{\partial f_{3}^{\mathrm {a}\mathrm {b}\mathrm {c}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \varvec{\mathcal {F}}_{\! 2^{\mathrm {b}} 3^{\mathrm {c}}} \!\cdot \! \frac{\partial f_{3}^{\mathrm {a}\mathrm {b}\mathrm {c}}}{\partial \varvec{v}_{2}^{\mathrm {b}}} \bigg ] = 0 \, , \end{aligned}$$
(6.112)
where we insist on the fact that Eq. (6.112) holds for both cases where “\(\mathrm {a}\)” and “\(\mathrm {b}\)” are equal or different, and that the sum over “\(\mathrm {c}\)” runs for all components. Equations (6.111) and (6.112) are the direct multi-component analogs of the single-component BBGKY equation (6.9).
Following Eqs. (2.101) and (2.102), one can now introduce the cluster representation of the DFs, which in the multi-component context takes the form
$$\begin{aligned} f_{2}^{\mathrm {a}\mathrm {b}} (\Gamma _{1}^{\mathrm {a}} , \Gamma _{2}^{\mathrm {b}}) = f_{1}^{\mathrm {a}} (\Gamma _{1}^{\mathrm {a}}) \, f_{1}^{\mathrm {b}} (\Gamma _{2}^{\mathrm {b}}) + g_{2}^{\mathrm {a}\mathrm {b}} (\Gamma _{1}^{\mathrm {a}} , \Gamma _{2}^{\mathrm {b}}) \, , \end{aligned}$$
(6.113)
and
$$\begin{aligned} f_{3}^{\mathrm {a}\mathrm {b}\mathrm {c}} (\Gamma _{1}^{\mathrm {a}} , \Gamma _{2}^{\mathrm {b}} , \Gamma _{3}^{\mathrm {c}}) =&\, f_{1}^{\mathrm {a}} (\Gamma _{1}^{\mathrm {a}}) \, f_{1}^{\mathrm {b}} (\Gamma _{2}^{\mathrm {b}}) \, f_{1}^{\mathrm {c}} (\Gamma _{3}^{\mathrm {c}}) \nonumber \\&\, + f_{1}^{\mathrm {a}} (\Gamma _{1}^{\mathrm {a}}) \, g_{2}^{\mathrm {b}\mathrm {c}} (\Gamma _{2}^{\mathrm {b}} , \Gamma _{3}^{\mathrm {c}}) + f_{1}^{\mathrm {b}} (\Gamma _{2}^{\mathrm {b}}) \, g_{2}^{\mathrm {a}\mathrm {c}} (\Gamma _{1}^{\mathrm {a}} , \Gamma _{3}^{\mathrm {c}}) + f_{1}^{\mathrm {c}} (\Gamma _{3}^{\mathrm {c}}) \, g_{2}^{\mathrm {a}\mathrm {b}} (\Gamma _{1}^{\mathrm {a}} , \Gamma _{2}^{\mathrm {b}}) \nonumber \\&\, + g_{3}^{\mathrm {a}\mathrm {b}\mathrm {c}} (\Gamma _{1}^{\mathrm {a}} , \Gamma _{2}^{\mathrm {b}} , \Gamma _{3}^{\mathrm {c}}) \, . \end{aligned}$$
(6.114)
As obtained in Eq. (2.103), let us assume that \(g_{2}^{\mathrm {a}\mathrm {b}}\) scales like the inverse of the number of particles, while \(g_{3}^{\mathrm {a}\mathrm {b}\mathrm {c}}\) scales like the square of the inverse of the number of particles. Relying on the decompositions from Eqs. (6.113) and (6.114), and keeping only terms of order \({ 1/N_{\mathrm {a}} }\) (where “\(\mathrm {a}\)” runs over all components), one can rewrite the first equation (6.111) of the multi-component BBGKY hierarchy as
$$\begin{aligned} \frac{\partial f_{1}^{\mathrm {a}}}{\partial t} + \varvec{v}_{1}^{\mathrm {a}} \!\cdot \! \frac{\partial f_{1}^{\mathrm {a}}}{\partial \varvec{x}_{1}^{\mathrm {a}}} + \bigg [ M_{\bullet } \varvec{\mathcal {F}}_{\!1^{\mathrm {a}} 0} + M_{\star } \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} \mathrm {r}} + \sum _{\mathrm {b}} \!\! \int \!\! \mathrm {d}\Gamma _{2}^{\mathrm {b}} \, \varvec{\mathcal {F}}_{\!1^{\mathrm {a}} 2^{\mathrm {b}}} f_{1}^{\mathrm {b}} (\Gamma _{2}^{\mathrm {b}})\bigg ] \!\cdot \! \frac{\partial f_{1}^{\mathrm {a}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \sum _{\mathrm {b}} \!\! \int \!\! \mathrm {d}\Gamma _{2}^{\mathrm {b}} \, \varvec{\mathcal {F}}_{\!1^{\mathrm {a}} 2^{\mathrm {b}}} \!\cdot \! \frac{\partial g_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} = 0 \, , \end{aligned}$$
(6.115)
while the second equation (6.112) becomes
$$\begin{aligned}&\, \frac{\partial g_{2}^{\mathrm {a}\mathrm {b}}}{\partial t} + \varvec{v}_{1}^{\mathrm {a}} \!\cdot \! \frac{\partial g_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{x}_{1}^{\mathrm {a}}} + \varvec{v}_{2}^{\mathrm {b}} \!\cdot \! \frac{\partial g_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{x}_{2}^{\mathrm {b}}} + \mu _{\mathrm {b}} \varvec{\mathcal {F}}_{\!1^{\mathrm {a}} 2^{\mathrm {b}}} \!\cdot \! \frac{\partial f_{1}^{\mathrm {a}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} f_{1}^{\mathrm {b}} (\Gamma _{2}^{\mathrm {b}}) + \mu _{\mathrm {a}} \varvec{\mathcal {F}}_{\!2^{\mathrm {b}} 1^{\mathrm {a}}} \!\cdot \! \frac{\partial f_{1}^{\mathrm {b}}}{\partial \varvec{v}_{2}^{\mathrm {b}}} f_{1}^{\mathrm {a}} (\Gamma _{1}^{\mathrm {a}}) \nonumber \\&\, + \bigg [ M_{\bullet } \varvec{\mathcal {F}}_{\!1^{\mathrm {a}} 0} + M_{\star } \varvec{\mathcal {F}}_{\!1^{\mathrm {a}} \mathrm {r}} \bigg ] \!\cdot \! \frac{\partial g_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{1}^{\mathrm {a}} } + \bigg [ M_{\bullet } \varvec{\mathcal {F}}_{\!2^{\mathrm {b}}0} + M_{\star } \varvec{\mathcal {F}}_{\!2^{\mathrm {b}} \mathrm {r}} \bigg ] \!\cdot \! \frac{\partial g_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{2}^{\mathrm {b}}} \nonumber \\&\, + \bigg [ \sum _{\mathrm {c}} \!\! \int \!\! \mathrm {d}\Gamma _{3}^{\mathrm {c}} \, \varvec{\mathcal {F}}_{\!1^{\mathrm {a}} 3^{\mathrm {c}}} f_{1}^{\mathrm {c}} (\Gamma _{3}^{\mathrm {c}}) \bigg ] \!\cdot \! \frac{\partial g_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \bigg [ \sum _{\mathrm {c}} \!\! \int \!\! \mathrm {d}\Gamma _{3}^{\mathrm {c}} \, \varvec{\mathcal {F}}_{\! 2^{\mathrm {b}} 3^{\mathrm {c}}} f_{1}^{\mathrm {c}} (\Gamma _{3}^{\mathrm {c}}) \bigg ] \!\cdot \! \frac{\partial g_{2}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{2}^{\mathrm {b}}} \nonumber \\&\, + \bigg [ \sum _{\mathrm {c}} \!\! \int \!\! \mathrm {d}\Gamma _{3}^{\mathrm {c}} \, \varvec{\mathcal {F}}_{\!1^{\mathrm {a}} 3^{\mathrm {c}}} g_{2}^{\mathrm {b}\mathrm {c}} (\Gamma _{2}^{\mathrm {b}} , \Gamma _{3}^{\mathrm {c}}) \bigg ] \!\cdot \! \frac{\partial f_{1}^{\mathrm {a}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \bigg [ \sum _{\mathrm {c}} \!\! \int \!\! \mathrm {d}\Gamma _{3}^{\mathrm {c}} \, \varvec{\mathcal {F}}_{\!2^{\mathrm {b}} 3^{\mathrm {c}}} g_{2}^{\mathrm {a}\mathrm {c}} (\Gamma _{1}^{\mathrm {a}} , \Gamma _{3}^{\mathrm {c}}) \bigg ] \!\cdot \! \frac{\partial f_{1}^{\mathrm {b}}}{\partial \varvec{v}_{2}^{\mathrm {b}}} = 0 \, . \end{aligned}$$
(6.116)
Similarly to Eq. (6.12), let us now introduce the system’s \({1-}\)body DF \(F^{\mathrm {a}}\) and \({2-}\)body autocorrelation \(\mathcal {C}^{\mathrm {a}\mathrm {b}}\) as
$$\begin{aligned} F^{\mathrm {a}} = \frac{f_{1}^{\mathrm {a}}}{M_{\star }} \;\;\; ; \;\;\; \mathcal {C}^{\mathrm {a}\mathrm {b}} = \frac{g_{2}^{\mathrm {a}\mathrm {b}}}{M_{\star }^{2}} \, . \end{aligned}$$
(6.117)
In Eq. (6.117), one should pay attention to the slight change in the normalisation of \(\mathcal {C}^{\mathrm {a}\mathrm {b}}\). This ensures a symmetric rescaling w.r.t. “\(\mathrm {a}\)” and “\(\mathrm {b}\)”. Let us now follow Eqs. (6.13) and (6.14) to rescale the pairwise interaction potential as well as the relativistic corrections. Following these various renormalisations, Eq. (6.115) becomes
$$\begin{aligned}&\, \frac{\partial F^{\mathrm {a}}}{\partial t} + \varvec{v}_{1}^{\mathrm {a}} \!\cdot \! \frac{\partial F^{\mathrm {a}}}{\partial \varvec{x}_{1}^{\mathrm {a}}} + \varvec{\mathcal {F}}_{\!1^{\mathrm {a}}0} \!\cdot \! \frac{\partial F^{\mathrm {a}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \varepsilon \bigg [ \sum _{\mathrm {b}} \!\! \int \!\! \mathrm {d}\Gamma _{2}^{\mathrm {b}} \, \varvec{\mathcal {F}}_{\!1^{\mathrm {a}}2^{\mathrm {b}}} F^{\mathrm {b}} (\Gamma _{2}^{\mathrm {b}}) \bigg ] \!\cdot \! \frac{\partial F^{\mathrm {a}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} \nonumber \\&\, + \varepsilon \varvec{\mathcal {F}}_{\!1^{\mathrm {a}} \mathrm {r}} \!\cdot \! \frac{\partial F^{\mathrm {a}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \varepsilon \sum _{\mathrm {b}} \!\! \int \!\! \mathrm {d}\Gamma _{2}^{\mathrm {b}} \, \varvec{\mathcal {F}}_{\!1^{\mathrm {a}} 2^{\mathrm {b}}} \!\cdot \! \frac{\partial \mathcal {C}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} = 0 \, , \end{aligned}$$
(6.118)
where we introduced the small parameter \({ \varepsilon \!=\! M_{\star } / M_{\bullet } \!=\! (\sum _{\mathrm {a}} \!M_{\star }^{\mathrm {a}})/M_{\bullet } }\). Similarly, Eq. (6.116) becomes
$$\begin{aligned}&\, \frac{\partial \mathcal {C}^{\mathrm {a}\mathrm {b}}}{\partial t} + \varvec{v}_{1}^{\mathrm {a}} \!\cdot \! \frac{\partial \mathcal {C}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{x}_{1}^{\mathrm {a}}} + \varvec{v}_{2}^{\mathrm {b}} \!\cdot \! \frac{\partial \mathcal {C}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{x}_{2}^{\mathrm {b}}} + \varvec{\mathcal {F}}_{\!1^{\mathrm {a}}0} \!\cdot \! \frac{\partial \mathcal {C}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \varvec{\mathcal {F}}_{\!2^{\mathrm {b}}0} \!\cdot \! \frac{\partial \mathcal {C}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{2}^{\mathrm {b}}} + \varepsilon \varvec{\mathcal {F}}_{\!1^{\mathrm {a}} \mathrm {r}} \!\cdot \! \frac{\partial \mathcal {C}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \varepsilon \varvec{\mathcal {F}}_{\! 2^{\mathrm {b}} \mathrm {r}} \!\cdot \! \frac{\partial \mathcal {C}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{2}^{\mathrm {b}}} \nonumber \\&\, + \varepsilon \eta _{\mathrm {b}} \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} 2^{\mathrm {b}}} \!\cdot \! \frac{\partial F^{\mathrm {a}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} F^{\mathrm {b}}(\Gamma _{2}^{\mathrm {b}}) + \varepsilon \eta _{\mathrm {a}} \varvec{\mathcal {F}}_{\! 2^{\mathrm {b}} 1^{\mathrm {a}}} \!\cdot \! \frac{\partial F^{\mathrm {b}}}{\partial \varvec{v}_{2}^{\mathrm {b}}} F^{\mathrm {a}} (\Gamma _{1}^{\mathrm {a}}) \nonumber \\&\, + \varepsilon \bigg [\! \sum _{\mathrm {c}} \!\! \int \!\!\! \mathrm {d}\Gamma _{3}^{\mathrm {c}} \varvec{\mathcal {F}}_{\! 1^{\mathrm {a}} 3^{\mathrm {c}}} F^{\mathrm {c}} (\Gamma _{3}^{\mathrm {c}}) \bigg ] \!\cdot \! \frac{\partial \mathcal {C}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \varepsilon \bigg [\! \sum _{\mathrm {c}} \!\! \int \!\!\! \mathrm {d}\Gamma _{3}^{\mathrm {c}} \varvec{\mathcal {F}}_{\!2^{\mathrm {b}} 3^{\mathrm {c}}} F^{\mathrm {c}} (\Gamma _{3}^{\mathrm {c}}) \bigg ] \!\cdot \! \frac{\partial \mathcal {C}^{\mathrm {a}\mathrm {b}}}{\partial \varvec{v}_{2}^{\mathrm {b}}} \nonumber \\&\, + \varepsilon \bigg [ \sum _{\mathrm {c}} \!\! \int \!\! \mathrm {d}\Gamma _{3}^{\mathrm {c}} \, \varvec{\mathcal {F}}_{\!1^{\mathrm {a}} 3^{\mathrm {c}}} \mathcal {C}^{\mathrm {b}\mathrm {c}} (\Gamma _{2} , \Gamma _{3}^{\mathrm {c}}) \bigg ] \!\cdot \! \frac{\partial F^{\mathrm {a}}}{\partial \varvec{v}_{1}^{\mathrm {a}}} + \varepsilon \bigg [ \sum _{\mathrm {c}} \!\! \int \!\! \mathrm {d}\Gamma _{3}^{\mathrm {c}} \, \varvec{\mathcal {F}}_{\! 2^{\mathrm {b}} 3^{\mathrm {c}}} \mathcal {C}^{\mathrm {a}\mathrm {c}} (\Gamma _{1}^{\mathrm {a}} , \Gamma _{3}^{\mathrm {c}}) \bigg ] \!\cdot \! \frac{\partial F^{\mathrm {b}}}{\partial \varvec{v}_{2}^{\mathrm {b}}} = 0 \, , \end{aligned}$$
(6.119)
where we introduced in the second line the small parameter \({ \eta _{\mathrm {a}} \!=\! \mu _{\mathrm {a}}/M_{\star } }\) of order \({ 1/N_{\mathrm {a}}}\). Equations (6.118) and (6.119) are the direct multi-component equivalents of Eqs. (6.15) and (6.16).
As presented in Sect. 6.3, let us now rewrite the two previous BBGKY equations within the angle-action coordinates appropriate for the Keplerian motion induced by the central BH. Let us consequently perform the degenerate angle-average from Eq. (6.26) and assume that \(F^{\mathrm {a}}\) and \(\mathcal {C}^{\mathrm {a}\mathrm {b}}\) satisfy the crucial assumptions from Eqs. (6.34) and (6.39). One can then rewrite Eq. (6.118) as
$$\begin{aligned} \frac{\partial \overline{F^{\mathrm {a}}}}{\partial \tau } + \big [ \overline{F^{\mathrm {a}}}, \overline{\Phi }+ \overline{\Phi }_{\mathrm {r}} \big ] + \sum _{\mathrm {b}} \!\! \int \!\! \mathrm {d}\varvec{\mathcal {E}}_{2} \, \big [ \overline{\mathcal {C}^{\mathrm {a}\mathrm {b}}}(\varvec{\mathcal {E}}_{1} , \varvec{\mathcal {E}}_{2}) , \overline{U}_{12} \big ]_{(1)} = 0 \, , \end{aligned}$$
(6.120)
where we introduced the rescaled time \({ \tau \!=\! (2 \pi )^{d-k} \varepsilon t }\) from Eq. (6.44) with \( \varepsilon \!=\! M_{\star } /M_{\bullet } \). Following Eq. (6.36), we also introduced the total averaged self-consistent potential \(\overline{\Phi }\) as
$$\begin{aligned} \overline{\Phi }= \sum _{\mathrm {a}} \overline{\Phi ^{\mathrm {a}}}\, , \end{aligned}$$
(6.121)
where the averaged potential \(\overline{\Phi ^{\mathrm {a}}}\) due to the component “\(\mathrm {a}\)” follows from Eq. (6.36) and reads
$$\begin{aligned} \overline{\Phi ^{\mathrm {a}}}(\varvec{\mathcal {E}}_{1}) = \!\! \int \!\! \mathrm {d}\varvec{\mathcal {E}}_{2} \, \overline{F^{\mathrm {a}}}(\varvec{\mathcal {E}}_{2}) \, \overline{U}_{12} (\varvec{\mathcal {E}}_{1} , \varvec{\mathcal {E}}_{2}) \, . \end{aligned}$$
(6.122)
In Eq. (6.122), we relied on the averaged wire-wire interaction potential \(\overline{U}_{12}\) from Eq. (6.37). Following the same approach, Eq. (6.119) can be rewritten as
$$\begin{aligned}&\, \frac{\partial \overline{\mathcal {C}^{\mathrm {a}\mathrm {b}}}}{\partial \tau } + \big [ \overline{\mathcal {C}^{\mathrm {a}\mathrm {b}}}(\varvec{\mathcal {E}}_{1} , \varvec{\mathcal {E}}_{2}), \overline{\Phi }(\varvec{\mathcal {E}}_{1}) + \overline{\Phi }_{\mathrm {r}} (\varvec{\mathcal {E}}_{1}) \big ]_{(1)} + \big [ \overline{\mathcal {C}^{\mathrm {a}\mathrm {b}}}(\varvec{\mathcal {E}}_{1} , \varvec{\mathcal {E}}_{2}) , \overline{\Phi }(\varvec{\mathcal {E}}_{2}) + \overline{\Phi }_{\mathrm {r}} (\varvec{\mathcal {E}}_{2}) \big ]_{(2)} \nonumber \\&\, + \sum _{\mathrm {c}} \!\! \int \!\! \mathrm {d}\varvec{\mathcal {E}}_{3} \, \overline{\mathcal {C}^{\mathrm {b}\mathrm {c}}}(\varvec{\mathcal {E}}_{2} , \varvec{\mathcal {E}}_{3}) \big [ \overline{F^{\mathrm {a}}}(\varvec{\mathcal {E}}_{1}) , \overline{U}_{13} \big ]_{(1)} + \sum _{\mathrm {c}} \!\! \int \!\! \mathrm {d}\varvec{\mathcal {E}}_{3} \, \overline{\mathcal {C}^{\mathrm {a}\mathrm {c}}}(\varvec{\mathcal {E}}_{1} , \varvec{\mathcal {E}}_{3}) \big [ \overline{F^{\mathrm {b}}}(\varvec{\mathcal {E}}_{2}) , \overline{U}_{23} \big ]_{(2)} \nonumber \\&\, + \frac{1}{(2 \pi )^{d - k}} \bigg \{ \eta _{\mathrm {b}} \big [ \overline{F^{\mathrm {a}}}(\varvec{\mathcal {E}}_{1}) \overline{F^{\mathrm {b}}}(\varvec{\mathcal {E}}_{2}) , \overline{U}_{12} \big ]_{(1)} + \eta _{\mathrm {a}} \big [ \overline{F^{\mathrm {a}}}(\varvec{\mathcal {E}}_{1}) \overline{F^{\mathrm {b}}}(\varvec{\mathcal {E}}_{2}) , \overline{U}_{21} \big ]_{(2)} \bigg \} = 0 \, . \end{aligned}$$
(6.123)
The two coupled evolution equations (6.120) and (6.123) are the direct multi-component equivalents of Eqs. (6.45) and (6.46). The main differences here are the changes in the mass prefactors in the last term (the source term) of Eq. (6.123). Indeed, it mixes the two small parameters \({ \eta _{\mathrm {a}} \!=\! \mu _{\mathrm {a}} / M_{\star } }\) and \({ \eta _{\mathrm {b}} \!=\! \mu _{\mathrm {b}} / M_{\star } }\). This change is the one which allows for mass segregation in multi-component systems, as briefly discussed in Sect. 6.5.2. Starting from Eqs. (6.120) and (6.123), one can then follow the method presented in Sect. 6.5 to derive the associated kinetic equation for \(\overline{F^{\mathrm {a}}}\). This is the multi-component inhomogeneous degenerate Balescu–Lenard equation (6.63).
C From Fokker–Planck to Langevin
The degenerate inhomogeneous Balescu–Lenard equation (6.53) is a self-consistent integro-differential equation describing the evolution of the system’s DF as a whole under the effect of its own graininess. Instead of describing the statistical dynamics of the full system’s DF, one could be interested in characterising the individual dynamics of one test particle in this system. Following Risken (1996), let us recall how one may obtain the stochastic Langevin equation describing such an individual dynamics. Let us start from the generic writing of the degenerate Balescu–Lenard equation (6.58) written as an anisotropic Fokker–Planck equation. It reads
$$\begin{aligned} \frac{\partial \overline{F}}{\partial \tau } = \frac{\partial }{\partial \varvec{J}^{\mathrm {s}}} \!\cdot \! \bigg [ \varvec{A} (\varvec{J} , \tau ) \, \overline{F}(\varvec{J} , \tau ) + \varvec{D} (\varvec{J} , \tau ) \!\cdot \! \frac{\partial \overline{F}}{\partial \varvec{J}^{\mathrm {s}}} \bigg ] \, , \end{aligned}$$
(6.124)
where, following the notations from Eq. (6.59), we introduced the system’s total drift vector \({ \varvec{A} (\varvec{J} , \tau ) }\) and diffusion tensor \({ \varvec{D} (\varvec{J} , \tau ) }\) as
$$\begin{aligned} \varvec{A} (\varvec{J} , \tau ) = \sum _{\varvec{m}^{\mathrm {s}}} \varvec{m}^{\mathrm {s}} \, A_{\varvec{m}^{\mathrm {s}}} (\varvec{J} , \tau ) \;\;\; ; \;\;\; \varvec{D} (\varvec{J} , \tau ) = \sum _{\varvec{m}^{\mathrm {s}}} \varvec{m}^{\mathrm {s}} \!\otimes \! \varvec{m}^{\mathrm {s}} \, D_{\varvec{m}^{\mathrm {s}}} (\varvec{J} , \tau ) \, . \end{aligned}$$
(6.125)
Let us recall here that the Balescu–Lenard equation being self-consistent, the drift and diffusion coefficients depend secularly on the system’s DF, \(\overline{F}\), but this was not written out explicitly to shorten the notations. Following the notations from Eq. (4.94a) in Risken (1996), let us rewrite Eq. (6.124) as
$$\begin{aligned} \frac{\partial \overline{F}}{\partial \tau } = \frac{\partial }{\partial \varvec{J}^{\mathrm {s}}} \!\cdot \! \bigg [ - \varvec{D}^{(1)} (\varvec{J} , \tau ) \, \overline{F}(\varvec{J} , \tau ) + \frac{\partial }{\partial \varvec{J}^{\mathrm {s}}} \!\cdot \! \bigg [ \varvec{D}^{(2)} (\varvec{J} , \tau ) \, \overline{F}(\varvec{J} , \tau ) \bigg ] \bigg ] \, , \end{aligned}$$
(6.126)
where we introduced the first- and second-order diffusion coefficients as
$$\begin{aligned} \varvec{D}^{(1)} (\varvec{J} , \tau ) = - \varvec{A} (\varvec{J} , \tau ) + \frac{\partial }{\partial \varvec{J}^{\mathrm {s}}} \!\cdot \! \varvec{D} (\varvec{J} , \tau ) \;\;\; ; \;\;\; \varvec{D}^{(2)} (\varvec{J} , \tau ) = \varvec{D} (\varvec{J} , \tau ) \, . \end{aligned}$$
(6.127)
Here, let us emphasise that the diffusion of the Keplerian wires takes place in the full action domain \(\varvec{J}\), while Eq. (6.126) only involves gradients w.r.t. the slow actions \(\varvec{J}^{\mathrm {s}}\). This leads, amongst others, to the conservation of the fast actions \(\varvec{J}^{\mathrm {f}}\) during the resonant diffusion, as noted in Eq. (6.62). Of course, by enlarging the diffusion coefficients \(\varvec{D}^{(1)}\) and \(\varvec{D}^{(2)}\) with zero coefficients for all the adiabatically conserved fast actions \(\varvec{J}^{\mathrm {f}}\), it is straightforward to rewrite Eq. (6.126) as a diffusion equation in the full action space involving derivatives w.r.t. all action coordinates \(\varvec{J}\).
Let us now focus on the dynamics of a given test Keplerian wire. We denote as \({ \varvec{\mathcal {J}} (\tau ) }\) its position in action space a time \(\tau \). On secular timescales, this test particle undergoes an individual stochastic diffusion consistent with the system’s averaged diffusion captured by the diffusion equation (6.126). This diffusion follows a stochastic Langevin equation reading
$$\begin{aligned} \frac{\mathrm {d}\varvec{\mathcal {J}}}{\mathrm {d}\tau } = \varvec{h} (\varvec{\mathcal {J}} , \tau ) + {\varvec{g}} (\varvec{\mathcal {J}} , \tau ) \!\cdot \! \varvec{\Gamma } (\tau ) \, , \end{aligned}$$
(6.128)
where we introduced the Langevin vector and tensor \(\varvec{h}\) and \(\varvec{g}\), as well as the stochastic Langevin forces \({ \varvec{\Gamma } (\tau ) }\), whose statistics are given by
$$\begin{aligned} \big< \varvec{\Gamma } (\tau ) \big> = 0 \;\;\; ; \;\;\; \big < \varvec{\Gamma } (\tau ) \!\otimes \! \varvec{\Gamma } (\tau ') \big > = 2 \varvec{I} \delta _{\mathrm {D}} (\tau \!-\! \tau ') \, , \end{aligned}$$
(6.129)
with \(\varvec{I}\) the identity matrix. Following Eq. (3.124) of Risken (1996), let us finally express the Langevin coefficients from Eq. (6.128) as a function of the drift and diffusion coefficients appearing in Eq. (6.126). The second-order diffusion tensor \(\varvec{D}^{(2)}\) is definite positive, so that we may introduce as \({ \sqrt{\varvec{D}}^{(2)} }\) one of its square root. One then has the components relations
$$\begin{aligned} \varvec{h}_{i} = \varvec{D}_{i}^{(1)} - \sum _{j , k} \big ( \sqrt{\varvec{D}}^{(2)} \big )_{kj} \frac{\partial \big ( \sqrt{\varvec{D}}^{(2)} \big )_{ij}}{\partial x_{k}} \;\;\; ; \;\;\; \varvec{g}_{ij} = \big ( \sqrt{\varvec{D}}^{(2)} \big )_{ij} \, . \end{aligned}$$
(6.130)
Thanks to Eq. (6.130), one can fully specify the detailed characteristics of the diffusion of an individual orbit as described by the Langevin equation (6.128). The self-consistency of the diffusion imposes to the diffusion coefficients \(\varvec{D}^{(1)}\) and \(\varvec{D}^{(2)}\), and therefore to the Langevin coefficients \(\varvec{h}\) and \(\varvec{g}\), to be updated as the system’s DF \(\overline{F}\) secularly changes. Let us finally emphasise that the previous presentation of the associated Langevin equation was made for quasi-Keplerian systems governed by the degenerate Balescu–Lenard equation (6.53). It is straightforward to follow the same approach to write the Langevin equation associated with the non-degenerate Balescu–Lenard equation (2.67), which can indeed also be cast as an anisotropic Fokker–Planck equation, as in Eq. (2.68).
