A&A 556, A93 (2013) Astronomy DOI: 10.1051/0004-6361/201220607 & c ESO 2013 Astrophysics

Kinetic theory of spatially inhomogeneous stellar systems without collective effects

P.-H. Chavanis

Laboratoire de Physique Théorique (IRSAMC), CNRS and UPS, Université de Toulouse, 31062 Toulouse Cedex 4, France e-mail: [email protected]

Received 21 October 2012 / Accepted 27 March 2013

ABSTRACT

We review and complete the kinetic theory of spatially inhomogeneous stellar systems when collective effects (dressing of the stars by their polarization cloud) are neglected. We start from the BBGKY hierarchy issued from the Liouville equation and consider an expansion in powers of 1/N in a proper thermodynamic limit. For N → +∞, we obtain the Vlasov equation describing the evolution of collisionless stellar systems like elliptical galaxies. This corresponds to the mean field approximation. At the order 1/N, we obtain a kinetic equation describing the evolution of collisional stellar systems like globular clusters. This corresponds to the weak coupling approximation. This equation coincides with the generalized Landau equation derived from a more abstract projection operator formalism. This equation does not suffer logarithmic divergences at large scales since spatial inhomogeneity is explicitly taken into account. Making a local approximation, and introducing an upper cut-off at the Jeans length, it reduces to the Vlasov- Landau equation which is the standard kinetic equation of stellar systems. Our approach provides a simple and pedagogical derivation of these important equations from the BBGKY hierarchy which is more rigorous for systems with long-range interactions than the two-body encounters theory. Making an adiabatic approximation, we write the generalized Landau equation in angle-action variables and obtain a Landau-type kinetic equation that is valid for fully inhomogeneous stellar systems and is free of divergences at large scales. This equation is less general than the recently derived Lenard-Balescu-type kinetic equation since it neglects collective effects, but it is substantially simpler and could be useful as a first step. We discuss the evolution of the system as a whole and the relaxation of a test star in a bath of field stars. We derive the corresponding Fokker-Planck equation in angle-action variables and provide expressions for the diffusion coefficient and friction force. Key words. gravitation – methods: analytical – globular clusters: general

1. Introduction equation coupled to the Poisson equation. This purely mean field description applies to large groups of stars such as elliptical In its simplest description, a stellar system can be viewed as a galaxies whose ages are much less than the collisional relax- collection of N classical point mass stars in Newtonian gravita- ation time. A similar equation was introduced by Vlasov (1938) tional interaction (Spitzer 1987; Heggie & Hut 2003; Binney & in plasma physics to describe the collisionless evolution of a Tremaine 2008). As understood early on by Hénon (1964), self- system of electric charges interacting by the Coulomb force. gravitating systems experience two successive types of relax- The collisionless Boltzmann equation coupled self-consistently ation, a rapid collisionless relaxation towards a quasi-stationary to the Poisson equation is often called the Vlasov equation2,or state (QSS) that is a virialized state in mechanical equilibrium the Vlasov-Poisson system. but not in thermodynamical equilibrium, followed by a slow The concept of collisionless relaxation was first understood collisional relaxation. One might think that, because of the de- by Hénon (1964) and King (1966). Lynden-Bell (1967)devel- velopment of stellar encounters, the system will reach, at suf- oped a statistical theory of this process and coined the term “vi- ficiently long times, a statistical equilibrium state described by olent relaxation”. In the collisionless regime, the evolution of the the Maxwell-Boltzmann distribution. However, it is well-known star cluster is described by the Vlasov-Poisson system. Starting that unbounded stellar systems cannot be in strict statistical equi- from an unsteady or unstable initial condition, the Vlasov- librium1 because of the permanent escape of high energy stars Poisson system develops a complicated mixing process in phase (evaporation) and the gravothermal catastrophe (core collapse). space. Because the Vlasov equation is time-reversible, it never Therefore, the statistical mechanics of stellar systems is essen- achieves a steady state but develops filaments at smaller and tially an out-of-equilibrium problem which must be approached smaller scales. However, the coarse-grained distribution func- through kinetic theories. tion obtained by locally averaging the fine-grained distribution The first kinetic equation was written by Jeans (1915). function over the filaments usually achieves a steady state on Neglecting encounters between stars, he described the dynam- a few dynamical times. Lynden-Bell (1967) tried to predict the ical evolution of stellar systems by the collisionless Boltzmann QSS resulting from violent relaxation by developing a statisti- Appendices are available in electronic form at cal mechanics of the Vlasov equation. He derived a distribution http://www.aanda.org 1 For reviews of the statistical mechanics of self-gravitating systems 2 See Hénon (1982) for a discussion about the name that should be see e.g. Padmanabhan (1990), Katz (2003), and Chavanis (2006). given to this equation.

Article published by EDP Sciences A93, page 1 of 27 A&A 556, A93 (2013) function formally equivalent to the Fermi-Dirac distribution (or distance l (see also Jeans 1929; and Spitzer 1940). However, to a superposition of Fermi-Dirac distributions). However, when since the interparticle distance is smaller than the Debye length, coupled to the Poisson equation, these distributions have an in- the same arguments should also apply in plasma physics, which finite mass. Therefore, the Vlasov-Poisson system has no sta- is not the case. Therefore, the conclusions of Chandrasekhar and tistical equilibrium state (in the sense of Lynden-Bell). This is von Neumann are usually taken with circumspection. In partic- clear evidence that violent relaxation is incomplete (Lynden-Bell ular, Cohen et al. (1950) argued that the logarithmic divergence 1967). Incomplete relaxation is caused by inefficient mixing and should be cut-off at the Jeans length which gives an estimate of non-ergodicity. In general, the fluctuations of the gravitational the system’s size. While in neutral plasmas the effective interac- potential δΦ(r, t) that drive the collisionless relaxation last only tion distance is limited to the Debye length, in a self-gravitating for a few dynamical times and die out before the system has system the distance between interacting particles is only limited mixed efficiently (Tremaine et al. 1986). Understanding the ori- by the system’s size. Therefore, the Jeans length is the gravita- gin of incomplete relaxation, and developing models of incom- tional analogue of the Debye length. These kinetic theories lead plete violent relaxation to predict the structure of galaxies, is a to a collisional relaxation time scaling as tR ∼ (N/ln N)tD,where very difficult problem (Arad & Johansson 2005). Some models tD is the dynamical time and N the number of stars in the system. of incomplete violent relaxation have been proposed based on Chandrasekhar (1949) also developed a Brownian theory of stel- different physical arguments (Bertin & Stiavelli 1984; Stiavelli lar dynamics and showed that, from a qualitative point of view, & Bertin 1987;Hjorth&Madsen1991; Chavanis et al. 1996; the results of kinetic theory can be understood very simply in that Chavanis 1998; Levin et al. 2008). framework6. In particular, he showed that a dynamical friction On longer timescales, stellar encounters (sometimes referred is necessary to maintain the Maxwell-Boltzmann distribution of to as collisions by an abuse of language) must be taken into statistical equilibrium and that the coefficients of friction and account. This description is particularly important for small diffusion are related to each other by an Einstein relation which groups of stars such as globular clusters whose ages are of the is a manifestation of the fluctuation-dissipation theorem. This order of the collisional relaxation time. Chandrasekhar (1942, relation is confirmed by his more precise kinetic theory based 1943a,b) developed a kinetic theory of stellar systems to de- on two-body encounters. It is important to emphasize, however, termine the timescale of collisional relaxation and the rate of that Chandrasekhar did not derive the kinetic equation for the escape of stars from globular clusters3. To simplify the kinetic evolution of the system as a whole. Indeed, he considered the theory, he considered an infinite homogeneous system of stars. Brownian motion of a test star in a fixed distribution of field He started from the general Fokker-Planck equation and deter- stars (bath) and derived the corresponding Fokker-Planck equa- mined the diffusion coefficient and the friction force (first and tion. This equation has been used by Chandrasekhar (1943b), second moments of the velocity increments) by considering the Spitzer & Härm (1958), Michie (1963), King (1965), and more mean effect of a succession of two-body encounters4. This ap- recently by Lemou & Chavanis (2010) to study the evaporation proach was based on Jeans (1929) demonstration that the cu- of stars from globular clusters in a simple setting. mulative effect of the weak deflections resulting from the rel- King (1960) noted that if we were to describe the dynam- atively distant encounters is more important than the effect of ical evolution of the cluster as a whole, the distribution of the occasional large deflections produced by relatively close en- field stars should evolve in time in a self-consistent manner so counters. However, this approach leads to a logarithmic diver- that the kinetic equation must be an integro-differential equa- gence at large scales that is more difficult to remove in stel- tion. The kinetic equation obtained by King, from the results lar dynamics than in plasma physics because of the absence of Rosenbluth et al. (1957), is equivalent to the Landau equa- of Debye shielding for the gravitational force5. Chandrasekhar tion, although written in a different form. It is interesting to note, & von Neumann (1942) developed a completely stochastic for historical reasons, that none of the previous authors seemed formalism of gravitational fluctuations and showed that the fluc- to be aware of the work of Landau (1936) in plasma physics. tuations of the gravitational force are given by the Holtzmark There is, however, an important difference between stellar dy- distribution (a particular Lévy law) in which the nearest neigh- namics and plasma physics. Neutral plasmas are spatially ho- bor plays a prevalent role. From these results, they argued that mogeneous because of electroneutrality and Debye shielding. the logarithmic divergence has to be cut-off at the interparticle By contrast, stellar systems are spatially inhomogeneous. The

3 above-mentioned kinetic theories developed for an infinite ho- Early estimates of the relaxation time of stellar systems were mogeneous system can be applied to an inhomogeneous system made by Schwarzschild (1924), Rosseland (1928), Jeans (1929), Smart (1938), and Spitzer (1940). On the other hand, the evaporation time was first estimated by Ambartsumian (1938) and Spitzer (1940). 6 The stochastic evolution of a star is primarily due to many small 4 Later, Cohen et al. (1950), Gasiorowicz et al. (1956), and Rosenbluth deflections produced by relatively distant encounters. The problem of et al. (1957) proposed a simplified derivation of the coefficients of dif- treating particles undergoing numerous weak deflections was originally fusion and friction. encountered in relation to the Brownian motion of large molecules 5 A few years earlier, Landau (1936) had developed a kinetic theory of which are thermally agitated by the smaller field molecules. The Coulombian plasmas taking two-body encounters into account. Starting stochastic character of the many small impulses which act on a sus- from the Boltzmann (1872) equation, and making a weak deflection ap- pended particle is usually described by the Fokker-Planck equation. proximation, he derived a kinetic equation for the collisional evolution Chandrasekhar (1943c) noted the analogy between stellar dynamics and of neutral plasmas. His approach leads to a divergence at large scales Brownian theory and employed a Fokker-Planck equation to describe that he removed heuristically by introducing a cut-off at the Debye the evolution of the velocity distribution function of stars. Indeed, he length λD (Debye & Hückel 1923) which is the size over which the argued that a given star undergoes many small-angle (large impact pa- electric field produced by a charge is screened by the cloud of op- rameter) collisions in a time that is small compared with that in which posite charges. Later, Lenard (1960)andBalescu(1960) developed a its position or velocity changes appreciably. Since the Newtonian po- more precise kinetic theory taking collective effects into account. They tential makes the cumulative effect of these small momentum trans- derived a more elaborate kinetic equation, free of divergence at large fers dominant, the stochastic methods of the Fokker-Planck equation scales, in which the Debye length appears naturally. This justifies the should be more appropriate for a stellar system than the Boltzmann heuristic procedure of Landau. approach.

A93, page 2 of 27 P.-H. Chavanis: Kinetic theory of spatially inhomogeneous stellar systems without collective effects only if we make a local approximation. In that case, the collision for in the Lenard-Balescu equation. They eliminate the loga- term is calculated as if the system were spatially homogeneous rithmic divergence that occurs at large scales in the Landau or as if the collisions could be treated as local. Then, the ef- equation. For self-gravitating systems, they lead to anti-shielding fect of spatial inhomogeneity is only retained in the advection and are more difficult to analyze because the system is spatially (Vlasov) term which describes the evolution of the system due inhomogeneous8. If we consider a finite homogeneous system, to mean field effects. This leads to the Vlasov-Landau equation and take collective effects into account, one finds a severe di- which is the standard kinetic equation of stellar dynamics. To vergence of the diffusion coefficient when the size of the domain our knowledge, this equation has been first written (in a differ- reaches the Jeans scale (Weinberg 1993). This divergence, which ent form), and studied, by Hénon (1961). Hénon also exploited is related to the Jeans instability, does not occur in a stable spa- the timescale separation between the dynamical time tD and the tially inhomogeneous stellar system. Some authors like Thorne relaxation time tR  tD to derive a simplified kinetic equation (1968), Miller (1968), Gilbert (1968, 1970), and Lerche (1971) for f (,t), where  = v2/2 +Φ(r, t) is the individual energy of attempted to take collective effects and spatial inhomogeneity a star by unit of mass, called the orbit-averaged Fokker-Planck into account. They obtained very complicated kinetic equations equation. In this approach, the distribution function f (r, u, t), av- that have not found application until now. They managed, how- eraged over a short timescale, is a steady state of the Vlasov ever, to show that collective effects are equivalent to increasing equation of the form f (,t) which slowly evolves in time, on a the effective mass of the stars, hence diminishing the relaxation long timescale, due to the development of collisions (i.e. corre- time. Since, on the other hand, the effect of spatial inhomogene- lations caused by finite N effects or graininess). Hénon used this ity is to increase the relaxation time (Parisot & Severne 1979), equation to obtain a more relevant value for the rate of evapo- the two effects act in opposite directions and may balance each ration from globular clusters, valid for inhomogeneous systems. other. Cohn (1980) solved the orbit-averaged Fokker-Planck equation Recently, Heyvaerts (2010) derived from the BBGKY hi- numerically to describe the collisional evolution of star clus- erarchy issued from the Liouville equation a kinetic equation ters. His treatment accounts both for the escape of high energy in angle-action variables that takes both spatial inhomogeneity stars put forward by Spitzer (1940) and for the phenomenon of and collective effects into account. To calculate the collective re- core collapse resulting from the gravothermal catastrophe dis- sponse, he used Fourier-Laplace transforms and introduced a bi- covered by Antonov (1962) and Lynden-Bell & Wood (1968) orthogonal basis of pairs of density-potential functions (Kalnajs on the basis of thermodynamics and statistical mechanics. The 1971a). The kinetic equation derived by Heyvaerts is the coun- local approximation, which is a crucial step in the kinetic the- terpart for spatially inhomogeneous self-gravitating systems of ory, is supported by the stochastic approach of Chandrasekhar & the Lenard-Balescu equation for plasmas. Following his work, von Neumann (1942) showing the preponderance of the nearest we showed that this equation could be obtained equivalently neighbor. However, this remains a simplifying assumption which from the Klimontovich equation by making the so-called quasi- is not easily controllable. In particular, as we have already indi- linear approximation (Chavanis 2012a). We also developed a test cated, the local approximation leads to a logarithmic divergence particle approach and derived the corresponding Fokker-Planck at large scales that is difficult to remove. This divergence would equation in angle-action variables, taking collective effects into not have occurred if full account of spatial inhomogeneity had account. This provides general expressions of the diffusion co- been given from the start. efficient and friction force for spatially inhomogeneous stellar The effect of spatial inhomogeneity was investigated by systems. Severne & Haggerty (1976), Parisot & Severne (1979), Kandrup In a sense, these equations solve the problem of the kinetic (1981), and Chavanis (2008a,b). In particular, Kandrup (1981) theory of stellar systems since they take into account both spatial derived a generalized Landau equation from the Liouville equa- inhomogeneity and collective effects. However, the drawback is tion by using projection operator technics. This generalized that they are extremely complicated to solve (in addition of be- Landau equation is interesting because it takes into account ef- ing complicated to derive). In an attempt to reduce the complex- fects of spatial inhomogeneity which were neglected in pre- ity of the problem, we shall derive in this paper a kinetic equa- vious approaches. Since the finite extension of the system is tion that is valid for spatially inhomogeneous stellar systems but properly accounted for, there is no divergence at large scales7. that neglects collective effects. Collective effects may be less Furthermore, this approach clearly shows which approximations crucial in stellar dynamics than in plasma physics. In plasma are needed in order to recover the traditional Landau equa- physics, they must be taken into account in order to remove tion. Unfortunately, the generalized Landau equation remains the divergence at large scales that appears in the Landau equa- extremely complicated for practical applications. tion. In the case of stellar systems, this divergence is removed In addition, this equation is still approximate as it neglects by the spatial inhomogeneity of the system, not by collective collective effects and considers binary collisions between naked effects. Actually, previous kinetic equations based on the local particles. As in any weakly coupled system, the particles en- approximation ignore collective effects and already give satis- gaged in collisions are dressed by the polarization clouds caused factory results. We shall therefore ignore collective effects and by their own influence on other particles. Collisions between derive a kinetic equation (in position-velocity and angle-action dressed particles have quantitatively different outcomes than col- lisions between naked ones. In the case of plasmas, collective ef- 8 In a plasma, since the Coulomb force between electrons is repulsive, fects are responsible for Debye shielding and they are accounted each particle of the plasma tends to attract to it particles of opposite charge and to repel particles of like charge, thereby creating a kind 7 There remains a logarithmic divergence at small scales as in the orig- of cloud of opposite charge which screens the interaction at the scale inal Landau (1936) theory due to the neglect of strong collisions. This of the Debye length. In the gravitational case, since the Newton force divergence can be cured heuristically by introducing a cut-off at the is attractive, the situation is considerably different. The test star draws Landau length λL corresponding to the impact parameter leading to a neighboring stars into its vicinity and these add their gravitational force deflection at 90◦ (Landau 1936). This divergence does not occur in the to that of the test star itself. The bare gravitational force of the test star theory of Chandrasekhar (1942, 1943a,b) which takes strong collisions is thus augmented rather than shielded. The polarization acts to increase into account. the effective gravitational mass of a test star.

A93, page 3 of 27 A&A 556, A93 (2013) variables) that is the counterpart for spatially inhomogeneous 2. Evolution of the system as a whole self-gravitating systems of the Landau equation for plasmas. Our approach has three main interests. First, the derivation of this 2.1. The BBGKY hierarchy Landau-type kinetic equation is considerably simpler than the We consider an isolated system of N stars with identical mass m derivation of the Lenard-Balescu-type kinetic equation, and it in Newtonian interaction. Their dynamics is fully described by can be done in the physical space without having to introduce the Hamilton equations Laplace-Fourier transforms nor bi-orthogonal basis of pairs of density-potential functions. This offers a more physical deriva- dr ∂H du ∂H m i = , m i = − , tion of kinetic equations of stellar systems that may be of in- dt ∂u dt ∂r ffi i i terest for astrophysicists. Secondly, our approach is su cient to N 1 1 remove the large-scale divergence that occurs in kinetic theories H = mv2 − Gm2 · (1) based on the local approximation. It represents, therefore, a con- 2 i |r − r | i= i< j i j ceptual progress in the kinetic theory of stellar systems. Finally, 1 this equation is simpler than the Lenard-Balescu-type kinetic This Hamiltonian system conserves the energy E = H,themass equation derived by Heyvaerts (2010), and it could be useful as a M = Nm, and the angular momentum L = mr × u .Asre- ff i i i first step before considering more complicated e ects. Its draw- called in the Introduction, stellar systems cannot reach a statisti- ff back is to ignore collective e ects but this may not be crucial cal equilibrium state in a strict sense. In order to understand their as we have explained (as suggested by Weinberg’s work, collec- evolution, it is necessary to develop a kinetic theory. tive effects become important only when the system is close to We introduce the N-body distribution function PN (r1, u1,..., instability). This Landau-type equation was previously derived rN , uN , t) giving the probability density of finding at time t for systems with arbitrary long-range interactions9 in various di- the first star with position r1 and velocity u1, the second star mensions of space (Chavanis 2007, 2008a,b, 2010) but we think with position r2 and velocity u2, etc. Basically, the evolution of that it is important to discuss these results in the specific case of the N-body distribution function is governed by the Liouville self-gravitating systems with complements and amplification. equation The paper is organized as follows. In Sect. 2, we study the dynamical evolution of a spatially inhomogeneous stellar system N ∂P ∂P ∂P as a whole. Starting from the BBGKY hierarchy issued from the N + u · N + F · N = 0, (2) ∂t i ∂r i ∂u Liouville equation, and neglecting collective effects, we derive a i=1 i i general kinetic equation valid at the order 1/N in a proper ther- modynamic limit. For N → +∞, it reduces to the Vlasov equa- where tion. At the order 1/N we recover the generalized Landau equa- ∂Φ ri − r j tion derived by Kandrup (1981) from a more abstract projection F = − d = −Gm = F( j → i)(3) i ∂r |r − r |3 operator formalism. This equation is free of divergence at large i ji i j ji scales since spatial inhomogeneity has been properly accounted for. Making a local approximation and introducing an upper cut- is the gravitational force by unit of mass experienced by the ff o at the Jeans length, we recover the standard Vlasov-Landau ith star due to its interaction with the other stars. Here, Φd(r)de- equation which is usually derived from a kinetic theory based notes the exact gravitational potential produced by the discrete on two-body encounters. Our approach provides an alternative distribution of stars and F( j → i) denotes the exact force by derivation of this fundamental equation from the more rigorous unit of mass created by the jth star on the ith star. The Liouville Liouville equation. It has therefore some pedagogical interest. In equation (Eq. (2)), which is equivalent to the Hamilton equations Sect. 3, we study the relaxation of a test star in a steady distri- (Eq. (1)), contains too much information to be exploitable. In bution of field stars. We derive the corresponding Fokker-Planck practice, we are only interested in the evolution of the one-body ff equation and determine the expressions of the di usion and fric- distribution P1(r, u, t). tion coefficients. We emphasize the difference between the fric- From the Liouville equation we can construct the complete tion by polarization and the total friction (this difference may BBGKY hierarchy for the reduced distribution functions have been overlooked in previous works). For a thermal bath, we derive the Einstein relation between the diffusion and fric- = tion coefficients and obtain the explicit expression of the diffu- P j(x1, ..., xj, t) PN (x1, ..., xN , t)dx j+1...dxN , (4) sion tensor. This returns the standard results obtained from the two-body encounters theory but, again, our presentation is dif- where the notation x stands for (r, u). The generic term of this ferent and offers an alternative derivation of these important re- hierarchy reads sults. For that reason, we give a short review of the basic formu- lae. In Sect. 4, we derive a Landau-type kinetic equation written j j j ∂P j ∂P j ∂P j in angle-action variables and discuss its main properties. This + u · + F(k → i) · ∂t i ∂r ∂u equation, which does not make the local approximation, applies i=1 i i=1 k=1,ki i to fully inhomogeneous stellar systems and is free of divergence j ∂P j+1 at large scales. We also develop a test particle approach and de- + (N − j) F( j + 1 → i) · dx + = 0. (5) ∂u j 1 rive the corresponding Fokker-Planck equation in angle-action i=1 i variables. Explicit expressions are given for the diffusion tensor and friction force, and they are compared with previous expres- This hierarchy of equations is not closed since the equation for sions obtained in the literature. the one-body distribution P1(x1, t) involves the two-body dis- tribution P2(x1, x2, t), the equation for the two-body distribution 9 For a review on the dynamics and thermodynamics of systems with P2(x1, x2, t) involves the three-body distribution P3(x1, x2, x3, t), long-range interactions, see Campa et al. (2009). andsoon.

A93, page 4 of 27 P.-H. Chavanis: Kinetic theory of spatially inhomogeneous stellar systems without collective effects

   It is convenient to introduce a cluster representation of the 1 ∂P ∂P ∂P 2 (1, 2) + u · 2 (1, 2) + F(2 → 1) · 2 (1, 2) distribution functions. Specifically, we can express the reduced 2 ∂t 1 ∂r ∂u 1  1 distribution P j(x1, ..., x j, t) in terms of products of distribu- ∂P + − → · 2 tion functions P j< j(x1, ..., xj , t) of lower order plus a correla- (N 2) F(3 1)P1(3) dx3 (1, 2)  ∂u tion function P (x , ..., x , t)(seee.g.Eqs.(8)and(9)below). 1 j 1 j ∂P Considering the scaling of the terms in each equation of the + → − → · 1 F(2 1) F(3 1)P1(3) dx3 u (1)P1(2) BBGKY hierarchy, we can see that there exist solutions of the ∂ 1   ∂P1 whole BBGKY hierarchy such that the correlation functions P j + (N − 2) F(3 → 1)P (2, 3) dx · (1) − 2 3 u scale as 1/N j 1 in the proper thermodynamic limit N → +∞ ∂ 1 defined in Appendix A. This implicitly assumes that the initial − ∂ · →  F(3 1)P2(1, 3)P1(2) dx3 condition has no correlation, or that the initial correlations re- ∂u1 10 spect this scaling . If this scaling is satisfied, we can consider + − ∂ · →  + ↔ = (N 2) F(3 1)P3(1, 2, 3) dx3 (1 2) 0. (11) an expansion of the BBGKY hierarchy in terms of the small pa- ∂u1 rameter 1/N. This is similar to the expansion of the BBGKY hi- erarchy in plasma physics in terms of the small parameter 1/Λ, Equations (10)and(11) are exact for all N, but they are not where Λ  1 represents the number of charges in the Debye closed. As explained previously, we shall close these equations sphere (Balescu 2000). However, in plasma physics, the system at the order 1/N in the thermodynamic limit N → +∞.Inthis ∼  ∼  ∼ 2 is spatially homogeneous (because of Debye shielding which re- limit P1 1, P2 1/N,andP3 1/N . On the other hand, we stricts the range of interaction) while, for stellar systems, spa- can introduce dimensionless variables such that |r|∼|u|∼t ∼ tial inhomogeneity must be taken into account. This brings addi- m ∼ 1and|F(i → j)|∼G ∼ 1/N (see Appendix A). V = V + V tional terms in the BBGKY hierarchy that are absent in plasma The advection term 0 m.f. in the left-hand side physics. (l.h.s.) of Eq. (10) is of order 1, and the collision term C in the right-hand side (r.h.s.) is of order 1/N. Let us now consider the terms in Eq. (11) one by one. The first four terms corre- 2.2. The truncation of the BBGKY hierarchy at the order 1/N spond to the Liouville equation. The Liouville operator L =  L0+L +Lm.f. describes the complete two-body problem, includ- The first two equations of the BBGKY hierarchy are ing the inertial motion, the interaction between the stars (1, 2) and the mean field produced by the other stars. The terms L0 L L 2 ∂P1 + u · ∂P1 and m.f. are of order 1/N while the term is of order 1/N . (1) 1 (1) L 11 ∂t ∂r1 Therefore, the interaction term can a priori be neglected in the Liouville operator. This corresponds to the weak coupling ∂P L + L + − → · 2 = approximation where only the mean field term 0 m.f. is re- (N 1) F(2 1) (1, 2) dx2 0, (6) S ∂u1 tained. The fifth term in Eq. (11) is a source term express- ible in terms of the one-body distribution; it is of order 1/N.If 1 ∂P ∂P ∂P 2 2 2 we consider only the mean field Liouville operator L0 + Lm.f. (1, 2) + u1 · (1, 2) + F(2 → 1) · (1, 2) 2 ∂t ∂r1 ∂u1 and the source term S, as we shall do in this paper, we can ob- tain a kinetic equation for stellar systems that is the counter- ∂P3 part of the Landau equation in plasma physics. The sixth term C + (N − 2) F(3 → 1) · (1, 2, 3) dx3 +(1 ↔ 2)=0. (7) ∂u1 is of order 1/N and it corresponds to collective effects (i.e. the dressing of the particles by the polarization cloud). In plasma We decompose the two- and three-body distributions in the form physics, this term leads to the Lenard-Balescu equation. It takes into account dynamical screening and regularizes the divergence P (x , x ) = P (x )P (x ) + P (x , x ), (8) at large scales that appears in the Landau equation. In the case 2 1 2 1 1 1 2 2 1 2 of stellar systems, there is no large-scale divergence because of = +  the spatial inhomogeneity of the system. Therefore, collective P3(x1, x2, x3) P1(x1)P1(x2)P1(x3) P2(x1, x2)P1(x3) effects are less crucial in the kinetic theory of stellar systems +  +  +  P2(x1, x3)P1(x2) P2(x2, x3)P1(x1) P3(x1, x2, x3), (9) than in plasma physics. However, this term has been properly taken into account by Heyvaerts (2010) who obtained a kinetic where P (x , ..., x , t) is the correlation function of order j. equation of stellar systems that is the counterpart of the Lenard- j 1 j T Substituting Eqs. (8)and(9)inEqs.(6)and(7), and simplifying Balescu equation in plasma physics. The last two terms are of the order 1/N2 and they will be neglected. In particular, the some terms, we obtain  2 three-body correlation function P3,oforder1/N , can be ne- glected at the order 1/N. In this way, the hierarchy of equations is ∂P1 ∂P1 (1) + u1 · (1) closed and a kinetic equation involving only two-body encoun- ∂t ∂r1 ters can be obtained. ∂P1 + − → · = 11 (N 1) F(2 1)P1(2) dx2 (1) Actually, the interaction term becomes large at small scales |r2 − ∂u1 r1|→0, so its effect is not totally negligible (in other words, the − − ∂ · →  expansion in terms of 1/N is not uniformly convergent). In particu- (N 1) F(2 1)P2(1, 2) dx2, (10) lar, the interaction term L must be taken into account in order to ∂u1 describe strong collisions with small impact parameter that lead to large deflections very different from the mean field trajectory cor- 10 If there are non-trivial correlations in the initial state, like primordial responding to L0 + Lm.f.. This is important to regularize the diver- binary stars, the kinetic theory will be different from the one developed gence at small scales that appears in the Landau equation. We shall in the sequel. return to this problem in Sect. 2.4.

A93, page 5 of 27 A&A 556, A93 (2013)

If we introduce the notations f = NmP1 (distribution func- phase mixing and violent relaxation leading to a QSS on a = 2 2  tion) and g N m P2 (two-body correlation function), we get very short timescale, of the order of a few dynamical times tD. at the order 1/N Elliptical galaxies are in such QSSs. Lynden-Bell (1967)devel- − oped a statistical mechanics of the Vlasov equation in order to ∂ f + u · ∂ f + N 1 · ∂ f = describe this process of violent relaxation and predict the QSS (1) 1 (1) F (1) u (1) ∂t ∂r1 N ∂ 1 achieved by the system. Unfortunately, the predictions of his sta- 1 ∂ tistical theory are limited by the problem of incomplete relax- − · F(2 → 1)g(1, 2) dx2, (12) m ∂u1 ation. Kinetic theories of violent relaxation, which may account 1 ∂g ∂g ∂g for incomplete relaxation, have been developed by Kadomtsev & (1, 2) + u1 · (1, 2) + F (1) · (1, 2) Pogutse (1970), Severne & Luwel (1980), and Chavanis (1998, ∂t ∂r ∂u 2 1 1 2008a,b). ∂ f ∂g + F˜ (2 → 1) · (1) f (2) + F(2 → 1) · (1, 2) ∂u1 ∂u1 2.4. The order O(1/N): the generalized Landau equation 1 ∂ f + F(3 → 1)g(2, 3) dx3 · (1) + (1 ↔ 2) = 0. (13) (collisional regime) m ∂u1 We have introduced the mean force (by unit of mass) created on If we neglect strong collisions and collective effects, the first star 1 by all the other stars two equations of the BBGKY hierarchy (Eqs. (12)and(13)) re- duce to = → f (2) = −∇Φ F (1) F(2 1) dx2 (1), (14) ∂ f ∂ f N − 1 ∂ f m (0) + u · (0) + F (0) · (0) = ∂t ∂r N ∂u and the fluctuating force (by unit of mass) created by star 2 on star 1 1 ∂ − · F(1 → 0)g(0, 1) dx , (18) 1 m ∂u 1 F˜ (2 → 1) = F(2 → 1) − F (1). (15) N Equations (12)and(13) are exact at the order 1/N. They form 1 ∂g ∂g ∂g (0, 1) + u · (0, 1) + F (0) · (0, 1) + (0 ↔ 1) = the right basis to develop the kinetic theory of stellar systems at 2 ∂t ∂r ∂u this order of approximation. Since the collision term in the r.h.s. of Eq. (12)isoforder1/N, we expect that the relaxation time of ∂ f −F˜ (1 → 0) · (0) f (1) + (0 ↔ 1). (19) stellar systems scales as ∼ NtD where tD is the dynamical time. ∂u As we shall see, the discussion is more complicated because of the presence of logarithmic corrections in the relaxation time and The first equation gives the evolution of the one-body distribu- the absence of a strict statistical equilibrium state. tion function. The l.h.s. corresponds to the (Vlasov) advection term. The r.h.s. takes into account correlations (finite N effects, graininess, discreteness effects) between stars that develop as a 2.3. The limit N → +∞: the Vlasov equation (collisionless result of their interactions. These correlations correspond to en- regime) counters (collisions). Equation (19) may be viewed as a linear first order differen- In the limit N → +∞,foranyfixedintervaloftime[0, T], the tial equation in time. It can be symbolically written as correlations between stars can be neglected. Therefore, the mean ∂g field approximation becomes exact and the N-body distribution + Lg = S[ f ], (20) function factorizes in N one-body distribution functions ∂t

N where L = L0 +Lm.f. is a mean field Liouville operator and S[ f ] PN (x1, ..., xN , t) = P1(xi, t). (16) is a source term S expressible in terms of the one-body distribu- i=1 tion. This equation may be solved by the method of characteris- tics. Introducing the Green function Substituting this factorization in the Liouville equation, and in- tegrating over x2, x3, ..., xN , we find that the smooth distribution t  function f (r, u, t) = NmP1(r, u, t) is the solution of the Vlasov G(t, t ) = exp − L(τ)dτ , (21) equation t constructed with the mean field Liouville operator L, we obtain ∂ f + u · ∂ f + ·∂ f = F u 0, ∂t ∂r ∂ t = − − F = −∇Φ, ΔΦ = 4πG f du. (17) g(x, x1, t) dτ G(t, t τ) 0 This equation also results from Eq. (12)ifweneglectthecorre- ∂ ∂ − × F˜ (1 → 0) · + F˜ (0 → 1) · lation function g(1, 2) in the r.h.s. and replace N 1byN. ∂u ∂u1 The Vlasov equation describes the collisionless evolution of ∼ stellar systems for times shorter than the relaxation time NtD. × f (x, t − τ) f (x1, t − τ), (22) In practice N  1 so that the domain of validity of the Vlasov equation is huge (see the end of Sect. 2.8). As recalled in the where we have assumed that no correlation is present initially Introduction, the Vlasov-Poisson system develops a process of so that g(x, x1, t = 0) = 0; if correlations are present initially,

A93, page 6 of 27 P.-H. Chavanis: Kinetic theory of spatially inhomogeneous stellar systems without collective effects it can be shown that they are rapidly washed out12. Substituting adiffusion term and a friction term. The coefficients of diffusion Eq. (22)inEq.(18), we obtain and friction are given by generalized Kubo formulae, i.e. they involve the time integral of the auto-correlation function of the ∂ f ∂ f N − 1 ∂ f + u · + · = fluctuating force. The kinetic equation (Eq. (24)) bears some re- F u ∂t ∂r N ∂ semblance with the Fokker-Planck equation. However, it is more ∂ t ff dτ dr du Fμ(1 → 0)G(t, t − τ) complicated since it is an integro-di erential equation, not a dif- μ 1 1 ferential equation (see Sect. 3). ∂v 0 ∂ ∂ Equation (24) may be viewed as a generalized Landau equa- × F˜ν(1 → 0) + F˜ν(0 → 1) tion. Since the spatial inhomogeneity and the finite extension of ∂vν ∂vν 1 the system are properly taken into account, there is no divergence f × f (r, u, t − τ) (r , u , t − τ). (23) at large scales. There is, however, a logarithmic divergence at m 1 1 small scales which is due to the neglect of the interaction term L In writing this equation, we have adopted a Lagrangian point of in the Liouville operator (see Sect. 2.6). At large scales (i.e. for large impact parameters), this term can be neglected and the tra- view. The coordinates ri appearing after the Green function must τ r t − τ = r t − s u t − s s u t − τ = jectories of the stars are essentially due to the mean field. Thus, be viewed as i( ) i( ) 0 d i( )d and i( ) τ we can make the weak coupling approximation leading to the u t − s F r t − s , t − s s i( ) 0 d ( i( ) )d . Therefore, in order to evaluate Landau equation. This approximation describes weak collisions. the integral of Eq. (23), we must move the stars following the However, at small scales (i.e. for small impact parameters), we trajectories determined by the self-consistent mean field. can no longer ignore the interaction term L in the Liouville op- The kinetic equation (Eq. (23)) is valid at the order 1/N so it erator and we have to solve the classical two-body problem. This describes the collisional evolution of the system (ignoring col- is necessary to correctly describe strong collisions for which the ff lective e ects) on a timescale of order NtD. Equation (23)is trajectory of the particles deviates strongly from the mean field a non-Markovian integro-differential equation. It takes into ac- motion. When the mean field Green function G (constructed  count delocalizations in space and time (i.e. spatial inhomogene- with L0 + Lm.f.) is replaced by the total Green function G (con- ity and memory effects). Actually, the Markovian approximation structed with L +L +L ), taking into account the interaction → +∞ ∼ 0 m.f. is justified in the N limit because the timescale NtD term, the generalized Landau equation (Eq. (24)) is replaced by over which f (r, u, t) changes is long compared to the correla- a more complicated equation which can be viewed as a gener- ∼ tion time τcorr tD over which the integrand in Eq. (23)has alized Boltzmann equation. This equation is free of divergence 13 significant support . Therefore, we can compute the correlation (since it takes both spatial inhomogeneity and strong collisions function (Eq. (22)) by assuming that the distribution function is into account), but it is unnecessarily complicated because it does frozen at time t. This corresponds to the Bogoliubov ansatz in not exploit the dominance of weak collisions over rare strong u − u − plasma physics. If we replace f (r, , t τ)and f (r1, 1, t τ) collisions for the gravitational potential. Indeed, a star suffers a u u by f (r, , t)and f (r1, 1, t)inEq.(23) and extend the time inte- large number of weak distant encounters and very few close en- gral to +∞, we obtain counters. A better practical procedure is to use the generalized +∞ ff ∂ f ∂ f N − 1 ∂ f ∂ Landau equation (Eq. (24)) with a cut-o at small scales in order +u · + F · = dτ dr du Fμ(1→0) u μ 1 1 to take into account our inability to describe strong collisions by ∂t ∂r N ∂ ∂v 0 this approach (see Sect. 2.6). × − ˜ν → ∂ + ˜ν → ∂ The generalized Landau equation (Eq. (24)) was derived G(t, t τ) F (1 0) ν F (0 1) ν ∂v ∂v1 by Kandrup (1981) from the Liouville equation by using the f projection operator formalism. It can also be derived from the × f (r, u, t) (r1, u1, t). (24) Liouville equation by using the BBGKY hierarchy or from m the Klimontovich equation by making a quasilinear approxima- Similarly, we can compute the trajectories of the stars by as- tion (Chavanis 2008a,b). suming that the mean field is independent on τ and equal to τ t r t − τ = r t − s u t − s s 2.5. The Vlasov-Landau equation its value at time so that i( ) i( ) 0 d i( )d τ u t − τ = u t − s F r t − s , t s and i( ) i( ) 0 d ( i( ) )d . Self-gravitating systems are intrinsically spatially inhomoge- The structure of the kinetic equation (Eq. (24)) has a clear neous. However, the collision operator at position r can be physical meaning. The l.h.s. corresponds to the Vlasov advection simplified by making a local approximation and performing term arising from mean field effects. The r.h.s. can be viewed as the integrals as if the system were spatially homogeneous with a collision operator CN [ f ] taking finite N effects into account. the density ρ = ρ(r). This amounts to replacing f (r , u , t) → +∞ 1 1 For N , it vanishes and we recover the Vlasov equation. by f (r, u1, t)andF˜ (i → j)byF(i → j)inEq.(24). This lo- For finite N, it describes the cumulative effect of binary colli- cal approximation is motivated by the work of Chandrasekhar sions between stars. The collision operator is a sum of two terms, & von Neumann (1942) who showed that the distribution of the

12 gravitational force is a Lévy law (called the Holtzmark distri- The term corresponding to g(x, x1, t = 0) in the kinetic equation has been called the “destruction term” by Prigogine and Résibois because it bution) dominated by the contribution of the nearest neighbor. describes the destruction of the effect of the initial correlations (Balescu With this local approximation, Eq. (24) becomes 2000). ∂ f ∂ f N − 1 ∂ f 13 It is sometimes argued that the Markovian approximation is not jus- + u · + F · = tified for stellar systems because the force auto-correlation decreases ∂t ∂r N ∂u +∞ slowly as 1/t (Chandrasekhar 1944). However, this result is only ∂ dτ dr du Fμ(1 → 0)G (t, t − τ) true for an infinite homogeneous system (see Appendix G). For spa- ∂vμ 1 1 0 tially inhomogeneous distributions, the correlation function decreases 0 ∂ ∂ f more rapidly and the Markovian approximation is justified (Severne & × Fν(1 → 0) − f (r, u, t) (r, u , t), (25) Haggerty 1976). ν ν 1 ∂v ∂v1 m A93, page 7 of 27 A&A 556, A93 (2013) wherewehaveusedF(0 → 1) = −F(1 → 0). The Green func- homogeneous and the advection term is absent in Eq. (28). In tion G0 corresponds to the free motion of the particles associated the case of stellar systems, when we make the local approxima- with the Liouville operator L0. Using Eqs. (C.1)and(C.2), the tion, the spatial inhomogeneity of the system is only retained in foregoing equation can be rewritten as the advection term of Eq. (28). This is why this kinetic equation ∂ f ∂ f N − 1 ∂ f is referred to as the Vlasov-Landau equation. This is the funda- + u · + F · = mental kinetic equation of stellar systems. ∂t ∂r N ∂u +∞ ∂ u μ → ν → − μ dτ dr1d 1F (1 0, t)F (1 0, t τ) 2.6. Heuristic regularization of the divergences ∂v 0 ∂ ∂ f To obtain the Vlasov-Landau equation (Eq. (28)), we have made × − f (r, u, t) (r, u , t), (26) ∂vν ∂vν m 1 a local approximation. This amounts to calculating the collision 1 operator at each point as if the system were spatially homoge- where F(1 → 0, t − τ) is expressed in terms of the Lagrangian neous. As a result of this homogeneity assumption, a logarithmic coordinates. The integrals over τ and r1 can be calculated explic- divergence appears at large scales in the Coulombian logarithm itly (see Appendices C and D). We then find that the evolution (Eq. (30)). In plasma physics, this divergence is cured by the of the distribution function is governed by the Vlasov-Landau Debye shielding. A charge is surrounded by a polarization cloud equation of opposite charges which reduces the range of the interaction. ∂ f ∂ f N − 1 ∂ f When collective effects are properly taken into account, as in the + u · + F · = π(2π)3m ∂t ∂r N ∂u Lenard-Balescu equation, no divergence appears at large scales and the Debye length arises naturally. Heuristically, we can use ∂ μ ν 2 ∂ f ∂ f1 ff × k k δ(k · w)ˆu (k) f1 − f du1dk, (27) the Landau equation and introduce an upper cut-o at the Debye μ ν ν 2 1/2 ∂v ∂v ∂v1 length λD ∼ (kBT/ne ) . For self-gravitating systems, there is no shielding and the divergence is cured by the finite extent of where we have noted w = u − u1, f = f (r, u, t), f1 = f (r, u1, t), and where (2π)3uˆ(k) = −4πG/k2 represents the Fourier trans- the system. The interaction between two stars is only limited form of the gravitational potential. Under this form, we see that by the size of the system. When spatial inhomogeneity is taken the collisional evolution of a stellar system is due to a condi- into account, as in the generalized Landau equation (Eq. (24)), tion of resonance k · u = k · u (with u  u) encapsulated in the no divergence occurs at large scales. Heuristically, we can use δ-function. This δ-function expresses the conservation of energy. the Vlasov-Landau equation (Eq. (28)) and introduce an upper ff ∼ 2 1/2 The Vlasov-Landau equation may also be written as (see cut-o at the Jeans length λJ (kBT/Gm n) which gives an Appendix C) estimate of the system’s size R. The Coulombian logarithm (Eq. (30)) also diverges at small − ∂ f + u · ∂ f + N 1 ·∂ f = scales. As explained previously, this is due to the neglect of F u ∂t ∂r N ∂ strong collisions that produce important deflections. Indeed, for ∂ ∂ f ∂ f1 collisions with small impact parameter, the mean field approx- Kμν f − f du , (28) μ 1 ν ν 1 imation is clearly irrelevant and it is necessary to solve the ∂v ∂v ∂v1 two-body problem exactly (see Appendix H). Accordingly, the w2δμν − wμwν Kμν = A , A = 2πmG2 ln Λ, (29) small-scale divergence is cured by the proper treatment of strong w3 collisions. Heuristically, we can use the Landau equation and in- where troduce a lower cut-off at the gravitational Landau length λL ∼ 2 ∼ 2 kmax Gm/vm Gm /(kBT) (the gravitational analogue of the Landau 2 2 2 ln Λ= dk/k (30) length λL ∼ e /mv ∼ e /kBT in plasma physics) which corre- m ◦ kmin sponds to the impact parameter leading to a deflection at 90 . is the Coulombian logarithm that has to be regularized with ap- Introducing a large-scale cut-off at the Jeans length λJ and a propriate cut-offs (see Sect. 2.6). The r.h.s. of Eq. (28)isthe small-scale cut-off at the Landau length λL, and noting that λL ∼ 2 Λ ∼ original form of the collision operator given by Landau (1936) 1/(nλJ ), we find that the Coulombian logarithm scales as ln 14 ∼ 3 ∼ Λ ∼ 3 ∼ 3 ∼ for the Coulombian interaction . It applies to weakly coupled ln (λJ/λL) ln (nλJ ) ln N where nλJ nR N is the plasmas. We note that the potential of interaction only appears number of stars in the cluster16. in the constant A which merely determines the relaxation time. The structure of the Landau equation is independent of the po- tential. The Landau equation was originally derived from the 2.7. Properties of the Vlasov-Landau equation |Δu| Boltzmann equation in the limit of weak deflections 1 The Vlasov-Landau equation conserves the mass M = f drdu (Landau 1936)15. In the case of plasmas, the system is spatially = v2 u+ 1 Φ and the energy E f 2 drd 2 ρ dr. It also monotonically 14 2 ↔ The case of plasmas is recovered by making the substitution Gm increases the Boltzmann entropy S = − ( f /m)ln(f /m)drdu −e2 leading to A = (2πe4/m3)lnΛ. 15 in the sense that S˙ ≥ 0(H-theorem). As a result of the lo- We note that Eq. (25) with G0 replaced by the total Green func- 17 tion G taking into account the interaction term is equivalent to cal approximation, the proof of these properties is the same the Boltzmann equation. Indeed, for spatially homogeneous systems, the Boltzmann equation can be derived from the BBGKY hierarchy 16 Similarly, in plasma physics, introducing a large-scale cut-off at the  ff (Eqs. (10)and(11)) by keeping the Liouville operator L = L0 + L de- Debye length λD and a small-scale cut-o at the Landau length λL,and ∼ 2 scribing the two-body problem exactly and the source S (Balescu 2000). noting that λL 1/(nλD), we find that the Coulombian logarithm scales Λ ∼ ∼ 3 Λ ∼ 3 Therefore, the procedure used by Landau which amounts to expanding as ln ln (λD/λL) ln (nλD), where nλD is the number of the Boltzmann equation in the limit of weak deflections is equivalent electrons in the Debye sphere. to the one presented here that starts from the BBGKY hierarchy and 17 We recall that the Vlasov advection term conserves mass, energy, and neglects L in the Liouville operator. entropy.

A93, page 8 of 27 P.-H. Chavanis: Kinetic theory of spatially inhomogeneous stellar systems without collective effects as for the spatially homogeneous Landau equation (Balescu & Wood (1968), stellar systems may evolve more rapidly be- 2000). Because of these properties, we might expect that a stel- cause of gravothermal catastrophe. In that case, the Michie-King lar system will relax towards the Boltzmann distribution which distribution changes significatively as a result of core collapse. maximizes the entropy at fixed mass and energy. However, we This evolution has been described by Cohn (1980), and it leads know that there is no maximum entropy state for an unbounded ultimately to the formation of a binary star surrounded by a hot self-gravitating system (the Boltzmann distribution has infinite halo. A configuration of this type can have arbitrarily large en- mass). Therefore, the Vlasov-Landau equation does not relax to- tropy. Cohn (1980) found that the entropy increases permanently wards a steady state and the entropy does not reach a stationary during core collapse, confirming that the Vlasov-Landau equa- value. Actually, the entropy increases permanently as the system tion has no equilibrium state. evaporates. But since evaporation is a slow process, the system Even if the system were confined within a small box so as to may achieve a QSS that is close to the Boltzmann distribution. prevent both the evaporation and the gravothermal catastrophe, A typical quasi-stationary distribution is the Michie-King model there would be no statistical equilibrium state in a strict sense because there is no global entropy maximum (Antonov 1962). − 2 2 − − f = Ae βmj /(2ra) e βm − e βmm , (31) A configuration in which some subset of the particles are tightly bound together (e.g. a binary star), and in which the remaining where  = v2/2 +Φ(r) is the energy and j = r × u the angu- particles share the energy thereby released, may have an arbi- lar momentum. This distribution takes into account the escape trarily large entropy. However, such configurations, which re- of high energy stars and the anisotropy of the velocity distri- quire strong correlations, are generally reached very slowly (on bution. It can be derived, under some approximations, from the a timescale much larger than (N/ln N)tD) as a result of encoun- Vlasov-Landau equation by using the condition that f = 0if ters involving many particles. To describe these configurations, the energy of the star is larger than the escape energy m (Michie one would have to take high order correlations into account in 1963;King1965). The Michie-King distribution reduces to the the kinetic theory. These configurations may be relevant in sys- isothermal distribution f ∝ e−βm for low energies. In this sense, tems with a small number of stars (Chabanol et al. 2000), but we can define a relaxation time for a stellar system. From the when N is large the picture is different. On a timescale of the or- Vlasov-Landau equation (Eq. (28)), we find that the relaxation der of (N/ln N)tD the one-body distribution function is expected time scales as to reach the Boltzmann distribution which is a local entropy maximum. This state is metastable, but its lifetime is expected v3 to be very large, scaling as eN, so that it is stable in practice t ∼ m , (32) R nm2G2 ln N (Chavanis 2005, 2006). In this sense, there exist true statistical equilibrium states for self-gravitating systems confined within a where vm is the root mean square (rms) velocity of the stars. small box. However, we may argue that this situation is highly ∼ ∼ Introducing the dynamical time tD λJ/vm R/vm, we obtain artificial. the scaling Finally, using very different methods based on ergodic the- N ory, Gurzadyan & Savvidy (1986) argue that, because of collec- tR ∼ tD. (33) tive behaviour, the relaxation time for stellar systems scales like ln N ∗ 1/3 t ∼ N tD intermediate between the dynamical time tD (violent R ∼ The fact that the ratio between the relaxation time and the dy- relaxation) and the binary relaxation time tR (N/ln N)tD. namical time depends only on the number of stars and scales as N/ln N was first noted by Chandrasekhar (1942). 2.8. Dynamical evolution of stellar systems: a short review A simple estimate of the evaporation time was calculated ff by Ambartsumian (1938) and Spitzer (1940) and gives tevap  Using the kinetic theory, we can identify di erent phases in the 136tR. More precise values have been obtained by study- dynamical evolution of stellar systems. ing the evaporation process in an artificially uniform medium A self-gravitating system initially out-of-mechanical equi- (Chandrasekhar 1943b; Spitzer & Härm 1958;Michie1963; librium undergoes a process of violent collisionless relaxation King 1965; Lemou & Chavanis 2010) or in a more realistic inho- towards a virialized state. In this regime, the dynamical evolu- mogeneous cluster (Hénon 1961). Since tevap  tR, we can con- tion of the cluster is described by the Vlasov-Poisson system. sider that the system relaxes towards a steady distribution given The phenomenology of violent relaxation has been described by by Eq. (31) on a timescale tR and that this distribution slowly Hénon (1964), King (1966), and Lynden-Bell (1967). Numerical evolves on a longer timescale as the stars escape18. The charac- simulations that start from cold and clumpy initial conditions 1/4 teristic time in which the system’s stars evaporate is tevap.The generate a QSS that fits the de Vaucouleurs R law for the evaporation is one reason for the evolution of stellar systems. surface brightness of elliptical galaxies quite well (van Albada However, as demonstrated by Antonov (1962) and Lynden-Bell 1982). The inner core is almost isothermal (as predicted by Lynden-Bell 1967) while the velocity distribution in the en- 18 This distribution is not steady in the sense that the coefficients A, velope is radially anisotropic and the density profile decreases −4 β,andra slowly vary in time as the stars escape and the system loses as r . One success of Lynden-Bell’s statistical theory of violent mass and energy. However, the distribution keeps the same form during relaxation is to explain the isothermal core of elliptical galax- the evaporation process (King 1965). Hénon (1961) previously found ies without recourse to collisions. In contrast, the structure of a self-similar solution of the orbit-averaged Fokker-Planck equation. the halo cannot be explained by Lynden-Bell’s theory as it re- He showed that the core contracts as the cluster evaporates and that sults from an incomplete relaxation. Models of incompletely re- the central density is infinite (the structure of the core resembles the singular isothermal sphere with a density ρ ∝ r−2). He argued that the laxed stellar systems have been elaborated by Bertin & Stiavelli concentration of energy, without concentration of mass, at the center (1984), Stiavelli & Bertin (1987), and Hjorth & Madsen (1991). of the system is attributable to the formation of tight binary stars. The These theoretical models nicely reproduce the results of obser- invariant profile found by Hénon does not exactly coincide with the vations and numerical simulations (Londrillo et al. 1991;Trenti Michie-King distribution, but is reasonably close. et al. 2005). In the simulations, the initial condition needs to be

A93, page 9 of 27 A&A 556, A93 (2013) sufficiently clumpy and cold to generate enough mixing required can release sufficient energy to stop the collapse and even drive for a successful application of the statistical theory of violent a re-expansion of the cluster in a post-collapse regime (Inagaki relaxation. Numerical simulations starting from homogeneous & Lynden-Bell 1983; Goodman 1984). Then, in principle, a se- spheres (see e.g. Roy & Perez 2004; Levin et al. 2008;Joyce ries of gravothermal oscillations should follow (Bettwieser & et al. 2009) show little angular momentum mixing and lead to Sugimoto 1984). different results. In particular, they display a larger amount of At the present epoch, small groups of stars such as globular 5 5 10 10 mass loss (evaporation) than simulations starting from clumpy clusters (N ∼ 10 , tD ∼ 10 yr, age ∼ 10 yr, tR ∼ 10 yr) are initial conditions. Clumps thus help the system to reach a uni- in the collisional regime. They are either in QSSs described by versal final state from a variety of initial conditions, which can the Michie-King model or experiencing core collapse. By con- explain the similarity of the density profiles observed in elliptical trast, large clusters of stars like elliptical galaxies (N ∼ 1011, 8 10 19 galaxies. tD ∼ 10 yr, age ∼ 10 yr, tR ∼ 10 yr) are still in the colli- On longer timescales, encounters between stars must be sionless regime and their apparent organization is a result of an taken into account and the dynamical evolution of the cluster is incomplete violent relaxation. governed by the Vlasov-Landau-Poisson system. The first stage of the collisional evolution is driven by evaporation. Because 3. Test star in a thermal bath of a series of weak encounters, the energy of a star can grad- ually increase until it reaches the local escape energy; in that 3.1. The Fokker-Planck equation case the star leaves the system. Numerical simulations (Spitzer We now consider the relaxation of a test star (i.e. a tagged parti- 1987; Binney & Tremaine 2008) show that during this regime cle) evolving in a steady distribution of field stars21. Because of the system reaches a QSS that slowly evolves in amplitude as the encounters with the field stars, the test star has a stochastic a result of evaporation as the system loses mass and energy. motion. We call P(r, u, t) the probability density of finding the This quasi-stationary distribution function is close to the Michie- test star at position r with velocity u at time t. The evolution of King model (Eq. (31)). The system has a core-halo structure. The P(r, u, t) can be obtained from the generalized Landau equation core is isothermal while the stars in the outer halo move in pre- (Eq. (24)) by considering that the distribution function of the dominantly radial orbits. Therefore, the distribution function in field stars is fixed. Therefore, we replace f (r, u, t)byP(r, u, t) the halo is anisotropic. The density follows the isothermal law and f (r , u , t)by f (r , u )wheref (r , u ) is the steady distri- ρ ∼ r−2 in the central region (with a core of almost uniform 1 1 1 1 1 1 − bution of the field stars. This procedure transforms the integro- density) and decreases as ρ ∼ r 7/2 in the halo (for an isolated ff ff 3/2 di erential equation (Eq. (24)) into the di erential equation cluster with m = 0) or tends to zero as ρ ∝ (rt − r) (for a cluster with a finite tidal radius). Because of evaporation, the − ∂P + u · ∂P + N 1 · ∂P = halo expands while the core shrinks as required by energy con- F u ∂t ∂r N ∂ servation. During the evaporation process, the central density in- ∂ +∞ creases permanently. At some point in the evolution, the system u μ → μ dτ dr1d 1F (1 0) undergoes an instability related to the Antonov (1962) instabil- ∂v 0 ity19 and the gravothermal catastrophe sets in (Lynden-Bell & ∂ ∂ × G(t, t − τ) F˜ν(1 → 0) + F˜ν(0 → 1) Wood 1968). This instability is due to the negative specific heat ∂vν ∂vν of the inner system that evolves by losing energy and thereby 1 f growing hotter. The energy lost is transferred outward by stellar × P(r, u, t) (r1, u1), (34) encounters. Hence the temperature always decreases outward, m and the center continually loses energy, shrinks, and heats up. where F (r) = −∇Φ(r) is the static mean force created by the This leads to core collapse. Mathematically speaking, core col- field stars with density ρ(r1). This equation does not present lapse would generate a finite time singularity. When the evolu- any divergence at large scales. We can understand the above tion is modeled by the orbit-averaged Fokker-Planck equation, procedure as follows (see Chavanis 2012b, for more details). Cohn (1980) finds that the collapse is self-similar, that the cen- Equations (24)and(34) govern the evolution of the distribution tral density becomes infinite in a finite time, and that the density function of a test star (described by the coordinates r and u)in- ∼ −2.23 behaves as ρ r . The invariant profile found by Cohn dif- teracting with field stars (described by the running coordinates fers from the Michie-King distribution (for which ρ ∼ r−2)be- r1 and u1). In Eq. (24), all the stars are equivalent so the distri- yond a radius of about 10r .Larson(1970) and Lynden-Bell core bution of the field stars f (r1, u1, t) changes with time exactly like & Eggleton (1980) find similar results by modeling the evolu- the distribution of the test star f (r, u, t). In Eq. (34), the test star tion of the system by fluid equations. In all cases, the authors and the field stars are not equivalent since the field stars form find a singular density profile at the collapse time that is inte- a bath. The field stars have a steady given distribution f (r , u ) = 1 1 grable at r 0. This means that “the singularity contains no while the distribution of the test star P(r, u, t) changes with time. mass”20. In reality, if we come back to the N-body system, there is no singularity and core collapse is arrested by the formation 21 In plasma physics, this steady distribution is the Boltzmann distribu- of binary stars caused by three-body collisions. These binaries tion of statistical equilibrium which is the steady state of the Landau equation. In the case of stellar systems, there is an intrinsic diffi- 19 The collisional evolution of the system can be measured precisely culty since no statistical equilibrium state exists in a strict sense: the = Φ 2 in terms of the scale escape energy x0 3 (0)/vm(0). The onset of Boltzmann distribution has infinite mass, and the Vlasov-Landau equa- instability corresponds to x0  9.3 which is the value at which the King tion has no steady state because of the escape of high-energy stars. model becomes thermodynamically unstable (Cohn 1980). However, we have seen that the system can reach a quasi-steady dis- 20 Actually, the number of stars in the core decreases as the collapse tribution (e.g. a Michie-King distribution) and that this distribution proceeds. When <102 stars remain in the core, strong collisions become changes on an evaporation timescale that is long with respect to the important. Goodman (1983) showed that the evolution caused by strong collisional relaxation time. Therefore, we can consider that this distribu- collisions is very similar to that caused by numerous weak ones, except tion is steady on the collisional timescale over which the Fokker-Planck for the rate of evaporation which is greatly increased. approach applies.

A93, page 10 of 27 P.-H. Chavanis: Kinetic theory of spatially inhomogeneous stellar systems without collective effects

If we make a local approximation and use the Vlasov-Landau Chavanis (2008b). The friction force has also been calculated equation (Eq. (26)), we obtain by Tremaine & Weinberg (1984), Bekenstein & Maoz (1992), ff ∂P ∂P N − 1 ∂P Maoz (1993), and Nelson & Tremaine (1999) using di erent ap- + u · + F · = proaches. ∂t ∂r N ∂u +∞ Since the diffusion tensor depends on the velocity u of the ∂ u μ → ν → − test star, it is useful to rewrite Eq. (36) in a form that is fully μ dτ dr1d 1F (1 0, t)F (1 0, t τ) ∂v 0 consistent with the general Fokker-Planck equation ∂ ∂ f 2 × − P(r, u, t) (r, u ). (35) dP ∂ μν ∂ μ ∂vν ∂vν m 1 = (D P) − PF , (44) 1 dt ∂vμ∂vν ∂vμ friction Denoting the advection operator by d/dt,Eq.(35) can be written with in the form of a Fokker-Planck equation ΔvμΔvν Δvμ Dμν = , Fμ = · (45) dP ∂ ∂P 2Δt friction Δt = Dμν − PFμ (36) dt ∂vμ ∂vν pol By identification, we find that μν involving a diffusion tensor μ μ ∂D F = F + · (46) +∞ friction pol ∂vν μν 1 μ D = dτ dr1du1F (1 → 0, t) ff ffi m Therefore, when the di usion coe cient depends on the veloc- 0 ity, the total friction is different from the friction by polarization. × Fν(1 → 0, t − τ) f (r, u ) (37) 1 Substituting Eqs. (40)and(41)inEq.(46), and using an integra- and a friction force tion by parts, we find that the diffusion and friction coefficients +∞ may be written as μ 1 μ = u → μ ν Fpol dτ dr1d 1F (1 0, t) Δv Δv m 0 = π(2π)3m kμkνδ(k · w)ˆu(k)2 f du dk, (47) 2Δt 1 1 × ν → − ∂ f u F (1 0, t τ) (r, 1). (38) Δ μ ν v μ ν ∂ ∂ ∂v1 = π(2π)3m k k f − Δt 1 ∂vν ∂vν If we start directly from Eq. (27), which amounts to performing 1 × δ(k · w)ˆu(k)2 du dk. (48) the integrals over τ and r1 in the previous expressions, we obtain 1 the Fokker-Planck equation These expressions can be obtained directly from the equations − of motion by expanding the trajectories of the stars in pow- ∂P + u · ∂P + N 1 · ∂P = 3 F u π(2π) m ers of 1/N in the limit N → +∞ (Chavanis 2008b). We re- ∂t ∂r N ∂ ∂ ∂P ∂ f callthatEqs.(47)and(48) display a logarithmic divergence × kμkνδ(k · w)ˆu(k)2 f − P 1 du dk, (39) at small and large scales that must be regularized by introduc- ∂vμ 1 ∂vν ∂vν 1 1 ing proper cut-offs as explained in Sect. 2.6. In astrophysics, ff ffi with the diffusion and friction coefficients the di usion and friction coe cients of a star were first cal- culated by Chandrasekhar (1943a) from a two-body encounters μν 3 μ ν 2 D = π(2π) m k k δ(k · w)ˆu(k) f1 du1dk, (40) theory (see also Cohen et al. 1950; Gasiorowicz et al. 1956;and Rosenbluth et al. 1957). The expressions obtained by these au- ∂ f1 thors are different from those given above, but they are equiv- Fμ = π(2π)3m kμkνδ(k · w)ˆu(k)2 du dk. (41) pol ν 1 alent (see Sect. 3.5). In plasma physics, the diffusion and fric- ∂v1 tion coefficients of a charge were first calculated by Hubbard Using Eq. (40), we can easily establish the identity (1961a) who took collective effects into account, thereby elimi- ∂Dμν ∂D nating the divergence at large scales. When collective effects are = , (42) neglected, his expressions reduce to Eqs. (47)and(48). On the ∂vν ∂vμ other hand, strong collisions have been taken into account by where D = Dμμ = tr(D). Chandrasekhar (1943a) in astrophysics and by Hubbard (1961b) The diffusion tensor Dμν results from the fluctuations of the in plasma physics. In that case, there is no divergence at small gravitational force caused by the granularities in the distribution impact parameters in the diffusion and friction coefficients, and of the field stars. It can be derived directly from the formula (see the Landau length appears naturally (see Appendix H). Appendix G) The two forms of the Fokker-Planck equation (Eqs. (36) +∞ and (44)) have their own interest. The expression (Eq. (44)) Dμν = Fμ(t)Fν(t − τ) dτ, (43) where the diffusion coefficient is placed after the two deriva- 2 0 tives ∂ (DP) involves the total friction force Ffriction and the expression (Eq. (36)) where the diffusion coefficient is placed obtained from Eq. (45-a). The friction force Fpol results from the retroaction of the field stars to the perturbation caused by between the derivatives ∂D∂P involves the friction by polar- the test star, as in a polarization process. It can be derived ization Fpol. Astrophysicists are used to the form in Eq. (44). from a linear response theory (Marochnik 1968;Kalnajs1971b; However, it is the form in Eq. (36) that stems from the Landau Kandrup 1983; Chavanis 2008b). It will be called the friction equation (Eq. (26)). We shall come back to this observation in by polarization to distinguish it from the total friction (see be- Sect. 3.5. low). Equations (35)–(41) have been derived within the local From Eqs. (40)and(41), we easily obtain approximation. More general formulae, valid for fully inho- ∂Dμν = Fμ . (49) mogeneous stellar systems, are given in Kandrup (1983)and ∂vν pol A93, page 11 of 27 A&A 556, A93 (2013)

Combining Eq. (46) with Eq. (49), we get isotropic distribution function (e.g. the Maxwellian), it can be put in the form Ffriction = 2Fpol. (50) μ ν μν 1 v v 1 μν D = D − D⊥ + D⊥δ , (56) Therefore, the friction force Ffriction is equal to twice the fric- 2 v2 2 tion by polarization Fpol (for a test star with mass m interact- ff ffi ing with field stars with mass mf, this factor two is replaced by where D and D⊥ are the di usion coe cients in the directions (m + mf)/m; see Appendix F). This explains the difference of parallel and perpendicular to the velocity of the test star. We note μμ μν ν μ factor 2 in the calculations of Chandrasekhar (1943a) who de- that D ≡ D = D + D⊥.UsingD v = Dv , the friction by termined Ffriction and in the calculations of Kalnajs (1971b)and polarization (Eq. (53)) can be written as Kandrup (1983) who determined Fpol. Fpol = −Dβmu. (57) The friction is proportional and opposite to the velocity of the 3.2. The Einstein relation test star, and the friction coefficient is given by the Einstein rela- In the central region of the system, the distribution of the field tion ξ = Dβm. The total friction is 22 stars is close to the Maxwell-Boltzmann distribution = − u Ffriction 2Dβm . (58) 3/2 v2 βm −βm 1 f (r1, u1) = ρ(r1)e 2 , (51) This is the Chandrasekhar dynamical friction. 2π The steady state of the Fokker-Planck equation (Eq. (55)) where β = 1/(kB T) is the inverse temperature and ρ(r1) ∝ is the Maxwell-Boltzmann distribution (Eq. (51)). Since the − Φ e βm (r1) is the density given by the Boltzmann law. Therefore, Fokker-Planck equation admits an H-theorem for the Boltzmann the field stars form a thermal bath. Substituting the identity free energy (Risken 1989), one can prove that P(r, u, t) con- verges towards the Maxwell-Boltzmann distribution (Eq. (51)) ∂ f1 for t → +∞. In other words, the test star acquires the distribution = −βmf1u1, (52) ∂u1 of the field stars (thermalization). We recall, however, that for self-gravitating systems, these results cannot hold everywhere in · u · u in Eq. (41), using the δ-function to replace k 1 by k ,and the cluster since the Maxwell-Boltzmann distribution does not comparing the resulting expression with Eq. (40), we find that exist globally. μ If we assume that the distribution P(r, u, t)oftheteststaris F = −βmDμν(u)vν. (53) pol isotropic, the Fokker-Planck equation becomes The friction coefficient is given by an Einstein relation express- dP ∂ ∂P 23 = D(v) + βmPu , (59) ing the fluctuation-dissipation theorem . We emphasize that dt ∂u ∂u the Einstein relation is valid for the friction force by polariza- or tion Fpol, not for the total friction Ffriction (we do not have this subtlety in standard Brownian theory where the diffusion coeffi- dP 1 ∂ 2 ∂P = v D(v) + βmPv . (60) cient is constant). Using Eq. (43), we can rewrite Eq. (53)inthe dt v2 ∂v ∂v form +∞ It can be obtained from an effective Langevin equation μ = − ν μ ν − Fpol βmv dτ F (t)F (t τ) . (54) du 1 ∂D 0 = −∇Φ − D(v)βmu + + 2D(v)R(t), (61) dt 2 ∂u This relation is usually called the Kubo formula. More gen- where R(t) is a Gaussian white noise satisfying R(t) = 0 eral expressions of the Kubo formula valid for fully inhomoge- μ ν  = μν −  ff ffi neous stellar systems are given in Kandrup (1983)andChavanis and R (t)R (t ) δ δ(t t ). Since the di usion coe cient (2008b). Using the Einstein relation, the Fokker-Planck equation depends on the velocity, the noise is multiplicative. The force acting on the star consists of two parts, a part derivable from (Eq. (36)) takes the form −∇Φ a smoothed distribution of matter , and a residual random dP ∂ ∂P part attributable to the fluctuations of that distribution. In turn, = Dμν(u) + βmPvν , (55) dt ∂vμ ∂vν the random part can be described by a Gaussian white noise mul- tiplied by a velocity-dependent factor plus a friction force equal where the diffusion coefficient is given by Eq. (40) with Eq. (51). to the friction by polarization Fpol = −D(v)βmu and a spurious This equation is similar to the Kramers equation in Brownian drift Fspurious = (1/2)(∂D/∂u) attributable to the multiplicative theory (Kramers 1940) except that the diffusion coefficient is a noise (we have used the Stratonovich representation). If we ne- tensor and that it depends on the velocity of the test star. For an glect the velocity dependence of the diffusion coefficient, we re- cover the results of Chandrasekhar (1943a, 1949) obtained in his 22 This isothermal distribution cannot hold in the whole cluster since it Brownian theory. has an infinite mass. 23 ff The collisional evolution of a star, under the e ect of two-body en- 3.3. The diffusion tensor for isothermal systems counters, can be understood as an interplay between two competing ef- fects. The fluctuations of the gravitational field induce a diffusion in When the velocity distribution of the field stars is given by velocity space which tends to increase the speed of the star. This effect the Maxwellian distribution (Eq. (51)), the diffusion tensor is counterbalanced by a friction (dissipation) which results in a sys- (Eq. (40)) may be calculated as follows. If we introduce the rep- tematic deceleration along the direction of motion. As emphasized by resentation Chandrasekhar (1943a, 1949) in his Brownian theory of stellar motion, +∞ the Einstein relation (Eq. (53)) guarantees that the Maxwell distribution dt δ(x) = eitx , (62) (Eq. (51)) is a steady state of the Fokker-Planck equation (Eq. (36)). −∞ 2π A93, page 12 of 27 P.-H. Chavanis: Kinetic theory of spatially inhomogeneous stellar systems without collective effects for the δ-function in Eq. (40), the diffusion tensor can be 10 rewritten as +∞ 1 8 Dμν = (2π)6 dt dk kμkνuˆ(k)2eik·ut fˆ(kt), (63) 2 −∞ G⊥(x) where fˆ is the three-dimensional Fourier transform of the ve- 6 locity distribution. This equation can be directly obtained from

Eq. (37)orEq.(43) (see Appendix G). This shows that the aux- G(x) iliary integration variable t in Eq. (63) represents the time. For 4 G (x) the Maxwellian distribution (Eq. (51)), fˆ(kt) is a Gaussian. If || we perform the integration over t (which is the one-dimensional ff 2 Fourier transform of a Gaussian), we find that the di usion ten- xG (x) sor can be expressed as || 1/2 0 μ ν (k·u)2 02468 10 μν 3 βm k k 2 −βm D = π(2π) ρm uˆ(k) e 2k2 dk. (64) x 2π k Fig. 1. Normalized diffusion coefficients G(x), G⊥(x) and friction force Alternatively, this expression can be obtained from Eq. (40)by xG(x) for a thermal bath. The friction is maximum for x  0.97, i.e. introducing a cartesian system of coordinates for u1 with the when the velocity of the test star is approximately equal to 0.792 times z-axis taken along the direction of k, and performing the inte- the rms velocity of the field stars (its value is (xG)max = 2.383). gration. With the notation kˆ = k/k,Eq.(64) can be rewritten as ⎛  ⎞ +∞ βm 1/2 ⎜ βm ⎟ We have the asymptotic behaviours Dμν = π(2π)3 ρm k3uˆ(k)2dkGμν ⎝⎜ u⎠⎟ , (65) 2π 0 2 4π 8π G(0) = , G⊥(0) = , (73) where 3 3 π3/2 2π3/2 2 G(x) ∼+∞ , G⊥(x) ∼+∞ · (74) Gμν(x) = kˆμkˆνe−(kˆ·x) dkˆ. (66) x3 x μν μν 1 μν    ⊥ | |→ We note that G (x) G (0)δ 2 G (0)δ when x 0. The We note that the potential of interaction only appears in a mul- longitudinal and transverse diffusion coefficients and the friction tiplicative constant that fixes the relaxation time (see below). force are plotted in Fig. 1. Using Eq. (C.7), we get ⎛  ⎞ 1/2 2 ⎜ ⎟ 3 2ρmG ln Λ ⎜ 3 u ⎟ 3.4. The relaxation time Dμν = Gμν ⎝⎜ ⎠⎟ , (67) 2π v 2 v m m We can use the preceding results to estimate the relaxation Λ 2 = time of the velocity distribution of the test particle towards where ln is the Coulombian logarithm (Eq. (30)) and vm = 3/(βm) is the mean square velocity of the field stars. The diffu- the Maxwellian distribution (thermalization). If we set x u sion tensor may be written as βm/2 , the Fokker-Planck equation (Eq. (55)) can be rewrit- ⎛  ⎞ ten as 2 ⎜ u ⎟ μν 2vm μν ⎜ 3 ⎟ D = G ⎝⎜ ⎠⎟ , (68) dP = 1 ∂ μν ∂P + ν 3t 2 v μ G (x) ν 2Px , (75) R m dt tR ∂x ∂x where tR is the local relaxation time defined below in Eq. (76). where tR is the local relaxation time ∼ 2 This relation emphasizes the scaling D vm/tR. Introducing a spherical system of coordinates with the z-axis 1 2π 1/2 v3 t = m · (76) in the direction of x, we can easily compute the integral in R 3 3 ρmG2 ln Λ Eq. (66). Then, we can write the normalized diffusion tensor in the form The prefactor is equal to 0.482 but, of course, this numerical μ ν factor may vary depending on the definition of the relaxation μν 1 x x 1 μν G = G − G⊥ + G⊥δ , (69) time. The relaxation time is inversely proportional to the local 2 x2 2 density ρ(r). Therefore, the relaxation time is shorter in regions where of high density (core) and longer in regions of low density (halo). Introducing the dynamical time tD = λJ/vm,weget 2π3/2 2π3/2 = = − 3 G G(x), G⊥ [erf(x) G(x)], (70) nλ N x x t ∼ J t ∼ t . (77) R ln Λ D ln N D with x   We note that the relaxation time of a test particle in a bath is of 2 1 2 1 2x 2 G x = √ t2 −t t = x − √ −x . the same order as the relaxation time of the system as a whole ( ) 2 e d 2 erf( ) e (71) π x 0 2x π (see Sect. 2.7). The error function is defined by We can also get an estimate of the relaxation time by the following argument (Spitzer 1987). If the diffusion coefficient x 2 −t2 were constant, the typical velocity of the test star (in one spatial erf(x) = √ e dt. (72) 2 Δu ∼  π 0 direction) would increase as ( ) /3 2D t. The relaxation A93, page 13 of 27 A&A 556, A93 (2013) time tr is the typical time at which the typical velocity of the test If the field particles have an isotropic velocity distribution, the 2 +∞ = = 2 star has reached its equilibrium value v ( ) 3/(mβ) vm Rosenbluth potentials take the particularly simple form (see e.g. 2 2 so that (Δu) (tr) = v (+∞). Since D depends on v, the de- Binney & Tremaine 2008) ff scription of the di usion process is more complex. However, +∞ = 2 1 v the formula tr vm/[6D(vm)] resulting from the previous ar- = 2 + h(v) 4π f (v1)v1dv1 f (v1)v1dv1 , (88) guments with D = D(vm) should provide a good estimate of v 0 ⎛ ⎞ v the relaxation time. Using Eq. (67) and comparing√ with Eq. (76)  v ⎜ 4 ⎟ 4πv ⎜ 2 v1 ⎟ we obtain tr = K3tR,whereK3 = 1/[4G( 3/2)]. Numerically, = ⎝⎜ + ⎠⎟ g(v) 3v1 2 f (v1)dv1 K = 0.13587547.... 3 0 v 3 ⎛ ⎞  = −1 +∞ 3  Finally, we can estimate the relaxation time by tr ξ ⎜3v ⎟ ⎜ 1 ⎟ where ξ is the friction coefficient. Using the Einstein relation ξ = + ⎝ + vv1⎠ f (v1)dv1 . (89) =  = v v Dβm (see Eq. (57)) with D D(vm)wefindthattr 2tr. When g = g(v), the diffusion tensor (Eq. (83)) can be put in the 3.5. The Rosenbluth potentials form of Eq. (56) with It is possible to obtain simple expressions of the diffusion and d2g 1 dg ffi D = A , D⊥ = 2A · (90) friction coe cients for any isotropic distribution of the bath. dv2 v dv If we start from the expression of the Vlasov-Landau equation (Eq. (28)), we find that the Fokker-Planck equation (Eq. (35)) Using Eq. (89), we obtain can be written as ⎡ ⎤ ⎢ v 4 +∞ ⎥ 8π 1 ⎢ v1 ⎥  = ⎣⎢ + ⎦⎥ dP ∂ μν ∂P ∂ f1 D A 2 f (v1)dv1 v v1 f (v1)dv1 , (91) = K f − P du , (78) 3 v 0 v v dt ∂vμ 1 ∂vν ∂vν 1 ⎡ ⎛ ⎞ 1 ⎢ v ⎜ 4 ⎟ 8π 1 ⎢ ⎜ 2 v1 ⎟ D⊥ = A ⎣⎢ ⎝⎜3v − ⎠⎟ f (v )dv where Kμν is defined by Eq. (29). Comparing Eq. (78) with 1 2 1 1 3 v 0 v Eqs. (36)and(44), the diffusion and friction coefficients are +∞ given by + 2v v1 f (v1)dv1 . (92) v 2 μν − μ ν μν μν w δ w w D = K f1 du1 = A f1 du1, (79) On the other hand, when h = h(v), the friction force (Eq. (84)) w3 can be written as ∂ f Fμ = 2Fμ = 2 Kμν 1 du friction pol ∂vν 1 1 dh 1 Ffriction = 2Fpol = 4A u. (93) ∂Kμν wμ v dv = 2 f du = −4A f du . (80) ∂vν 1 1 1 w3 1 Using Eq. (88), we get Using the identities u v F = 2F = −16πA f (v )v2dv . (94) friction pol v3 1 1 1 ∂2w 0 Kμν = A (81) ∂vμ∂vν This expression can be obtained directly from Eq. (84) by not- ing (Binney & Tremaine 2008)thath(u)inEq.(85) is similar and Φ to the gravitational potential (r) produced by a distribution of ∂Kμν wμ ∂ 1 mass ρ(r), where u plays the role of r and f (u) the role of ρ(r). = −2A = 2A , (82) ∂vν w3 ∂vμ w Therefore, if f (v) is isotropic, Eq. (94) is equivalent to the ex- pression of the gravitational field F = −GM(r)r/r3 produced the coefficients of diffusion and friction can be rewritten as by a spherically symmetric distribution of matter, where M(r) is the mass within the sphere of radius r (Gauss theorem). This ∂2g Dμν = A (u), (83) formula shows that the friction is due only to field stars with a ∂vμ∂vν velocity less than the velocity of the test star. This observation ∂h was first made by Chandrasekhar (1943a). F = 2F = 4A (u), (84) friction pol ∂u The previous expressions for the diffusion and friction coef- ficients are valid for any isotropic distribution of the field par- where ticles. When f (v) is the Maxwell distribution, we recover the u results of Sect. 3.2. If we substitute Eqs. (91)–(93) into Eq. (36) u = u |u − u | u u = f ( 1) u g( ) f ( 1) 1 d 1, h( ) |u − u |d 1, (85) or (44), we get a Fokker-Planck equation describing the evo- 1 lution of a test particle in a bath with a prescribed distribu- are the so-called Rosenbluth potentials (Rosenbluth et al. tion f (v)24. Alternatively, if we come back to the original Landau 1957). In terms of these potentials, the Fokker-Planck equations kinetic equation (Eq. (28)) which can be written as (Eqs. (36)and(44)) may be rewritten as ∂ f ∂ ∂ f = Dμν − fFμ , (95) dP ∂ ∂2g ∂P ∂h ∂t ∂vμ ∂vν pol = A − 2P , (86) dt ∂vμ ∂vμ∂vν ∂vν ∂vμ 24 Actually, this description is not self-consistent since the distribution dP ∂2 ∂2g ∂ ∂h f (v) is not steady on the relaxation time t unless it is the Maxwell = A P − 4 P · (87) R dt ∂vμ∂vν ∂vμ∂vν ∂vμ ∂vμ distribution. A93, page 14 of 27 P.-H. Chavanis: Kinetic theory of spatially inhomogeneous stellar systems without collective effects

μν ν μ assume an isotropic velocity distribution, use D v = Dv , In this paper we have proceeded the other way round. and substitute the general expressions of the diffusion and fric- Starting from the Liouville equation, using the BBGKY hier- tion coefficients (Eqs. (90)–(94)) with f = f (v, t), we obtain the archy, making a local approximation, and neglecting strong col- integro-differential equation lisions, we have derived the Vlasov-Landau equation (Eq. (28)) describing the evolution of the system as a whole. Then, mak- ∂ f 1 ∂ ∂2g ∂ f ∂h 25 = A v2 − 2 f , (96) ing a bath approximation , we have obtained the Fokker-Planck ∂t v2 ∂v ∂v2 ∂v ∂v equation in the form of Eq. (36) with the diffusion and friction coefficients given by Eqs. (79)and(80). This equation can then or, in more explicit form be transformed into Eq. (44) with Eqs. (83)and(84). Our ap- ∂ f 1 ∂ 1 ∂ f 1 v proach emphasizes the importance of the Landau equation in = 8πA v4 f (v , t)dv ∂t v2 ∂v 3 ∂v v 1 1 1 the kinetic theory of stellar systems, while this equation does 0 +∞ v not appear in the works of Chandrasekhar (1942), Rosenbluth + 2 + 2 et al. (1957), King (1960), and Hénon (1961), nor in the stan- v v1 f (v1, t)dv1 f f (v1, t)v1dv1 , (97) v 0 dard textbooks of stellar dynamics by Spitzer (1987), Heggie & Hut (2003), and Binney & Tremaine (2008). describing the evolution of the system as a whole. Under this Actually, the kinetic equation derived by these authors is form, Eq. (97) applies to an artificial infinite homogeneous dis- equivalent to the Landau equation26, but it is written in a dif- tribution of stars. This equation has been studied by King (1960) ferent form. They write the Fokker-Planck equation in the form in his investigations on the evaporation of globular clusters. of Eq. (44) with the diffusion coefficient placed after the second Within the local approximation, Eq. (97) also represents the sim- derivative (∂2D) while the Landau equation (Eq. (28)) is related plification of the collision operator that occurs in the r.h.s. of to the Fokker-Planck equation (Eq. (36)) in which the diffusion the Vlasov-Landau equation (Eq. (28)) when the velocity dis- coefficient is inserted between the first derivatives (∂D∂). This tribution of the stars is isotropic. In that case, we must restore difference is important on a physical point of view for two rea- the space variable and the advection term in Eq. (97). From this sons. First, the Landau equation isolates the friction by polariza- equation, implementing an adiabatic approximation, we can de- tion F while the equation derived by Chandrasekhar (1942) rive the orbit-averaged Fokker-Planck equation which was used pol and Rosenbluth et al. (1957) involves the total friction Ffriction. by Hénon (1961) and Cohn (1980) to study the collisional evo- Secondly, the Landau equation has a nice symmetric structure lution of globular clusters. It reads from which we can immediately deduce all the conservation  laws of the system (conservation of mass, energy, impulse, and ∂q ∂ f ∂q ∂ f ∂ ∂q1 − = 8πA f f1 d1 angular momentum) and the H-theorem for the Boltzmann en- ∂ ∂t ∂t ∂ ∂ −∞ ∂ 1 tropy (Balescu 2000). These properties are less apparent in the +∞ ∂ f  equations derived by King (1960) and Hénon (1961)fortheevo- + f1q1 d1 + q f1 d1 , (98) ∂ −∞  lution of the system as a whole. It is interesting to note that the symmetric structure of the kinetic equation was not realized by where early stellar dynamicists while the Landau equation was known 1 rmax long before in plasma physics. q(,t) = [2( − Φ(r, t))]3/2 r2 dr, (99) 3 Finally, the approach based on the Liouville equation and on 0 the BBGKY hierarchy is more rigorous than the two-body en- is proportional to the phase space volume available to stars with counters theory because it relaxes the assumption of locality and an energy less than . does not produce any divergence at large scales27. It leads to the generalized Landau equation (Eq. (24)) that is perfectly well- behaved at large scales contrary to the Vlasov-Landau equation 3.6. Comparison with the two-body encounters theory (Eq. (28)) in which a large-scale cut-off has to be introduced In the previous sections, we have derived the standard kinetic in order to avoid a divergence. This divergence is due to the equations of stellar systems from the the Liouville equation by long-range nature of the gravitational potential which precludes using the BBGKY hierarchy. In classical textbooks of astro- a rigorous application of the two-body encounters theory that physics (Spitzer 1987; Heggie & Hut 2003; Binney & Tremaine is valid for potentials with short-range interactions. Actually, 2008), these equations are derived in a different manner. One the two-body encounters theory is marginally applicable to the usually starts from the Fokker-Planck equation (Eq. (44)), gravitational force (it only generates a weak logarithmic diver- makes a local approximation, and evaluates the diffusion tensor gence at large scales) and this is why it is successful in practice. ΔvμΔvν and the friction force Δvμ by considering the mean ef- The generalized Landau equation (Eq. (24)) represents a concep- fect of a succession of two-body encounters. This two-body en- tual improvement of the Vlasov-Landau equation (Eq. (28)) be- counters theory was pioneered by Chandrasekhar (1942)andfur- cause it goes beyond the local approximation and takes fully into ther developed by Rosenbluth et al. (1957). This approach leads 25 directly to the expressions in Eqs. (83)and(84)ofthediffusion In the BBGKY hierarchy, this amounts to singling out a particular ffi test star and assuming that the distribution of the other stars is fixed. and friction coe cients of a test star in a bath of field stars. These 26 expressions are then substituted in the Fokker-Planck equation It is important to emphasize, however, that the two-body encoun- (Eq. (44)). Finally, arguing that the field stars and the test star ters theory of Chandrasekhar (1942) can take strong collisions into ac- count. Therefore, there is no divergence at small impact parameters and should evolve in the same manner, the Fokker-Planck equation ff the gravitational Landau length λL appears naturally in the Coulombian is transformed into the integro-di erential equation (Eq. (97)) logarithm ln Λ, contrary to the Landau theory which ignores strong col- describing the evolution of the system as a whole (King 1960). lisions (see Appendix H). When an adiabatic approximation is implemented (Hénon 1961), 27 It is moreover applicable with almost no modification to other sys- one finally obtains the orbit-averaged Fokker-Planck equation tems with long-range interactions for which a two-body encounters the- (Eq. (98)). ory is not justified (Chavanis 2010).

A93, page 15 of 27 A&A 556, A93 (2013) account the spatial inhomogeneity of the system. Unfortunately, 4.2. Evolution of the system as a whole: a Landau-type this equation is very complicated to be of much practical use. equation with angle-action variables It can, however, be simplified by using angle-action variables as w we show in the next section. Introducing angle-action variables and J, the generalized Landau equation (Eq. (24)) becomes28 +∞ ∂ f = ∂ · w → − 4. Kinetic equations with angle-action variables dτ d 1dJ1 F(1 0)G(t, t τ) ∂t ∂J 0 4.1. Adiabatic approximation ∂ ∂ f × F(1 → 0) · + F(0 → 1) · f (J, t) (J , t), (101) ∂J ∂J m 1 In order to deal with spatially inhomogeneous systems, it is 1 convenient to introduce angle-action variables (Goldstein 1956; with Binney & Tremaine 2008). Angle-action variables have been ∂u used by many authors in astrophysics in order to solve dynami- F(1 → 0) = −m r(J, w) − r(J1, w1) . (102) cal stability problems (Kalnajs 1977; Goodman 1988; Weinberg ∂w 1991; Pichon & Cannon 1997; Valageas 2006a)ortocom- To obtain Eq. (101), we have averaged Eq. (24) over w (to sim- ff ffi − pute the di usion and friction coe cients of a test star in a plify the expressions, the average . = (2π) 3 dw is implicit), cluster (Lynden-Bell & Kalnajs 1972; Tremaine & Weinberg written the scalar products as Poisson brackets, and used the 1984; Binney & Lacey 1988; Weinberg 1998, 2001;Nelson invariance of the Poisson brackets and of the phase space vol- &Tremaine1999; Pichon & Aubert 2006; Valageas 2006b; ume element on a change of canonical variables. Introducing the Chavanis 2007, 2010). By construction, the Hamiltonian H in Fourier transform of the potential with respect to the angles angle and action variables depends only on the actions J = (J1, ..., Jd) that are constants of the motion; the conjugate coor- = 1 w − w w = Ak, k1 (J, J1) u r(J, ) r(J1, 1) dinates (w1, ..., wd) are called the angles (see Appendix B). (2π)2d Therefore, any distribution of the form f = f (J) is a steady − ·w− ·w × e i(k k1 1) dwdw , (103) state of the Vlasov equation. According to the Jeans theorem, 1 this is not the general form of Vlasov steady states. However, so that if the potential is regular, for all practical purposes, any time- ·w− ·w w − w = i(k k1 1) independent solution of the Vlasov equation may be represented u r(J, ) r(J1, 1) Ak, k1 (J, J1)e , (104) by a distribution of the form f = f (J) (strong Jeans theorem). k, k1 We shall assume that the system has reached a QSS de- we get scribedbyadistribution f = f (J) as a result of a violent col- lisionless relaxation involving only mean field effects. Because i(k·w−k1·w1) F(1 → 0) = −im Ak, k (J, J1)ke . (105) of finite N effects, the distribution function f slowly evolves in 1 k, k time. Finite N effects are taken into account in the collision op- 1 erator appearing in the r.h.s. of Eq. (24). Since this term is of Substituting this expression in Eq. (101), we obtain ff ff order 1/N,thee ect of collisions (granularities, finite N e ects, +∞ correlations) is a very slow process that takes place on a relax- ∂ f = − ∂ · w m dτ d 1dJ1 Ak, k1 (J, J1) ation timescale tR ∼ (N/ln N)tD (see Sect. 2.7). Therefore, there ∂t ∂J 0 k, k1 l,l1 is a timescale separation between the dynamical time tD that is  ·w− ·w ·w− ·w ∂ the timescale during which the system reaches a steady state of × i(k k1 1) − i(l l1 1) · ke G(t, t τ) Al,l1 (J, J1)le the Vlasov equation through phase mixing and violent collision-  ∂J ·w − ·w ∂ less relaxation, and the collisional relaxation time tR that is the + i(l1 1 l ) · Al1,l(J1, J)l1e f (J, t) f (J1, t). (106) timescale during which the system reaches an almost isothermal ∂J1 distribution because of finite N effects. Because of this timescale separation, the distribution func- With angle-action variables, the equations of motion of a star tion is stationary on the dynamical timescale. It will evolve determined by the mean field take the very simple form (see through a sequence of QSSs that are steady states of the Vlasov Appendix B) equation, depending only on the actions J, slowly changing J(t − τ) = J(t) = J, in time as a result of the cumulative effect of encounters w t − τ = w t − Ω J, t τ = w − Ω J, t τ, (finite N effects). Indeed, the system re-adjusts itself dynam- ( ) ( ) ( ) ( ) (107) ically at each step of the collisional process. The distribution where Ω(J, t) is the angular frequency of the orbit with action J. function averaged over a short dynamical timescale can be ap- As explained previously, we have neglected the variation of the proximated by mean field on a timescale of the order of the dynamical time so it is considered to be frozen when we compute the stellar trajec- f (r, u, t)  f (J, t). (100) tories (adiabatic or Bogoliubov assumption). Substituting these

28 Therefore, the distribution function is a function f = f (J, t) The adiabatic assumption (Eq. (100)) is consistent with the of the actions alone that slowly evolves in time under the ef- Bogoliubov ansatz of kinetic theory. Since the correlation time of the fluctuations is of the order of the dynamical time or shorter, we can fect of collisions. This is similar to an adiabatic approxima- freeze the distribution function at time t to compute the integral over τ tion. The system is approximately in mechanical equilibrium (see Sect. 2.4). This distribution function, which is a steady state of the at each stage of the dynamics and the collisions slowly drive Vlasov equation, defines a set of angle-action variables w and J that we it towards an almost isothermal distribution, corresponding to a can use to perform the integral over τ. Then, the distribution function quasi-thermodynamical equilibrium state. f (J, t) evolves with time on a longer timescale according to Eq. (101).

A93, page 16 of 27 P.-H. Chavanis: Kinetic theory of spatially inhomogeneous stellar systems without collective effects relations in Eq. (106) and making the transformations l →−l (Chavanis 2007, 2010) and it is here specifically applied to and l1 →−l1 in the second term (friction term), we obtain stellar systems. Since collective effects are neglected, this ki- successively netic equation can be viewed as a Landau-type equation with +∞ angle-action variables describing the evolution of spatially in- ∂ f = − ∂ · w homogeneous stellar systems. The collisional evolution of these m dτ d 1dJ1 Ak, k1 (J, J1) ∂t ∂J 0 systems is due to a condition of resonance k · Ω(J, t) = k, k1 l,l1 ·w− ·w ·w − − ·w − k1 · Ω(J1, t) (with (k1, J1)  (k, J)) encapsulated in the × i(k k1 1) i(l (t τ) l1 1(t τ)) ke e δ-function. This δ-function expresses the conservation of en- × · ∂ − · ∂ ergy. It can be shown (Chavanis 2007) that the kinetic equation Al,l1 (J, J1)l A−l1,−l(J1, J)l1 ∂J ∂J1 (Eq. (114)) conserves the mass M = f dJ and the energy E = × f (J, t) f (J1, t) (108) f (J)dJ, and monotonically increases the Boltzmann en- and tropy S = − ( f /m)ln(f /m)dJ (H-theorem). However, as +∞ explained in Sect. 2.7, this equation does not reach a steady ∂ f ∂ = −m · dτ dw dJ A (J, J ) state because of the absence of statistical equilibrium for stel- 1 1 k, k1 1 29 ∂t ∂J 0 lar systems . k, k1 l,l1 i(k·w−k ·w ) i(l·w−l ·w ) −i(l·Ω(J,t)−l ·Ω(J ,t))τ × ke 1 1 e 1 1 e 1 1 4.3. Relaxation of a star in a thermal bath: a Fokker-Planck × · ∂ − · ∂ Al,l1 (J, J1)l A−l1,−l(J1, J)l1 equation with angle-action variables ∂J ∂J1 × f (J, t) f (J1, t). (109) Implementing a test particle approach as in Sect. 3,wefindthat the equation for P(J, t), the probability density of finding the test It is easy to establish that star with an action J at time t,is ∗ Ak ,k(J1, J) = A−k,−k (J, J1) = Ak, k (J, J1) . (110) 1 1 1 ∂P 3 ∂ 2 = π(2π) m · dJ1 k|Ak, k (J, J1)| Therefore, the kinetic equation can be rewritten as ∂t ∂J 1 k, k1 +∞ × δ[k · Ω(J) − k · Ω(J )] ∂ f = − ∂ · w 1 1 m dτ d 1dJ1 Ak, k1 (J, J1) ∂t ∂J 0 ∂ ∂ k, k1 l,l1 × k · − k1 · P(J, t) f (J1). (115) i(k+l)·w −i(k +l )·w −i(l·Ω(J, t)−l ·Ω(J , t))τ ∂J ∂J1 × ke e 1 1 1 e 1 1 ∂ ∂ The angular frequency Ω(J) is now a static function determined × A (J, J ) l · − l · f (J, t) f (J , t). (111) l,l1 1 ∂J 1 ∂J 1 by the distribution f (J) of the field stars. Equation (115) can be 1 written in the form of a Fokker-Planck equation Integrating over w1, and recalling that this expression has to be averaged over w, we obtain ∂P = ∂ μν ∂P − μ μ D ν PFpol , (116) +∞ ∂t ∂J ∂J ∂ f = 3 ∂ · | |2 (2π) m dτ dJ1 Ak, k1 (J, J1) involving a diffusion tensor ∂t ∂J 0 k, k1 − − ·Ω + ·Ω × i( k (J, t) k1 (J1, t))τ μν = 3 μ ν| |2 ke D π(2π) m dJ1 k k Ak, k1 (J, J1) × · ∂ − · ∂ k, k1 k k1 f (J, t) f (J1, t). (112) × · Ω − · Ω ∂J ∂J1 δ[k (J) k1 (J1)] f (J1), (117)

Making the transformation τ →−τ,then(k, k1) → (−k, −k1), and a friction by polarization and adding the resulting expression to Eq. (112), we get +∞ 3 2 Fpol = π(2π) m dJ1 k|Ak, k (J, J1)| ∂ f = 1 3 ∂ · | |2 1 (2π) m dτ dJ1 Ak, k1 (J, J1) k, k1 ∂t 2 ∂J −∞ k, k1 ∂ f ·Ω − ·Ω × · Ω − · Ω · × i(k (J, t) k1 (J1, t))τ δ[k (J) k1 (J1)]k1 (J1). (118) ke ∂J1 ∂ ∂ × k · − k1 · f (J, t) f (J1, t). (113) Writing the Fokker-Planck equation in the usual form ∂J ∂J1 ∂P ∂2 ∂ Finally, using the identity (Eq. (C.11)), we obtain the kinetic = (DμνP) − (PFμ ), (119) equation ∂t ∂Jμ∂Jν ∂Jμ friction ∂ f ∂ with = π(2π)3m · dJ k|A (J, J )|2 1 k, k1 1 Δ μΔ ν Δ ∂t ∂J μν J J J k, k1 D = , F = , (120) 2Δt friction Δt × δ k · Ω(J, t) − k1 · Ω(J1, t) 29 For self-gravitating systems in lower dimensions of space, or for ∂ ∂ systems with long-range interactions with a smooth potential like the × k · − k1 · f (J, t) f (J1, t). (114) ∂J ∂J1 HMF model, a statistical equilibrium state exists. In this case, it can be shown that the Landau-type equation (Eq. (114)) relaxes towards the This kinetic equation was previously derived for systems with Boltzmann distribution on a timescale NtD provided there are enough arbitrary long-range interactions in various dimensions of space resonances (see Chavanis 2007).

A93, page 17 of 27 A&A 556, A93 (2013) we find that the relation between the friction by polarization and 4.4. The impact theory and the wave theory the total friction is The approach developed in this paper has removed the problem- μν atic divergence at large scales that appears in the Vlasov-Landau μ μ ∂D F = F + · (121) equation when one makes a local approximation. However, this friction pol ∂Jν approach, which is based on a weak coupling approximation, Substituting Eqs. (117)and(118)inEq.(121) and using an in- yields a divergence at small scales because it does not take strong tegration by parts, we find that the diffusion and friction coeffi- collisions into account. Stellar systems are therefore described cients are given by by two complementary kinetic theories depending on the value of the impact parameter λ (see Appendix H). The two-body en- Δ μΔ ν counters theory, or impact theory, based on the Boltzmann equa- J J 3 μ ν = π(2π) m dJ1 f (J1) k k tion or on the Fokker-Planck equation is appropriate to describe 2Δt ∼ k, k1 strong collisions when λ λL. In this theory there is no diver- 2 gence at small scales. However, this theory does not take spatial ×|Ak, k (J, J1)| δ(k · Ω(J) − k1 · Ω(J1)), (122) 1 inhomogeneity and collective interactions between stars into ac- ΔJ ∂ ∂ = π(2π)3m dJ f (J ) k k · − k · count. As a result, it exhibits a logarithmic divergence at large Δt 1 1 ∂J 1 ∂J scales. This implies that the integrand in the Boltzmann equa- k, k 1 1 tion is valid only for impact parameters sufficiently smaller than ×|A (J, J )|2δ(k · Ω(J) − k · Ω(J )). (123) k, k1 1 1 1 the Jeans length (λ λJ). The wave theory based on the Landau (or Lenard-Balescu) equation written with angle-action variables These expressions can be obtained directly from the Hamiltonian is appropriate to describe spatial inhomogeneity and collective equations of motion by expanding the trajectories of the stars in effects when λ ∼ λJ. In this theory, there is no divergence at powers of 1/N in the limit N → +∞ (Valageas 2006a). large scales. However, this theory does not take into account the Let us assume that the field stars form a thermal bath with curvature of orbits at small impact parameters. As a result, it the Boltzmann distribution exhibits a logarithmic divergence at small scales. This implies that the integrand in the Landau (or Lenard-Balescu) equation −βm(J1) f (J1) = Ae , (124) with angle-action variables is valid only for wavelengths suffi- ciently larger than the Landau length (λ  λL). For N  1, where (J) is the energy of a star in an orbit with action J.Aswe the range of validity of these two theories greatly overlaps in have explained before, this distribution is not defined globally the intermediate region λL λ λJ corresponding to the do- for a self-gravitating system. However, it holds approximately main of validity of the Vlasov-Landau equation. This equation for stars with low energy30. Using the identity ∂/∂J = Ω(J) describes only weak collisions and presents logarithmic diver- (see Appendix B), we find that gences both at small and large scales. Combining the impact theory and the wave theory, we can motivate (but not rigor- ∂ f1 = − Ω ously justify) using the Vlasov-Landau equation with a small- βmf(J1) (J1). (125) ff ff ∂J1 scale cut-o at the Landau length and a large-scale cut-o at the Jeans length. The success of this equation is that the divergence Substituting this relation in Eq. (118), using the δ-function to of the integral over the impact parameter (or wavenumber) is replace k1 · Ω(J1)byk · Ω(J), and comparing the resulting ex- only logarithmic so that its value depends only weakly on the pression with Eq. (117), we finally get choice of the cut-offs. However, a better kinetic equation is the Landau (or Lenard-Balescu) equation written with angle-action μ = − μν Ων variables with a small-scale cut-off at the Landau length. Fpol D (J)βm (J), (126) which is the appropriate Einstein relation for our problem. For a thermal bath, using Eq. (126), the Fokker-Planck equation 5. Conclusion (Eq. (116)) can be written as Starting from the Liouville equation, using a truncation of the   BBGKY hierarchy at the order 1/N, and neglecting strong col- ∂P ∂ ∂P ff = Dμν(J) + βmPΩν(J) , (127) lisions and collective e ects, we have derived a kinetic equation ∂t ∂Jμ ∂Jν (Eq. (24)) in physical space that can be viewed as a general- ized Landau equation. This equation was previously derived by where Dμν(J)isgivenbyEq.(117) with Eq. (124). Recalling Kandrup (1981) using projection operator technics. A nice fea- that Ω(J) = ∂/∂J, this equation is similar to the Kramers equa- ture of this equation is that it does not present any divergence at tion in Brownian theory (Kramers 1940). This is a drift-diffusion large scales since the spatial inhomogeneity of the system and equation describing the evolution of the distribution P(J, t)of its finite extent are properly accounted for. When a local ap- the test star in an effective potential Ueff(J) = (J) produced by proximation is implemented, and a cut-off is introduced heuristi- the field stars. For t → +∞, the distribution of the test star re- cally at the Jeans length, we recover the Vlasov-Landau equation laxes towards the Boltzmann distribution (Eq. (124)). This takes (Eq. (27)) which is the standard equation of stellar dynamics. On place on a typical relaxation time tR ∼ (N/ln N)tD. Again, this is the other hand, using angle-action variables, we have derived a valid only in the part of the cluster where the Boltzmann distri- Landau-type equation (Eq. (114)) for fully inhomogeneous stel- bution holds approximately. lar systems. We have also developed a test particle approach and derived the corresponding Fokker-Planck equations (Eqs. (39) 30 This description is also valid for self-gravitating systems in lower and (115)) in position-velocity space and angle-action space re- dimensions of space, or for other systems with long-range interactions spectively. Explicit expressions have been given for the diffusion for which a statistical equilibrium state exists (Chavanis 2007, 2010). and friction coefficients. We have distinguished the friction by

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A93, page 20 of 27 P.-H. Chavanis: Kinetic theory of spatially inhomogeneous stellar systems without collective effects

Appendix A: The thermodynamic limit and the weak should choose the most convenient scaling which, in our opin- coupling approximation ion, is the Kac scaling33. We can also present the preceding results in the follow- In this Appendix, we discuss the proper thermodynamic limit ing manner, by analogy with plasma physics. A fundamental of self-gravitating systems and justify the weak coupling length scale in self-gravitating systems is the Jeans length λJ = −1/2 approximation. (4πGβmρ) . This is the counterpart of the Debye length λD = − We call vm the root mean square velocity of the stars and R (4πe2βρ/m) 1/2 in plasma physics. Since v2 ∼ GM/R,wefind ∼ 2 m the system’s size. The kinetic energy K Nmvm and the po- that λ ∼ R. Therefore, the Jeans length is of the order of the 2 2 J tential energy U ∼ N Gm /R in the Hamiltonian (Eq. (1)) are system’s size. On the other hand, a fundamental time scale in 2 ∼ comparable provided that vm GNm/R (this fundamental scal- self-gravitating systems is provided by the gravitational pulsa- 1/2 ing may also be obtained from the virial theorem). As a result, tion ωG = (4πGρ) . This is the counterpart of the plasma ∼ 2 ∼ 2 2 2 2 1/2 the energy scales as E Nmvm GN m /R and the kinetic tem- pulsation ωP = (4πe ρ/m ) in plasma physics. We can de- 2 2 − − perature, defined by kBT = mv /3, scales as kBT ∼ GNm /R ∼ = 1 = 1/2 = m fine the dynamical time by tD ωG (4πGρ) λJ/vm. E/N. Inversely, these relations may be used to define R and vm If we rescale the distances by λJ and the times by tD,wefind as a function of the energy E (conserved quantity in the micro- that the equations of the BBGKY hierarchy depend on a single canonical ensemble) or as a function of the temperature T (fixed 3 Λ= 3 ∼ dimensionless parameter 1/(nλJ ), where nλJ N gives quantity in the canonical ensemble). the number of stars in the Jeans sphere. This is the counterpart The proper thermodynamic limit of a self-gravitating sys- Λ= 3 of the number of electrons nλD in the Debye sphere in tem corresponds to N → +∞ in such a way that the normalized plasma physics. When Λ  1, we can expand the equations of 2 2 energy  = ER/(GN m ) and the normalized temperature η = the BBGKY hierarchy in terms of the small parameter 1/Λ. 2 βGNm /R are of order unity. Of course, the usual thermody- Finally, we show that 1/Λ may be interpreted as a cou- namic limit N, V → +∞ with N/V ∼ 1 is not applicable to pling parameter. The coupling parameter Γ is defined as the self-gravitating systems since these systems are spatially inho- ratio of the interaction strength at the mean interparticle dis- 2 1/3 2 1/3 mogeneous. tance Gm n (resp. e n ) to the thermal energy kBT.This √From R and vm, we define the dynamical time tD = R/vm ∼ Γ= 2 1/3 = 3 2/3 = Λ2/3 ∼ 2/3 leads to Gm n /kBT 1/(nλJ ) 1/ 1/N 1/ Gρ. If we rescale the distances by R, the velocities by vm,and Γ= 2 1/3 = 3 2/3 = Λ2/3 (resp. e n /kBT 1/(nλD) 1/ ). If we define the times by tD, we find that the equations of the BBGKY hierar- the coupling parameter g as the ratio of the interaction strength chy depend on a single dimensionless parameter η/N. This is the 2 2 at the Jeans (resp. Debye) length Gm /λJ (resp. e /λD)tothe equivalent of the plasma parameter in plasma physics. It is usu- thermal energy k T,wegetg = 1/Λ (or g = E /E = ally argued that the correlation functions scale as P ∼ 1/Nn−1. B pot kin n (Gm2/R)/k T = η/N with the initial variables). Therefore, the Therefore, when N → +∞, we can expand the equations of the B expansion of the BBGKY hierarchy in terms of the coupling pa- BBGKY hierarchy in powers of 1/N 1. This corresponds to rameter Γ or g is equivalent to an expansion in terms of the in- a weak coupling approximation (see below). We note that the verse of the number of particles in the Jeans sphere Λ=nλ3 ∼ N thermodynamic limit is well-defined for the out-of-equilibrium J Λ= 3 problem although there is no statistical equilibrium state (i.e. the (resp. Debye sphere nλD). The weak coupling approxima- Λ  density of state and the partition function diverge). tion is therefore justified when 1. By a suitable normalization of the parameters, we can take R ∼ vm ∼ tD ∼ m ∼ 1. In that case, we must impose G ∼ 1/N. Appendix B: Angle-action variables This is the Kac prescription (Kac et al. 1963; Messer & Spohn 1982). With this normalization, E ∼ N, S ∼ N and T ∼ 1. The In Sect. 4, we have explained that during its collisional evolu- energy and the entropy are extensive but they remain fundamen- tion a stellar system passes by a succession of QSSs that are tally non-additive (Campa et al. 2009). The temperature is in- steady states of the Vlasov equation slowly changing under the ff ff tensive. This normalization is very convenient since the length, e ect of close encounters (finite N e ects). The slowly varying u Φ velocity and time scales are of order unity. Furthermore, since distribution function f (r, ) determines a potential (r)anda = 2 +Φ the coupling constant G scales as 1/N, this immediately shows one-particle Hamiltonian  v /2 (r) that we assume to be that a regime of weak coupling holds when N  1. integrable. Therefore, it is possible to use angle-action variables Other normalizations of the parameters are possible. For ex- constructed with this Hamiltonian (Goldstein 1956; Binney & ample, Gilbert (1968) considers the limit N → +∞ with R ∼ Tremaine 2008). This construction is done adiabatically, i.e. the distribution function, and the angle-action variables, slowly vm ∼ tD ∼ G ∼ 1andm ∼ 1/N. In that case, E ∼ 1, S ∼ N, and T ∼ 1/N. On the other hand, de Vega & Sanchez (2002)de- change in time. u fine the thermodynamic limit as N → +∞ with m ∼ G ∼ v ∼ 1 A particle with coordinates (r, ) in phase space is de- m w and R ∼ N in order to have E ∼ N, S ∼ N,andT ∼ 1. This scribed equivalently by the angle-action variables ( , J). The u normalization is natural since m and G should not depend on N. Hamiltonian equations for the conjugate variables (r, )are However, in that case, the dynamical time t = R/v ∼ N di- dr ∂ du ∂ D m = = u, = − = −∇Φ(r). (B.1) verges with the number of particles. Therefore, this normaliza- dt ∂u dt ∂r tion is not very convenient to develop a kinetic theory of stellar u 32 In terms of the variables (r, ), the dynamics is complicated be- systems . If we impose G ∼ 1, E ∼ N, T ∼ 1andtD ∼ 1, we 1/5 −2/5 cause the potential explicitly appears in the second equation. get√R ∼ N and m ∼ N (this corresponds to ρ ∼ 1sincetD ∼ 1/ Gρ). We stress that all these scalings are equally valid. The 33 If we account for short-range interactions (quantum mechanics, important thing is that  and η are O(1) when N → +∞.One soften potential, hard spheres) there is a new dimensionless parameter in the problem that we shall generically call μ. In that case, the proper 32 If we take m ∼ G ∼ R ∼ 1thenT√∼ N, E ∼ N2 and S ∼ N. thermodynamic limit of self-gravitating systems with short-range inter- In that case, the dynamical time tD ∼ 1/ N tends to zero which is not actions corresponds to N → +∞ in such a way that , η,andμ are 1/3 convenient neither. If we take m ∼ G ∼ tD ∼ 1thenR ∼ N leading of order unity. Some examples of scalings are given by Chavanis & to T ∼ N2/3, E ∼ N5/3 and S ∼ N. Rieutord (2003).

A93, page 21 of 27 A&A 556, A93 (2013)

Therefore, this equation du/dt = −∇Φ cannot be easily inte- Using the equations of motion (C.1)and(C.2), and introducing grated except if Φ=0, i.e. for a spatially homogeneous sys- the notations x = r − r1 and w = u − u1, we obtain tem. In that case, the velocity u is constant and the unperturbed ik·(x−wτ) equations of motion reduce to r = ut + r0, i.e. to a rectilinear F(1 → 0, t − τ) = −im k e uˆ(k)dk. (C.9) motion at constant velocity. Now, the angle-action variables are constructed so that the Hamiltonian does not depend on the an- Therefore, w gles . Therefore, the Hamiltonian equations for the conjugate +∞ variables (w, J)are μν = −   K m dτ dx dk dk kμkν dw ∂ dJ ∂ 0 = = Ω(J), = − = 0, (B.2) w   dt ∂J dt ∂ × ei(k+k )·xe−ik ·wτuˆ(k)ˆu(k). (C.10) 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 = Using the identity Ω + w (J)t 0. Therefore, the equations of motion are very simple · dk in these variables. They extend naturally the trajectories at con- δ(x) = eik x , (C.11) (2π)d stant velocity for spatially homogeneous systems. This is why this choice of variables is relevant to develop the kinetic theory. and integrating over x and k,wefindthat Of course, even if the description of the motion becomes simple +∞ in these variables, the complexity of the problem has not com- Kμν = (2π)3m dτ dk kμkνeik·wτuˆ(k)2. (C.12) pletely disappeared. It is now embodied in the relation between 0 the position and momentum variables and the angle and action Performing the transformation τ →−τ,thenk →−k,and variables which can be very complicated. adding the resulting expression to Eq. (C.12), we get +∞ μν 1 Appendix C: Calculation of K Kμν = (2π)3m dτ dk kμkνeik·wτuˆ(k)2. (C.13) 2 −∞ In this Appendix, we compute the tensor Kμν that appears in the Using the identity (C.11), we finally obtain Vlasov-Landau equation (Eq. (28)). Within the local approxi- mation, we can proceed as if the system were spatially homo- μν = 3 μ ν · w 2 geneous. In that case, the mean field force vanishes, F = 0, K π(2π) m dk k k δ(k )ˆu(k) , (C.14) and the unperturbed equations of motion (i.e. for N → +∞)re- duce to whichleadstoEq.(27). Introducing a spherical system of coordinates in which the z u − = u = u (t τ) (t) , (C.1) axis is taken in the direction of w,wefindthat +∞ − = − u = − u π r(t τ) r(t) (t)τ r τ, (C.2) Kμν = π(2π)3m k2dk sin θ dθ corresponding to a rectilinear motion at constant velocity. The 0 0 collision term in the kinetic equation (Eq. (26)) can be written as 2π × μ ν 2 ∂ f ∂ ∂ ∂ dφ k k δ(kw cos θ)ˆu(k) . (C.15) = u μν − u u 0 μ d 1K ν ν f (r, , t) f (r, 1, t), (C.3) ∂t coll ∂v ∂v ∂v1 Using kx = k sin θ cos φ, ky = k sin θ sin φ and kz = k cos θ,itis with easy to see that only K , K and K can be non-zero. The other xx yy zz +∞ components of the matrix Kμν vanish by symmetry. Furthermore, μν 1 μ ν K = dτ dr1F (1 → 0, t)F (1 → 0, t − τ). (C.4) +∞ m 0 2 3 2 Kxx = Kyy = π (2π) m k dk The force by unit of mass created by particle 1 on particle 0 is 0 given by π ∂u × sin θ dθ k2δ(kw cos θ)ˆu(k)2 sin2 θ. (C.16) F(1 → 0) = −m (r − r ), (C.5) ∂r 1 0   = 1 where u(r − r ) = −G/|r − r | is the gravitational potential. The Using the identity δ(λx) |λ| δ(x), we get Fourier transform and the inverse Fourier transform of the po- +∞ 1 tential are defined by K = K = π2(2π)3m k3uˆ(k)2 dk xx yy w dx 0 uˆ(k) = e−ik·xu(x) , u(x) = eik·xuˆ(k)dk. (C.6) (2π)d π × sin3 θδ(cos θ)dθ. (C.17) For the gravitational interaction: 0 With the change of variables s = cos θ, we obtain 3 = −4πG · (2π) uˆ(k) 2 (C.7) k 1 +∞ K = K = π2(2π)3m k3uˆ(k)2 dk Substituting Eq. (C.6-b) in Eq. (C.5), and writing explicitly the xx yy w Lagrangian coordinates, we get 0 +1 · − − − F(1 → 0, t − τ) = −im k eik (r(t τ) r1(t τ))uˆ(k)dk. (C.8) × (1 − s2)δ(s)ds, (C.18) −1 A93, page 22 of 27 P.-H. Chavanis: Kinetic theory of spatially inhomogeneous stellar systems without collective effects so that, finally, Using the Plemelj formula 1 +∞ = = 5 3 2 1 = P 1 ∓ Kxx Kyy 8π m k uˆ(k) dk. (C.19) + iπδ(x), (D.6) w 0 x ± i0 x On the other hand, we get 1 +∞ u u +∞ = · w Kzz = 2π2(2π)3m k3uˆ(k)2 dk Im gˆ(k, , 1, ) πmuˆ(k)δ(k ) w 0 × · ∂ − ∂ u u k f ( , t) f ( 1, t). (D.7) π ∂u ∂u1 × 2 = sin θ cos θδ(cos θ)dθ 0. (C.20) Substituting Eq. (D.7)inEq.(D.3), we obtain the Landau equa- 0 tion (Eq. (27)). In conclusion, we obtain w2δμν − wμwν Kμν = A , Appendix E: Lenard-Balescu equation 3 (C.21) w for homogeneous stellar systems with +∞ If we assume that the system is spatially homogeneous (or ff A = 8π5m k3uˆ(k)2 dk. (C.22) make the local approximation), and take collective e ects 0 into account, the Vlasov-Landau Eq. (27) is replaced by the Vlasov-Lenard-Balescu equation Using Eq. (C.7), this leads to Eqs. (28)–(30). ∂ f ∂ f N − 1 ∂ f ∂ + u · + F · = π(2π)3m kμkν u μ Appendix D: Another derivation of the Landau ∂t ∂r N ∂ ∂v equation uˆ2(k) ∂ f ∂ f × δ(k · w) f − f 1 du dk, (E.1) | k, k · u |2 1 ∂vν ∂vν 1 For an infinite homogeneous system, the distribution function ( ) 1 and the two-body correlation function can be written as f (0) = where (k,ω) is the dielectric function f (u, t)andg(0, 1) = g(r − r1, u, u1, t). In that case, Eqs. (18) ∂ f and (19) become k · u (k,ω) = 1 + (2π)3uˆ(k) ∂ du. (E.2) ∂ f ∂ ∂u ω − k · u (u, t) = · g(x, u, u1, t)dxdu1, (D.1) ∂t ∂u ∂x The Landau equation is recovered by taking |(k, k · u)|2 = 1. ∂g ∂g The Lenard-Balescu equation generalizes the Landau equation (x, u, u1, t) + w · (x, u, u1, t) = ∂t ∂x by replacing the bare potential of interactionu ˆ(k) by a dressed ∂u ∂ ∂ potential of interaction m · − f (u, t) f (u , t), (D.2) ∂x ∂u ∂u 1 uˆ(k) 1 uˆ (k, k · u) = · (E.3) dressed |(k, k · u)| wherewehavedefinedx = r − r1 and w = u − u1.Usingthe Bogoliubov ansatz, we shall treat the distribution function f as The dielectric function in the denominator takes the dressing of a constant, and determine the asymptotic value g(x, u, u1, +∞) the particles by their polarization cloud into account. In plasma of the correlation function. Introducing the Fourier transforms physics, this term corresponds to a screening of the interactions. of the potential of interaction and of the correlation function, The Lenard-Balescu equation accounts for dynamical screen- Eq. (D.1) may be replaced by ing since the velocity u of the particles explicitly appears in the effective potential. However, for Coulombian interactions, ∂ f 3 ∂ = (2π) · kuˆ(k)Im gˆ(k, u, u1, +∞) dkdu1, (D.3) it is a good approximation to neglect the deformation of the ∂t ∂u polarization cloud due to the motion of the particles and use ∗ where we have used the reality conditiong ˆ(−k) = gˆ(k) .Onthe the static results on screening (Debye & Hückel 1923). This other hand, taking the Laplace-Fourier transform of Eq. (D.2) amounts to replacing the dynamic dielectric function |(k, k · u)| and assuming that no correlation is present initially (if there by the static dielectric function |(k, 0)|. In this approxima- ff are initial correlations, their e ect becomes rapidly negligible), tion,u ˆdressed(k, k · u) is replaced by the Debye-Hückel potential 3 = 2 2 2 + 2 = we get (2π) uˆDH(k) (4πe /m )/(k kD) corresponding to uDH(x) 2 2 −k r 1 (e /m )e D /r in physical space. If we make the same approx- g˜(k, u, u1,ω) = imuˆ(k) u k, k · u ω(k · w − ω) imation for stellar systems, we find that ˆdressed( )isre- placed by ∂ ∂ × k · − f (u, t) f (u , t). (D.4) u u 1 3 4πG ∂ ∂ 1 (2π) uˆDH(k) = − , (E.4) k2 − k2 Taking the inverse Laplace transform of Eq. (D.4), and using the J residue theorem, we find that the asymptotic value t → +∞ of corresponding to uDH(x) = −G cos(kJr)/r in physical space. the correlation function, determined by the pole ω = 0, is In this approximation, the Vlasov-Lenard-Balescu equation (Eq. (E.1)) takes the same form as the Vlasov-Landau equation u u +∞ = 1 = 2 gˆ(k, , 1, ) muˆ(k) + (Eq. (28)) except that A is now given by A 2πmG Q with k · w − i0 ∂ ∂ kL k3 × k · − f (u, t) f (u , t). (D.5) Q = dk, (E.5) u u 1 2 − 2 2 ∂ ∂ 1 1/R (k kJ ) A93, page 23 of 27 A&A 556, A93 (2013) where R is the system’s size. We see that Q diverges alge- remain in their steady state. The collisions of the test stars among −1 braically, as (λJ − R) ,whenR → λJ instead of yielding a finite themselves are also negligible, so they only evolve as a result of Coulombian logarithm ln Λ ∼ ln N when collective effects are collisions with the field stars. Therefore, if we call P(r, u, t)the neglected34. This naive approach shows that collective effects distribution function of the test stars (to have notations similar (which account for anti-shielding) tend to increase the diffusion to those of Sect. 3 with, however, a different interpretation), its coefficient and consequently tend to reduce the relaxation time. evolution is given by the Fokker-Planck equation obtained from This is the conclusion reached by Weinberg (1993) with a more Eq. (F.1) yielding precise approach. However, his approach is not fully satisfactory dP = 3 ∂ μ ν · w 2 since the system is assumed to be spatially homogeneous and π(2π) μ k k δ(k )ˆu (k) the ordinary Lenard-Balescu equation is used. In that case, the dt ∂v R → λ ∂P ∂ f1 divergence when J is a manifestation of the Jeans insta- × m f − mP du dk. (F.2) bility that a spatially homogeneous self-gravitating system ex- f 1 ν ν 1 ∂v ∂v1 periences when the size of the perturbation overcomes the Jeans The diffusion and friction coefficients are given by length. For inhomogeneous systems, the Jeans instability is sup- pressed so the results of Weinberg should be used with caution. Dμν = π(2π)3m kμkνδ(k · w)ˆu(k)2 f du dk, (F.3) Heyvaerts (2010)andChavanis(2012a) derived a more satisfac- f 1 1 tory Lenard-Balescu equation that is valid for spatially inhomo- μ 3 μ ν 2 ∂ f1 geneous stable self-gravitating systems. This equation does not F = π(2π) m k k δ(k · w)ˆu(k) du1dk. (F.4) pol ∂vν present any divergence at large scales. However, this equation is 1 complicated and it is difficult to measure the importance of col- We recall that the diffusion coefficient is due to the fluctuations lective effects. The approach of Weinberg (1993), and the argu- of the gravitational force produced by the field stars, while the ments given in this Appendix, suggest that collective effects re- friction by polarization is due to the perturbation on the distri- duce the relaxation time of inhomogeneous stellar systems. This bution of the field stars caused by the test stars. This explains reduction should be particularly strong for a system close to in- the occurrence of the masses mf and m in Eqs. (F.3)and(F.4), stability because of the enhancement of fluctuations (Monaghan respectively. 1978). Using Eq. (46) and noting that These considerations show that it is not possible to make a μν ∂D mf local approximation and simultaneously take collective effects = Fμ , (F.5) ∂vν m pol into account. The usual procedure is to ignore collective effects, we get make a local approximation, and introduce a large-scale cut-off m at the Jeans length. This is the procedure that is usually followed F = 1 + f F . (F.6) in stellar dynamics (Binney & Tremaine 2008). The only rigor- friction m pol ff ous manner to take collective e ects into account is to use the If we assume furthermore that m  mf,wefindthatFfriction  Lenard-Balescu equation written with angle-action variables. Fpol. 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. Appendix F: Multi-species systems If the field stars have an isothermal distribution, then μ = − μν u ν = − u μ It is straightforward to generalize the kinetic theory of stellar from Eq. (F.4), Fpol βmD ( )v βmD( )v and the Fokker-Planck equation (Eq. (F.2)) reduces to systems for several species of stars. The Vlasov-Landau equa- tion (Eq. (27)) is replaced by dP ∂ ∂P = Dμν(u) + βmPvν , (F.7) d f a ∂ dt ∂vμ ∂vν = π(2π)3 kμkνδ(k · w)ˆu2(k) dt ∂vμ where Dμν(u)isgivenbyEq.(F.3). Using the results of Sect. 3, ⎛ ⎞ b we get ⎜ ∂ f a ∂ f ⎟ × ⎝⎜m f b − m f a 1 ⎠⎟ du dk, (F.1) ⎛  ⎞ b 1 ∂vν a ∂vν 1 3 1/2 2ρ m G2 ln Λ ⎜ 3 u ⎟ b 1 Dμν = f f Gμν ⎝⎜ ⎠⎟ , (F.8) 2π vmf 2 vmf a u where f (r, , t) is the distribution function of species a normal- a 2 = ized such that f drdu = N m , and the sum runs over all where vmf 3/(βmf) is the velocity dispersion of the field stars a a b = Λ= species. We can use this equation to give a new interpretation of and ρf nfmf their density. On the other hand, ln ln (λJ/λL), ∼ 2 1/2 ∼ + the test particle approach developed in Sect. 3. We make three where λJ (kBT/Gmf nf) is the Jeans length and λL G(m 2 Λ ∼ 3 + assumptions: (i) We assume that the system is composed of two mf)/vmf is the Landau length. This yields nfλJ [mf/(m mf)]. μ = − + μν u ν = − + types of stars, the test stars with mass m and the field stars with The total friction is Ffriction β(m mf )D ( )v β(m − 2 mass mf; (ii) we assume that the number of test stars is much μ βmfv /2 mf)D(u)v . If the distribution of the field stars is f ∝ e , lower than the number of field stars; and (iii) we assume that the − 2 the equilibrium distribution of the test stars is P ∝ e βmv /2 ∝ field stars are in a steady distribution f (r, u). Because of assump- f m/mf .Whenm  m , the evolution is dominated by frictional tion (ii), the collisions between the field stars and the test stars f effects; when m m it is dominated by diffusion. The relax- do not alter the distribution of the field stars so that the field stars f ation time scales like tbath ∼ v2 /D where v2 is the r.m.s. velocity R m m k bath 34 Q = L k3/ k2 + k2 2 k of the test stars. Therefore, t ∼ 1/(Dβm) ∼ 1/ξ,whereξ In plasma physics, on the contrary, 0 [( D) ]d is R Λ2 is the friction coefficient associated to the friction by polariza- well-behaved as k → 0. Its value is Q = 1 [ln (1 +Λ2) − ], 2 1+Λ2 ffi = + where Λ=λ /λ . When collective effects are taken into account, there tion. The true friction coe cient is ξ∗ (1 mf/m)ξ.Weget D L bath ∼ 2 2 Λ ∼ 3 2 Λ is no divergence at large scales and the Debye length appears naturally. tR vmvmf/(ρfmfG ln ) vmf/(ρfmG ln ), where we have ∼ Λ Λ= 3 2 ∼ 2 ∼ In the dominant approximation Q ln with nλD. used mvm mfvmf 1/β. A93, page 24 of 27 P.-H. Chavanis: Kinetic theory of spatially inhomogeneous stellar systems without collective effects

Appendix G: Temporal correlation tensor different method. This result has also been obtained, and dis- of the gravitational force cussed, by Cohen et al. (1950)andLee(1968). According to Eqs. (43)and(G.7), the diffusion tensor is given by35 The diffusion of the stars is caused by the fluctuations of the μν 2 − μ ν gravitational force. For an infinite homogeneous system (or in μν 2  δ w w w ff D = 2πmG ln Λ du1 f (u1). (G.9) the local approximation), the di usion tensor can be derived w3 from Eq. (43) that is well-known in Brownian theory. This ex- pression involves the temporal auto-correlation tensor of the This returns Eq. (79) with ln Λ instead of ln Λ. gravitational force experienced by a star. It can be written as When f (u1) is the Maxwell distribution (Eq. (51)), we can compute the force auto-correlation tensor from Eq. (G.7)byus- μ ν 1 μ F (t)F (t − τ) = dr1du1 F (1 → 0, t) ing the Rosenbluth potentials as in Sect. 3.5. Alternatively, com- m bining Eqs. (G.1)and(G.3), we have ν × F (1 → 0, t − τ) f (u1). (G.1) μ ν − = 6 μ ν ik·uτ 2 ˆ We first compute the tensor F (t)F (t τ) (2π) m dk k k e uˆ(k) f (kτ), (G.10) μν 1 μ ν ˆ Q = F (1 → 0, t)F (1 → 0, t − τ)dr1. (G.2) where f is the three-dimensional Fourier transform of the dis- m tribution function. For the Maxwell distribution (Eq. (51)), we Proceeding as in Appendix C,wefind obtain Fμ(t)Fν(t − τ) = (2π)3mρ μν = 3 μ ν ik·wτ 2 Q (2π) m dk k k e uˆ(k) . (G.3) 2 2 μ ν k·uτ − k τ × dk k k ei uˆ(k)2e 2βm . (G.11) Introducing a system of spherical coordinates with the z-axis in the direction of w, and using Eq. (C.7), we obtain after some Introducing a system of spherical coordinates with the z-axis in calculations the direction of u, and using Eq. (C.7), we obtain after some 1 w2δμν − wμwν calculations Qμν = πmG2 · ⎛  ⎞ 2 3 (G.4) τ w 3 1/2 2ρmG2 1 ⎜ 3 v ⎟ Fμ(t)Fν(t − τ) = Gμν ⎝⎜ ⎠⎟ , (G.12) According to Eq. (C.4), we have 2π vm τ 2 vm +∞ μν Kμν = Qμν dτ. (G.5) where G (x) is defined in Sect. 3.3. In particular, 0 ⎛  ⎞ 4πρmG2 ⎜ 3 v ⎟ Therefore, Eqs. (G.4)and(G.5) lead to Eq. (29) with ln Λ re- F(t) · F(t − τ) = erf ⎝⎜ ⎠⎟ . (G.13) placed by vτ 2 vm tmax Λ = dτ Integrating Eq. (G.12) over time, we recover the expression ln , (G.6) (Eq. (67)) of the diffusion tensor for a Maxwellian distribution t τ min with ln Λ instead of ln Λ. where tmin and tmax are appropriate cut-offs. The upper cut-off Finally, the auto-correlation tensor of the gravitational field should be identified with the dynamical time tD. On the other at two different points (at the same time) is hand, the divergence at short times is due to the inadequacy of our assumption of straight-line trajectories to describe very close r2δμν − rμrν Fμ(0)Fν(r) = 2πnm2G2 · (G.14) encounters. If we take tmin = λL/vm and tmax = λJ/vm ∼ tD,we r3 find that ln Λ = ln Λ. In Appendix C, we have calculated Kμν by integrating first over time then over space. This yields the Coulombian logarithm (Eq. (30)). Here, we have integrated first Appendix H: The different kinetic equations over space then over time. This yields the Coulombian logarithm (Eq. (G.6)). As discussed by Lee (1968), these two approaches The standard kinetic equations (Boltzmann, Fokker-Planck, are essentially equivalent. We remark, however, that the calcula- Vlasov, Landau, and Lenard-Balescu) and their generalizations tions of Sect. C can be performed for arbitrary potentials, while can be derived from the BBGKY hierarchy. In this Appendix, the calculations of this section explicitly use the specific form of we show the connection between these different equations, and the gravitational potential. discuss their domains of validity, without entering into technical According to Eqs. (G.1), (G.2), and (G.4), the force auto- details. correlation function can be written as The first two equations of the BBGKY hierarchy may be written symbolically as 2πmG2 δμνw2 − wμwν Fμ(t)Fν(t − τ) = du f (u ). (G.7) τ 1 w3 1 ∂ f + (V0 + Vm.f.[ f ]) f = C[g], (H.1) In particular ∂t u · − = 2 1 f ( 1) u ∂g  F(t) F(t τ) 4πmG d 1. (G.8) + (L0 + L + Lm.f.[ f ])g + C[ f,g] + T [h] = S[ f ], (H.2) τ |u − u1| ∂t − The τ 1 decay of the auto-correlation function of the gravita- 35 Actually, when the force auto-correlation function decreases as t−1, tional force was first derived by Chandrasekhar (1944) with a Eq. (43) is no longer valid and (Δu)2 behaves as t ln t (Lee 1968).

A93, page 25 of 27 A&A 556, A93 (2013) where f is the one-body distribution function, g the two-body approach, completed by Rosenbluth et al. (1957), leads to the correlation function, and h the three-body correlation function. Fokker-Planck equation (Eq. (44)) with the expressions of the In the first equation, V = V0 + Vm.f. is the Vlasov operator diffusion tensor and friction force (Eqs. (83)and(84)) expressed taking into account the free motion V0 of the particles and the in terms of the Rosenbluth potentials (Eq. (85)) with V advection by the mean field m.f.. On the other hand, the colli- sion term C[g] describes the effect of two-body correlations on λ v2 Chandra = 2 max m · the evolution of the distribution function. In the second equation, A 2πmG ln (H.4)  Gm L = L0 + L + Lm.f. is a two-body Liouville operator where L0  describes the free motion of the particles, L describes the exact As shown in Sect. 3.5, the resulting Fokker-Planck equation can L ff two-body interaction, and m.f. takes into account the e ect of be transformed into the Landau equation (Eqs. (28)and(29)). the mean field in the two-body problem. The term C[ f,g]de- As in the Landau approach, the factor calculated in Eq. (H.4) scribes collective effects and the term T [h] describes three-body presents a logarithmic divergence at large scales. However, con- correlations. Finally, S[ f ] is a source term depending on the one- trary to the Landau approach, it does not present a logarithmic body distribution function. divergence at small scales since the effect of strong collisions is As explained in Sect. 2.2,forN  1 we can expand the taken into account explicitly in the Chandrasekhar approach36. equations of the BBGKY hierarchy in terms of the small param- = 2 As a result, the gravitational Landau length λL Gm/vm appears eter 1/N.ForN → +∞, the encounters are negligible and we naturally in the calculations of Chandrasekhar. This establishes get the Vlasov equation (Jeans 1915;Vlasov1938). This corre- that the relevant small-scale cut-off in Eq. (H.3) is the Landau sponds to the mean field approximation. At the order 1/N,we length. T = can neglect three-body correlations ( 0) and strong colli- The Landau and the Chandrasekhar kinetic theories make L = 2 sions ( 0) that are of order 1/N . This corresponds to the the same assumption: binary encounters and an expansion in weak coupling approximation. In fact, since the gravitational powers of the momentum transfer. They actually differ in the = potential is singular at r 0, the two-body correlation func- order in which these are introduced. Landau starts from the | − |→ tion g(r1, r2) becomes large when r1 r2 0 due to the ef- Boltzmann equation and considers a weak deflection approxima- fect of strong collisions. As a result, it is necessary to take the tion while Chandrasekhar directly starts from the Fokker-Planck L term g into account at small scales. When three-body corre- equation, but calculates the coefficients of diffusion and fric- T = lations are neglected ( 0), Eqs. (H.1)and(H.2) are closed. tion with the binary-collision picture. At that level, their theo- However, it does not appear possible to solve these equations ex- ries are equivalent37 since the Fokker-Planck equation can be plicitly without further approximation. The usual strategy is to precisely obtained from the Boltzmann equation in the limit of solve these equations for small, intermediate, and large impact weak deflections. The crucial difference is that Landau makes a parameters, and then connect these limits. weak coupling assumption and ignores strong collisions (i.e. the L = ff If we neglect strong collisions ( 0), collective e ects bending of the trajectories) while Chandrasekhar takes them into C = L = ( 0), and make a local approximation ( m.f. 0), we get the account. On the other hand, both theories ignore spatial inhomo- Landau equation (Eqs. (28)and(29)) with ff geneity and collective e ects. L = ff λmax If we neglect strong collisions ( 0) and collective e ects ALandau = 2πmG2 ln · (H.3) C = L  λ ( 0), but take spatial inhomogeneity ( m.f. 0) into account min we get the generalized Landau equation (Eqs. (24)and(114)) This factor presents a logarithmic divergence at small and large derived in this paper (see also Kandrup 1981;andChavanis scales. The Landau equation may be derived in different man- 2008a,b). This equation presents a logarithmic divergence at ners that are actually equivalent to the above procedure. Landau small scales since strong collisions are neglected, but not at large (1936) obtained his equation by starting from the Boltzmann scales since the finite extent of the system is taken into account. equation and using a weak deflection approximation |Δu| 1. This suggests that the relevant large-scale cut-off in Eqs. (H.3) In his calculations, he approximated the trajectories of the par- and (H.4) is the Jeans length. ticles by straight lines even for collisions with small impact pa- If we neglect strong collisions (L = 0) but take collective rameters. This is equivalent to starting from the Fokker-Planck effects (C  0) and spatial inhomogeneity (Lm.f.  0) into ac- equation (Eq. (44)) and calculating the first and second moments count, we get the generalized Lenard-Balescu equation derived of the velocity increments (Eq. (45)) resulting from a succession by Heyvaerts (2010)andChavanis(2012a). of binary encounters by using a straight line approximation. ff C = If we neglect collective e ects ( 0), make a local ap- 36 The approach of Chandrasekhar takes into account strong collisions proximation (Lm.f. = 0), but take strong collisions into account  with an impact parameter smaller than the Landau length λL that yield a (L  0), we get the Boltzmann equation (see Balescu 2000). deflection at an angle larger than 90◦. As we have seen, this can suppress This equation does not present any divergence at small scales. the divergence at small scales. However, the Chandrasekhar approach Since the system is dominated by weak encounters, we can ex- does not take into account the possibility of forming binary stars which pand the Boltzmann equation for weak deflections |Δu| 1 correspond to bound states with strong correlation between particles. In while taking into account the effect of strong collisions. This is other words, Chandrasekhar only considers hyperbolic trajectories and equivalent to the treatment of Chandrasekhar (1942, 1943a,b) not elliptical ones in the two-body problem. The effect of binary stars in the kinetic theory requires a special treatment (Heggie 1975). who started from the Fokker-Planck equation (Eq. (44)) and 37 calculated the first and second moments of the velocity incre- Actually, Landau studies the evolution of the system as a whole ments (Eq. (45)) resulting from a succession of binary colli- while Chandrasekhar studies the relaxation of a test star in a bath of field stars. Landau obtains an integro-differential equation (the Landau sions by taking strong collisions into account. In the calcula- equation) that conserves the energy (microcanonical description) while Δ μ Δ μΔ ν tion of v and v v , he used the exact trajectory of the Chandrasekhar obtains a differential equation (the Fokker-Planck or stars (i.e. he solved the two-body problem exactly) and took into Kramers equation) that does not conserve the energy and involves a account the strong deflections due to collisions with small im- fixed temperature (canonical description). However, these two equa- pact parameters. In the dominant approximation ln N  1, his tions are clearly related as explained in Sect. 3.

A93, page 26 of 27 P.-H. Chavanis: Kinetic theory of spatially inhomogeneous stellar systems without collective effects

 The ordinary Landau equation (L = Lm.f. = C = 0) is in- collisions (λ ∼ λL) and weak collisions (λ ∼ l). The general- termediate between the Boltzmann equation and the generalized ized Landau and generalized Lenard-Balescu equations are valid Lenard-Balescu (and generalized Landau) equation. It describes for λ  λL. They describe weak collisions (λ ∼ l), spatial inho- the effect of weak collisions but ignores strong collisions, spatial mogeneity and collective effects (λ ∼ λJ). The ordinary Landau inhomogeneity, and collective effects. It can be obtained from equation is valid for λL λ λJ. It describes weak collisions. the generalized Lenard-Balescu equation (L = 0) by making a When we go beyond the domains of validity of these equations, local approximation (Lm.f. = 0) and neglecting collective effects divergences occur and appropriate cut-offs must be introduced. (C = 0), or from the Boltzmann equation (Lm.f. = C = 0) by Connecting these different limits, we find that the best considering the limit of small deflections (L = 0). description of stellar systems is provided by the general- Actually, these kinetic equations describe the effect of ized Lenard-Balescu equation with a small-scale cut-off at the collisions at different scales (the scale λ may be interpreted as Landau length. If we neglect collective effects, we get the gener- the impact parameter). For λ ∼ λL (small impact parameters), alized Landau equation (Eqs. (24)and(114)) with a small-scale the collisions are strong and we must solve the two-body cut-off at the Landau length. Finally, if we make a local approx- problem exactly. For λ ∼ l (intermediate impact parameters), imation and neglect collective effects, the Vlasov-Landau equa- the collisions are weak and we can make a weak coupling tion (Eqs. (28)and(29)) with a small-scale cut-off at the Landau approximation. For λ ∼ λJ (large impact parameters), we must length and a large-scale cut-off at the Jeans length provides a rel- take spatial inhomogeneity and collective effects into account. evant description of stellar systems and has the advantage of the The Boltzmann equation is valid for λ λJ. It describes strong simplicity.

A93, page 27 of 27