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Appendix A: The thermodynamic limit and the weak coupling approximation

In this Appendix, we discuss the proper thermodynamic limit of self-gravitating systems and justify the weak coupling approximation.

We call v_m the root mean square velocity of the stars and R the system’s size. The kinetic energy and the potential energy U ~ N²Gm²/R in the Hamiltonian (Eq. (1)) are comparable provided that (this fundamental scaling may also be obtained from the virial theorem). As a result, the energy scales as  and the kinetic temperature, defined by , scales as k_BT ~ GNm²/R ~ E/N. Inversely, these relations may be used to define R and v_m as a function of the energy E (conserved quantity in the microcanonical ensemble) or as a function of the temperature T (fixed quantity in the canonical ensemble).

The proper thermodynamic limit of a self-gravitating system corresponds to N →  +∞ in such a way that the normalized energy ϵ = ER/(GN²m²) and the normalized temperature η = βGNm²/R are of order unity. Of course, the usual thermodynamic limit N,V →  +∞ with N/V ~ 1 is not applicable to self-gravitating systems since these systems are spatially inhomogeneous.

From R and v_m, we define the dynamical time . If we rescale the distances by R, the velocities by v_m, and the times by t_D, we find that the equations of the BBGKY hierarchy depend on a single dimensionless parameter η/N. This is the equivalent of the plasma parameter in plasma physics. It is usually argued that the correlation functions scale as . Therefore, when N →  +∞, we can expand the equations of the BBGKY hierarchy in powers of 1/N ≪ 1. This corresponds to a weak coupling approximation (see below). We note that the thermodynamic limit is well-defined for the out-of-equilibrium problem although there is no statistical equilibrium state (i.e. the density of state and the partition function diverge).

By a suitable normalization of the parameters, we can take R ~ v_m ~ t_D ~ m ~ 1. In that case, we must impose G ~ 1/N. This is the Kac prescription (Kac et al. 1963; Messer & Spohn 1982). With this normalization, E ~ N, S ~ N and T ~ 1. The energy and the entropy are extensive but they remain fundamentally non-additive (Campa et al. 2009). The temperature is intensive. This normalization is very convenient since the length, velocity and time scales are of order unity. Furthermore, since the coupling constant G scales as 1/N, this immediately shows that a regime of weak coupling holds when N ≫ 1.

Other normalizations of the parameters are possible. For example, Gilbert (1968) considers the limit N →  +∞ with R ~ v_m ~ t_D ~ G ~ 1 and m ~ 1/N. In that case, E ~ 1, S ~ N, and T ~ 1/N. On the other hand, de Vega & Sanchez (2002) define the thermodynamic limit as N →  +∞ with m ~ G ~ v_m ~ 1 and R ~ N in order to have E ~ N, S ~ N, and T ~ 1. This normalization is natural since m and G should not depend on N. However, in that case, the dynamical time t_D = R/v_m ~ N diverges with the number of particles. Therefore, this normalization is not very convenient to develop a kinetic theory of stellar systems³². If we impose G ~ 1, E ~ N, T ~ 1 and t_D ~ 1, we get R ~ N^1/5 and m ~ N^−2/5 (this corresponds to ρ ~ 1 since ). We stress that all these scalings are equally valid. The important thing is that ϵ and η are O(1) when N →  +∞. One should choose the most convenient scaling which, in our opinion, is the Kac scaling³³.

We can also present the preceding results in the following manner, by analogy with plasma physics. A fundamental length scale in self-gravitating systems is the Jeans length λ_J = (4πGβmρ)^−1/2. This is the counterpart of the Debye length λ_D = (4πe²βρ/m)^−1/2 in plasma physics. Since , we find that λ_J ~ R. Therefore, the Jeans length is of the order of the system’s size. On the other hand, a fundamental time scale in self-gravitating systems is provided by the gravitational pulsation ω_G = (4πGρ)^1/2. This is the counterpart of the plasma pulsation ω_P = (4πe²ρ/m²)^1/2 in plasma physics. We can define the dynamical time by . If we rescale the distances by λ_J and the times by t_D, we find that the equations of the BBGKY hierarchy depend on a single dimensionless parameter , where gives the number of stars in the Jeans sphere. This is the counterpart of the number of electrons in the Debye sphere in plasma physics. When Λ ≫ 1, we can expand the equations of the BBGKY hierarchy in terms of the small parameter 1/Λ.

Finally, we show that 1/Λ may be interpreted as a coupling parameter. The coupling parameter Γ is defined as the ratio of the interaction strength at the mean interparticle distance Gm²n^1/3 (resp. e) to the thermal energy k_BT. This leads to (resp. ). If we define the coupling parameter g as the ratio of the interaction strength at the Jeans (resp. Debye) length Gm²/λ_J (resp. e²/λ_D) to the thermal energy k_BT, we get g = 1/Λ (or g = E_pot/E_kin = (Gm²/R)/k_BT = η/N with the initial variables). Therefore, the expansion of the BBGKY hierarchy in terms of the coupling parameter Γ or g is equivalent to an expansion in terms of the inverse of the number of particles in the Jeans sphere (resp. Debye sphere ). The weak coupling approximation is therefore justified when Λ ≫ 1.

Appendix B: Angle-action variables

In Sect. 4, we have explained that during its collisional evolution a stellar system passes by a succession of QSSs that are steady states of the Vlasov equation slowly changing under the effect of close encounters (finite N effects). The slowly varying distribution function f(r, v) determines a potential Φ(r) and a one-particle Hamiltonian ϵ = v²/2 + Φ(r) that we assume to be integrable. Therefore, it is possible to use angle-action variables constructed with this Hamiltonian (Goldstein 1956; Binney & Tremaine 2008). This construction is done adiabatically, i.e. the distribution function, and the angle-action variables, slowly change in time.

A particle with coordinates (r, v) in phase space is described equivalently by the angle-action variables (w, J). The Hamiltonian equations for the conjugate variables (r, v) are (B.1)In terms of the variables (r, v), the dynamics is complicated because the potential explicitly appears in the second equation. Therefore, this equation dv/dt = −∇Φ cannot be easily integrated except if Φ = 0, i.e. for a spatially homogeneous system. In that case, the velocity v is constant and the unperturbed equations of motion reduce to r = vt + r₀, i.e. to a rectilinear motion at constant velocity. Now, the angle-action variables are constructed so that the Hamiltonian does not depend on the angles w. Therefore, the Hamiltonian equations for the conjugate variables (w, J) are (B.2)where Ω(J) is the angular frequency of the orbit with action J. From these equations, we find that J is a constant and that w = Ω(J)t + w₀. Therefore, the equations of motion are very simple in these variables. They extend naturally the trajectories at constant velocity for spatially homogeneous systems. This is why this choice of variables is relevant to develop the kinetic theory. Of course, even if the description of the motion becomes simple in these variables, the complexity of the problem has not completely disappeared. It is now embodied in the relation between the position and momentum variables and the angle and action variables which can be very complicated.

Appendix C: Calculation of K^μν

In this Appendix, we compute the tensor K^μν that appears in the Vlasov-Landau equation (Eq. (28)). Within the local approximation, we can proceed as if the system were spatially homogeneous. In that case, the mean field force vanishes,  ⟨ F ⟩  = 0, and the unperturbed equations of motion (i.e. for N →  +∞) reduce to corresponding to a rectilinear motion at constant velocity. The collision term in the kinetic equation (Eq. (26)) can be written as (C.3)with (C.4)The force by unit of mass created by particle 1 on particle 0 is given by (C.5)where u(r − r′) = −G/|r − r′| is the gravitational potential. The Fourier transform and the inverse Fourier transform of the potential are defined by (C.6)For the gravitational interaction: (C.7)Substituting Eq. (C.6-b) in Eq. (C.5), and writing explicitly the Lagrangian coordinates, we get (C.8)Using the equations of motion (C.1) and (C.2), and introducing the notations x = r − r₁ and w = v − v₁, we obtain (C.9)Therefore, (C.10)Using the identity (C.11)and integrating over x and k′, we find that (C.12)Performing the transformation τ → −τ, then k → −k, and adding the resulting expression to Eq. (C.12), we get (C.13)Using the identity (C.11), we finally obtain (C.14)which leads to Eq. (27).

Introducing a spherical system of coordinates in which the z axis is taken in the direction of w, we find that (C.15)Using k_x = ksinθcosφ, k_y = ksinθsinφ and k_z = kcosθ, it is easy to see that only K_xx, K_yy and K_zz can be non-zero. The other components of the matrix K^μν vanish by symmetry. Furthermore, (C.16)Using the identity , we get (C.17)With the change of variables s = cosθ, we obtain (C.18)so that, finally, (C.19)On the other hand, (C.20)In conclusion, we obtain (C.21)with (C.22)Using Eq. (C.7), this leads to Eqs. (28)–(30).

Appendix D: Another derivation of the Landau equation

For an infinite homogeneous system, the distribution function and the two-body correlation function can be written as f(0) = f(v, t) and g(0,1) = g(r − r₁,v, v₁, t). In that case, Eqs. (18) and (19) become where we have defined x = r − r₁ and w = v − v₁. Using the Bogoliubov ansatz, we shall treat the distribution function f as a constant, and determine the asymptotic value g(x, v, v₁, +∞) of the correlation function. Introducing the Fourier transforms of the potential of interaction and of the correlation function, Eq. (D.1) may be replaced by (D.3)where we have used the reality condition ĝ(−k) = ĝ(k) ^∗ . On the other hand, taking the Laplace-Fourier transform of Eq. (D.2) and assuming that no correlation is present initially (if there are initial correlations, their effect becomes rapidly negligible), we get (D.4)Taking the inverse Laplace transform of Eq. (D.4), and using the residue theorem, we find that the asymptotic value t →  +∞ of the correlation function, determined by the pole ω = 0, is (D.5)Using the Plemelj formula (D.6)we get (D.7)Substituting Eq. (D.7) in Eq. (D.3), we obtain the Landau equation (Eq. (27)).

Appendix E: Lenard-Balescu equation for homogeneous stellar systems

If we assume that the system is spatially homogeneous (or make the local approximation), and take collective effects into account, the Vlasov-Landau Eq. (27) is replaced by the Vlasov-Lenard-Balescu equation (E.1)where ϵ(k, ω) is the dielectric function (E.2)The Landau equation is recovered by taking |ϵ(k, k·v)|² = 1. The Lenard-Balescu equation generalizes the Landau equation by replacing the bare potential of interaction û(k) by a dressed potential of interaction (E.3)The dielectric function in the denominator takes the dressing of the particles by their polarization cloud into account. In plasma physics, this term corresponds to a screening of the interactions. The Lenard-Balescu equation accounts for dynamical screening since the velocity v of the particles explicitly appears in the effective potential. However, for Coulombian interactions, it is a good approximation to neglect the deformation of the polarization cloud due to the motion of the particles and use the static results on screening (Debye & Hückel 1923). This amounts to replacing the dynamic dielectric function |ϵ(k, k·v)| by the static dielectric function |ϵ(k, 0)|. In this approximation, û_dressed(k, k·v) is replaced by the Debye-Hückel potential corresponding to u_DH(x) = (e²/m²)e^−k_Dr/r in physical space. If we make the same approximation for stellar systems, we find that û_dressed(k, k·v) is replaced by (E.4)corresponding to u_DH(x) = −Gcos(k_Jr)/r in physical space. In this approximation, the Vlasov-Lenard-Balescu equation (Eq. (E.1)) takes the same form as the Vlasov-Landau equation (Eq. (28)) except that A is now given by A = 2πmG²Q with (E.5)where R is the system’s size. We see that Q diverges algebraically, as (λ_J − R)^-1, when R → λ_J instead of yielding a finite Coulombian logarithm lnΛ ~ lnN when collective effects are neglected³⁴. This naive approach shows that collective effects (which account for anti-shielding) tend to increase the diffusion coefficient and consequently tend to reduce the relaxation time. This is the conclusion reached by Weinberg (1993) with a more precise approach. However, his approach is not fully satisfactory since the system is assumed to be spatially homogeneous and the ordinary Lenard-Balescu equation is used. In that case, the divergence when R → λ_J is a manifestation of the Jeans instability that a spatially homogeneous self-gravitating system experiences when the size of the perturbation overcomes the Jeans length. For inhomogeneous systems, the Jeans instability is suppressed so the results of Weinberg should be used with caution. Heyvaerts (2010) and Chavanis (2012a) derived a more satisfactory Lenard-Balescu equation that is valid for spatially inhomogeneous stable self-gravitating systems. This equation does not present any divergence at large scales. However, this equation is complicated and it is difficult to measure the importance of collective effects. The approach of Weinberg (1993), and the arguments given in this Appendix, suggest that collective effects reduce the relaxation time of inhomogeneous stellar systems. This reduction should be particularly strong for a system close to instability because of the enhancement of fluctuations (Monaghan 1978).

These considerations show that it is not possible to make a local approximation and simultaneously take collective effects into account. The usual procedure is to ignore collective effects, make a local approximation, and introduce a large-scale cut-off at the Jeans length. This is the procedure that is usually followed in stellar dynamics (Binney & Tremaine 2008). The only rigorous manner to take collective effects into account is to use the Lenard-Balescu equation written with angle-action variables.

Appendix F: Multi-species systems

It is straightforward to generalize the kinetic theory of stellar systems for several species of stars. The Vlasov-Landau equation (Eq. (27)) is replaced by (F.1)where f^a(r, v, t) is the distribution function of species a normalized such that , and the sum  ∑ _b runs over all species. We can use this equation to give a new interpretation of the test particle approach developed in Sect. 3. We make three assumptions: (i) We assume that the system is composed of two types of stars, the test stars with mass m and the field stars with mass m_f; (ii) we assume that the number of test stars is much lower than the number of field stars; and (iii) we assume that the field stars are in a steady distribution f(r, v). Because of assumption (ii), the collisions between the field stars and the test stars do not alter the distribution of the field stars so that the field stars remain in their steady state. The collisions of the test stars among themselves are also negligible, so they only evolve as a result of collisions with the field stars. Therefore, if we call P(r, v, t) the distribution function of the test stars (to have notations similar to those of Sect. 3 with, however, a different interpretation), its evolution is given by the Fokker-Planck equation obtained from Eq. (F.1) yielding (F.2)The diffusion and friction coefficients are given by We recall that the diffusion coefficient is due to the fluctuations of the gravitational force produced by the field stars, while the friction by polarization is due to the perturbation on the distribution of the field stars caused by the test stars. This explains the occurrence of the masses m_f and m in Eqs. (F.3) and (F.4), respectively.

Using Eq. (46) and noting that (F.5)we get (F.6)If we assume furthermore that m ≫ m_f, we find that F_friction ≃ F_pol. However, in general, the friction force is different from the friction by polarization. The other results of Sect. 3 can be easily generalized to multi-species systems.

If the field stars have an isothermal distribution, then from Eq. (F.4), and the Fokker-Planck equation (Eq. (F.2)) reduces to (F.7)where D^μν(v) is given by Eq. (F.3). Using the results of Sect. 3, we get (F.8)where is the velocity dispersion of the field stars and ρ_f = n_fm_f their density. On the other hand, lnΛ = ln   (λ_J/λ_L), where is the Jeans length and is the Landau length. This yields . The total friction is . If the distribution of the field stars is f ∝ e^−βm_fv2/2, the equilibrium distribution of the test stars is P ∝ e^−βmv2/2 ∝ f^m/m_f. When m ≫ m_f, the evolution is dominated by frictional effects; when m ≪ m_f it is dominated by diffusion. The relaxation time scales like where is the r.m.s. velocity of the test stars. Therefore, , where ξ is the friction coefficient associated to the friction by polarization. The true friction coefficient is ξ _∗  = (1 + m_f/m)ξ. We get , where we have used .

Appendix G: Temporal correlation tensor of the gravitational force

The diffusion of the stars is caused by the fluctuations of the gravitational force. For an infinite homogeneous system (or in the local approximation), the diffusion tensor can be derived from Eq. (43) that is well-known in Brownian theory. This expression involves the temporal auto-correlation tensor of the gravitational force experienced by a star. It can be written as (G.1)We first compute the tensor (G.2)Proceeding as in Appendix C, we find (G.3)Introducing a system of spherical coordinates with the z-axis in the direction of w, and using Eq. (C.7), we obtain after some calculations (G.4)According to Eq. (C.4), we have (G.5)Therefore, Eqs. (G.4) and (G.5) lead to Eq. (29) with lnΛ replaced by (G.6)where t_min and t_max are appropriate cut-offs. The upper cut-off should be identified with the dynamical time t_D. On the other hand, the divergence at short times is due to the inadequacy of our assumption of straight-line trajectories to describe very close encounters. If we take t_min = λ_L/v_m and t_max = λ_J/v_m ~ t_D, we find that lnΛ′ = lnΛ. In Appendix C, we have calculated K^μν by integrating first over time then over space. This yields the Coulombian logarithm (Eq. (30)). Here, we have integrated first over space then over time. This yields the Coulombian logarithm (Eq. (G.6)). As discussed by Lee (1968), these two approaches are essentially equivalent. We remark, however, that the calculations of Sect. C can be performed for arbitrary potentials, while the calculations of this section explicitly use the specific form of the gravitational potential.

According to Eqs. (G.1), (G.2), and (G.4), the force auto-correlation function can be written as (G.7)In particular (G.8)The τ^-1 decay of the auto-correlation function of the gravitational force was first derived by Chandrasekhar (1944) with a different method. This result has also been obtained, and discussed, by Cohen et al. (1950) and Lee (1968). According to Eqs. (43) and (G.7), the diffusion tensor is given by³⁵(G.9)This returns Eq. (79) with lnΛ′ instead of lnΛ.

When f(v₁) is the Maxwell distribution (Eq. (51)), we can compute the force auto-correlation tensor from Eq. (G.7) by using the Rosenbluth potentials as in Sect. 3.5. Alternatively, combining Eqs. (G.1) and (G.3), we have (G.10)where is the three-dimensional Fourier transform of the distribution function. For the Maxwell distribution (Eq. (51)), we obtain (G.11)Introducing a system of spherical coordinates with the z-axis in the direction of v, and using Eq. (C.7), we obtain after some calculations (G.12)where G^μν(x) is defined in Sect. 3.3. In particular, (G.13)Integrating Eq. (G.12) over time, we recover the expression (Eq. (67)) of the diffusion tensor for a Maxwellian distribution with lnΛ′ instead of lnΛ.

Finally, the auto-correlation tensor of the gravitational field at two different points (at the same time) is (G.14)

Appendix H: The different kinetic equations

The standard kinetic equations (Boltzmann, Fokker-Planck, Vlasov, Landau, and Lenard-Balescu) and their generalizations can be derived from the BBGKY hierarchy. In this Appendix, we show the connection between these different equations, and discuss their domains of validity, without entering into technical details.

The first two equations of the BBGKY hierarchy may be written symbolically as where f is the one-body distribution function, g the two-body correlation function, and h the three-body correlation function. In the first equation, is the Vlasov operator taking into account the free motion of the particles and the advection by the mean field . On the other hand, the collision term C [g]  describes the effect of two-body correlations on the evolution of the distribution function. In the second equation, ℒ = ℒ₀ + ℒ′ + ℒ_m.f. is a two-body Liouville operator where ℒ₀ describes the free motion of the particles, ℒ′ describes the exact two-body interaction, and ℒ_m.f. takes into account the effect of the mean field in the two-body problem. The term describes collective effects and the term describes three-body correlations. Finally, is a source term depending on the one-body distribution function.

As explained in Sect. 2.2, for N ≫ 1 we can expand the equations of the BBGKY hierarchy in terms of the small parameter 1/N. For N →  +∞, the encounters are negligible and we get the Vlasov equation (Jeans 1915; Vlasov 1938). This corresponds to the mean field approximation. At the order 1/N, we can neglect three-body correlations () and strong collisions (ℒ′ = 0) that are of order 1/N². This corresponds to the weak coupling approximation. In fact, since the gravitational potential is singular at r = 0, the two-body correlation function g(r₁, r₂) becomes large when |r₁ − r₂| → 0 due to the effect of strong collisions. As a result, it is necessary to take the term ℒ′g into account at small scales. When three-body correlations are neglected (), Eqs. (H.1) and (H.2) are closed. However, it does not appear possible to solve these equations explicitly without further approximation. The usual strategy is to solve these equations for small, intermediate, and large impact parameters, and then connect these limits.

If we neglect strong collisions (ℒ′ = 0), collective effects (), and make a local approximation (ℒ_m.f. = 0), we get the Landau equation (Eqs. (28) and (29)) with (H.3)This factor presents a logarithmic divergence at small and large scales. The Landau equation may be derived in different manners that are actually equivalent to the above procedure. Landau (1936) obtained his equation by starting from the Boltzmann equation and using a weak deflection approximation |Δv| ≪ 1. In his calculations, he approximated the trajectories of the particles by straight lines even for collisions with small impact parameters. This is equivalent to starting from the Fokker-Planck equation (Eq. (44)) and calculating the first and second moments of the velocity increments (Eq. (45)) resulting from a succession of binary encounters by using a straight line approximation.

If we neglect collective effects (), make a local approximation (ℒ_m.f. = 0), but take strong collisions into account (ℒ′ ≠ 0), we get the Boltzmann equation (see Balescu 2000). This equation does not present any divergence at small scales. Since the system is dominated by weak encounters, we can expand the Boltzmann equation for weak deflections |Δv| ≪ 1 while taking into account the effect of strong collisions. This is equivalent to the treatment of Chandrasekhar (1942, 1943a,b) who started from the Fokker-Planck equation (Eq. (44)) and calculated the first and second moments of the velocity increments (Eq. (45)) resulting from a succession of binary collisions by taking strong collisions into account. In the calculation of  ⟨ Δv^μ ⟩  and  ⟨ Δv^μΔv^ν ⟩ , he used the exact trajectory of the stars (i.e. he solved the two-body problem exactly) and took into account the strong deflections due to collisions with small impact parameters. In the dominant approximation lnN ≫ 1, his approach, completed by Rosenbluth et al. (1957), leads to the Fokker-Planck equation (Eq. (44)) with the expressions of the diffusion tensor and friction force (Eqs. (83) and (84)) expressed in terms of the Rosenbluth potentials (Eq. (85)) with (H.4)As shown in Sect. 3.5, the resulting Fokker-Planck equation can be transformed into the Landau equation (Eqs. (28) and (29)). As in the Landau approach, the factor calculated in Eq. (H.4) presents a logarithmic divergence at large scales. However, contrary to the Landau approach, it does not present a logarithmic divergence at small scales since the effect of strong collisions is taken into account explicitly in the Chandrasekhar approach³⁶. As a result, the gravitational Landau length appears naturally in the calculations of Chandrasekhar. This establishes that the relevant small-scale cut-off in Eq. (H.3) is the Landau length.

The Landau and the Chandrasekhar kinetic theories make the same assumption: binary encounters and an expansion in powers of the momentum transfer. They actually differ in the order in which these are introduced. Landau starts from the Boltzmann equation and considers a weak deflection approximation while Chandrasekhar directly starts from the Fokker-Planck equation, but calculates the coefficients of diffusion and friction with the binary-collision picture. At that level, their theories are equivalent³⁷ since the Fokker-Planck equation can be precisely obtained from the Boltzmann equation in the limit of weak deflections. The crucial difference is that Landau makes a weak coupling assumption and ignores strong collisions (i.e. the bending of the trajectories) while Chandrasekhar takes them into account. On the other hand, both theories ignore spatial inhomogeneity and collective effects.

If we neglect strong collisions (ℒ′ = 0) and collective effects (), but take spatial inhomogeneity (ℒ_m.f. ≠ 0) into account we get the generalized Landau equation (Eqs. (24) and (114)) derived in this paper (see also Kandrup 1981; and Chavanis 2008a,b). This equation presents a logarithmic divergence at small scales since strong collisions are neglected, but not at large scales since the finite extent of the system is taken into account. This suggests that the relevant large-scale cut-off in Eqs. (H.3) and (H.4) is the Jeans length.

If we neglect strong collisions (ℒ′ = 0) but take collective effects () and spatial inhomogeneity (ℒ_m.f. ≠ 0) into account, we get the generalized Lenard-Balescu equation derived by Heyvaerts (2010) and Chavanis (2012a).

The ordinary Landau equation () is intermediate between the Boltzmann equation and the generalized Lenard-Balescu (and generalized Landau) equation. It describes the effect of weak collisions but ignores strong collisions, spatial inhomogeneity, and collective effects. It can be obtained from the generalized Lenard-Balescu equation (ℒ′ = 0) by making a local approximation (ℒ_m.f. = 0) and neglecting collective effects (), or from the Boltzmann equation () by considering the limit of small deflections (ℒ′ = 0).

Actually, these kinetic equations describe the effect of collisions at different scales (the scale λ may be interpreted as the impact parameter). For λ ~ λ_L (small impact parameters), the collisions are strong and we must solve the two-body problem exactly. For λ ~ l (intermediate impact parameters), the collisions are weak and we can make a weak coupling approximation. For λ ~ λ_J (large impact parameters), we must take spatial inhomogeneity and collective effects into account. The Boltzmann equation is valid for λ ≪ λ_J. It describes strong

collisions (λ ~ λ_L) and weak collisions (λ ~ l). The generalized Landau and generalized Lenard-Balescu equations are valid for λ ≫ λ_L. They describe weak collisions (λ ~ l), spatial inhomogeneity and collective effects (λ ~ λ_J). The ordinary Landau equation is valid for λ_L ≪ λ ≪ λ_J. It describes weak collisions. When we go beyond the domains of validity of these equations, divergences occur and appropriate cut-offs must be introduced.

Connecting these different limits, we find that the best description of stellar systems is provided by the generalized Lenard-Balescu equation with a small-scale cut-off at the Landau length. If we neglect collective effects, we get the generalized Landau equation (Eqs. (24) and (114)) with a small-scale cut-off at the Landau length. Finally, if we make a local approximation and neglect collective effects, the Vlasov-Landau equation (Eqs. (28) and (29)) with a small-scale cut-off at the Landau length and a large-scale cut-off at the Jeans length provides a relevant description of stellar systems and has the advantage of the simplicity.

© ESO, 2013