Violent relaxation in phase-space

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New Astronomy Reviews 52 (2008) 1–18 www.elsevier.com/locate/newastrev

Violent relaxation in phase-space D. Bindoni, L. Secco * Astronomy Department, Padova University, Vicolo dell’Osservatorio 2, 35122 Padova, Italy Accepted 9 November 2007 Available online 15 December 2007

Abstract The problem of violent relaxation mechanism in collisionless systems from the point of view of the distribution function (DF) in lspace is reviewed. The literature run starts from the seminal paper of Lynden-Bell [Lynden-Bell, D., 1967. MNRAS 136, 101] and is closed by that of the same author [Arad, I., Lynden-Bell, D., 2005. MNRAS 361, 385]. After some introductive sections on the stellar dynamical equilibria and on the Shannon’s information theory, the different approaches follow each accompanied with its criticism on the previous works. Different coarse-grained DFs proposed by different authors have been taken into account. It appears that for a collisionless gas of a unique mass specie there is not significant discrepancies among the different approaches which converge to the same DF at the end of relaxation process. The main problem is to avoid the non observed mass segregation in the case of multi-species composition, e.g., in a star-dominated galaxy component. On this topic the results are very different and are depending on the shape and size one chooses for l-space tiles. A great effort has been spent into the visualization of the different partitions in phase-space in order to understand clearly from what the differences arise. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Celestial mechanics; Stellar dynamics; Galaxies: clusters

Contents 1. 2. 3. 4. 5. 6.

7. 8. 9.

10. *

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Distribution function . . . . . . . . . . . . . . . . . . . . . Stellar dynamical equilibria – Boltzmann equation 3.1. Jeans theorem . . . . . . . . . . . . . . . . . . . . . . Information theory and statistical mechanics . . . . Lynden-Bell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Incomplete relaxation . . . . . . . . . . . . . . . . Shu’s criticism . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Shu’s approach . . . . . . . . . . . . . . . . . . . . . 6.2. Incomplete relaxation . . . . . . . . . . . . . . . . Mass segregation . . . . . . . . . . . . . . . . . . . . . . . . Kull, Treumann and Bo¨hringer’s criticism . . . . . . 8.1. Kull, Treumann and Bo¨hringer’s approach . Nakamura’s criticism . . . . . . . . . . . . . . . . . . . . . 9.1. Nakamura’s approach . . . . . . . . . . . . . . . . 9.2. To get Lynden-Bell statistics . . . . . . . . . . . Inconsistency in theories of violent relaxation . . . .

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Corresponding author. Tel.: +39 049 8278236; fax: +39 049 8278212. E-mail addresses: [email protected] (D. Bindoni), [email protected] (L. Secco).

1387-6473/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.newar.2007.11.001

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2 2 3 3 3 4 7 7 7 9 9 10 10 12 13 15 15

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11. 12.

The phase-space structure of dark matter halos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction It has long been realized that galaxies, and self-gravitating systems in general, follow a kind of organization despite the diversity of their initial conditions and their environment. At galaxy scale, this organization is illustrated by morphological classification schemes such as the Hubble sequence and by simple rules which govern their structure as individual self-gravitating systems. For example, elliptical galaxies display a quasi-universal luminosity profile described by de Vaucouleurs R1=4 law and on a different mass scale, a stellar system as a globular cluster is generally well fitted by the Michie-King model (Binney and Tremaine, 1987). The question that naturally emerges is, what determines the particular configuration to which a self-gravitating system settles? It is possible that their present configuration crucially depends on the conditions that prevail at their birth and on the details of their evolution. However, in view of their apparent regularity, it is tempting to investigate whether their organization can be favoured by some fundamental physical principles like those of thermodynamics and statistical physics. We wonder therefore if the actual states of self-gravitating systems are not simply more probable than any other possible configuration, i.e., if they cannot be considered as maximum entropy states (from, Chavanis, 2002). For most stellar systems, including the important class of elliptical galaxies, the relaxation time due to close twobody encounters is larger than the Hubble time by several orders of magnitude (Binney and Tremaine, 1987). Therefore, close encounters are negligible and the fundamental dynamics is that of a collisionless system in which the constituent particles (stars) move under the influence of the mean potential generated by all the other particles. Mathematically, the dynamics of stellar systems is described by the self-consistent Vlasov–Poisson system (see, Sections 2 and 3). The evolution of the Vlasov–Poisson system is extremely complicated. Although the dynamics is collisionless, the fluctuations of the gravitational potential are able to redistribute energy between stars and provide an effective relaxation mechanism on a very short timescale (less then the free-fall time). This process is referred to as violent relaxation. The end fate of the stellar collisionless structures after relaxation is strictly connected with their regularities translated into the existence of the Fundamental Plane, the main features of which the Clausius’ virial maximum theory (TCV) claims to explain (e.g., Secco, 2005 and references therein). The dynamical mechanism proposed by TCV may be generalized to all the virialized structures in a huge

16 17 17 18

range of mass: from globular clusters to galaxy clusters (the cosmic meta-plane; Burstein et al. (1997)). Lynden-Bell has been the first who introduced the violent relaxation theory in 1967. Despite his formulation at the first order, the main features of the mechanism have been outlined in his pioneering paper (Lynden-Bell, 1967). Due to the complexity of the problem, no analytical satisfactory treatment of the dynamics of violent relaxation has been successful. Instead, ideas from classical statistical mechanics have been used to calculate the most probable final state for a system of particles conserving quantities such as total mass M, energy E, and angular momentum J. The theoretical foundations of the statistical mechanics of violent relaxation were set again by Lynden-Bell (1967) in the same paper, using a continuum approach for the distribution function, and re-derived by Shu (1978) with a particulate approach to the same distribution. These analytical studies are now considered classical, despite the fact that the so-derived equilibrium distribution functions are far to be able in accounting for the properties of observed systems produced instead by more realistic Nbody simulations. Before analysing the contributions to the discussion about violent relaxation we wish to underline that in order to compare the approaches and to understand clearly the criticisms of the different authors a great help comes from the capability to visualize how they tile the phase-space. To this aim we have been inspired by the illuminating Fig. 10 of the relevant paper of Efthymiopoulos et al. (2006). We will use the contribution of these same authors also for the introductive next Section 2, for the beginning of Section 3 and to depict the derivation of Lynden-Bell statistics in Section 5. 2. Distribution function Following Efthymiopoulos et al. (2006), we introduce the most basic quantity in stellar systems, that is the finegrained distribution function (DF) or phase-space density: f ð~ x;~ v; tÞ ¼ lim 6

d l!0

dmð~ x;~ v; tÞ 3 d~ xd3~ v

ð1Þ

yielding the mass dmð~ x;~ v; tÞ (or the number of objects) contained at time t within an infinitesimal phase-space volume d6 l ¼ d3~ xd3~ v centered around any point ð~ x;~ vÞ of the sixdimensional phase-space of stellar motions (called the lspace in statistical mechanics). Clearly, f P 0, everywhere in phase-space. In the N-body approximation the mass dmð~ x;~ v; tÞ can be considered proportional to the number of particles, i.e., stars or fluid elements (e.g., of dark

D. Bindoni, L. Secco / New Astronomy Reviews 52 (2008) 1–18

matter), within the volume d3~ xd3~ v. Furthermore, it is often convenient to introduce a coarse-grained distribution function: Z 1 f ð~ x;~ v; tÞd3~ F ð~ x;~ v; tÞ ¼ 3 3 x d3~ v ð2Þ D~ xD ~ v D3~xD3~v which gives the average of the fine-grained distribution function f in a small, but not infinitesimal volume elements D3~ xD3~ v around the phase-space points ð~ x;~ vÞ. Contrary to the fine-grained distribution f, the value of the coarsegrained distribution F depends on the particular choice of partitioning the phase-space in which the volume elements D3~ xD3~ v are defined. The distribution function can be used to derive several other useful quantities. For example, the spatial mass density qð~ x; tÞ of the system is given by the integral of DF over velocities, e.g. (in Cartesian coordinates): Z 1 Z 1 Z 1 qð~ x; tÞ ¼ f ð~ x;~ v; tÞdvx dvy dvz ð3Þ 1

1

1

3.1. Jeans theorem Let now introduce the concept of constants and integrals of the motion. A constant of motion in a given force field is any function Cð~ x;~ v; tÞ that is constant along any stellar orbit; that is, if the position and velocity along an orbit are given by ~ xðtÞ and ~ vðtÞ ¼ d~ x=dt, then we have: vðt1 Þ; t1  ¼ C½~ xðt2 Þ;~ vðt2 Þ; t2  C½~ xðt1 Þ;~

I½~ xðt1 Þ;~ vðt1 Þ ¼ I½~ xðt2 Þ;~ vðt2 Þ d I½~ xðtÞ;~ vðtÞ ¼ 0 dt

r2 Uð~ x; tÞ ¼ 4pGqð~ x; tÞ

which reads:

~ p2 x; tÞ ð5Þ H ð~ x;~ p; tÞ  þ Uð~ 2 if the average mass particle is equal to unity, and ~ p ¼~ v in Cartesian coordinates. 3. Stellar dynamical equilibria – Boltzmann equation The basic equation governing the time evolution of the distribution function, DF, in collisionless stellar systems is (see, Efthymiopoulos et al., 2006): df of of oU of ¼ þ~ v  ¼0 dt ot o~ x o~ x o~ v

ð6Þ

otherwise called Boltzmann equation (or Vlasov equation in v plasma physics). In Eq. (6), d~ ¼  oU , to which the Euler dt o~ x equation reduces when pressure with collisionless system vanishes. This equation states that the mass contained within any infinitesimal volume d6 l that travels in phase-space along the orbits corresponding to the potential U, determined by Eq. (4), is preserved. Furthermore, the measure of the volume d6 l is also preserved (Liouville theorem). Now, the morphological regularity and the commonly observed characteristics of most galaxies suggest that the majority of these systems are close to a state of statistical equilibrium. Thus, we often look for steady-state solutions of Eq. (6) that do not have an explicit dependence of, DF, on time: of oU of ~ ¼0 ð7Þ v  o~ x o~ x o~ v It is relevant to underline as the stationary virial equation comes out from the stationary Vlasov equation (7) (see, Binney and Tremaine, 1987, Chapter 4).

ð9Þ

Every integral is a constant of the motion but the converse is not true. Eq. (9) becomes dI d~ x oI d~ x ¼ rI  þ  ¼0 dt dt o~ v dt

The orbits of stars are given by the Hamiltonian:

ð8Þ

for any t1 and t2 . An integral of motion Ið~ x;~ vÞ is any function only of the phase-space coordinates ð~ x;~ vÞ that is constant along any orbit, that is: ~ vðtÞ ¼ d~ x=dt,

The latter quantity, qð~ x; tÞ, can be used in turn to calculate the gravitational potential Uð~ x; tÞ via Poisson equation: ð4Þ

3

dI oI ¼~ v  rI  rU  ¼ 0 dt o~ v

ð10Þ

ð11Þ

as soon as Euler Equation is used. Comparing this with Eq. (6), we see that the condition for I to be an integral is identical with the condition for I to be a steady-state solution of the collisionless Boltzmann (or Vlasov) equation. It follows that f is necessarily a composite function of the phase-space variables ð~ x;~ vÞ through one or more of the integral functions I 1 ; I 2 ; . . . . That is x;~ vÞ; I 2 ð~ x;~ vÞ; . . . f ð~ x;~ vÞ  f ½I 1 ð~

ð12Þ

The last result is known as Jeans theorem of stellar dynamics (Jeans, 1915). In its complete form it reads as follows: ‘Any steady-state solution of the collisionless Boltzmann (or Vlasov) equation depends on the phase-space coordinates only through integrals of motion in the galactic potential, and any function of the integrals yields a steady-state solution of the collisionless Boltzmann equation’ (Binney and Tremaine, 1987, Chapter 4). 4. Information theory and statistical mechanics Information theory provides a constructive criterion for setting up probability distributions on the basis of partial knowledge, and leads to a type of statistical inference which is called the maximum-entropy estimate (Jaynes, 1957a). Some of the elementary properties of maximumentropy inference are defined and established in the following way. The quantity x is able of assuming the discrete values xi ði ¼ 1; 2; . . . ; nÞ. We do not know the corresponding probabilities pi to obtain xi . All we know is the expectation value of the function f ðxÞ:

4

hf ðxÞi ¼

D. Bindoni, L. Secco / New Astronomy Reviews 52 (2008) 1–18 n X

pi f ðxi Þ

ð13Þ

i¼1

We wish to know the probabilities pi that allows us to obtain the expectative value of a generic gðxÞ. At first glance the problem seems insoluble because the given information is insufficient to determine the probabilities pi . Eq. (13) and the normalization condition: X pi ¼ 1 ð14Þ would have to be supplemented by ðn  2Þ more conditions before pi and then hgðxÞi could be found (Jaynes, 1957a). The great advance provided by information theory lies in the discovery that there is a unique, unambiguous criterion for the ‘amount of uncertainty’ represented by a discrete probability distribution, which agrees with our intuitive notions that a broad distribution represents more uncertainty than does a sharply peaked one, and satisfies all other conditions which make it reasonable. Claude Shannon in 1948 proved that the quantity which is positive, which increases with increasing uncertainty, and is additive for independent sources of uncertainty, is X Qðp1 ; p2 ; . . . ; pn Þ ¼ K pi ln pi ð15Þ

subject to constraints (13) and (14), one introduces Lagrangian multipliers k; l; in the usual way to obtain the result: pi ¼ eklf ðxi Þ

The constant k, l are determined by substituting into (13) and (14). The result may be written in the form: o ln ZðlÞ ol

hf ðxÞi ¼  k ¼ ln ZðlÞ where ZðlÞ ¼

X

i

where now k is the Boltzmann constant. So the maximum of this function corresponds to the most probable distribution, the equilibrium state of the system. Notice that Eq. (16) provides a very powerful definition of the entropy of the system, which can be used even if it has not attained an equilibrium state. Notice also how this definition quantifies the relation between entropy, disorder and information (Longair, 1984). This last is defined as follows: X information ¼ k pi ln pi ð17Þ i

Obviously as higher is the information as lower is the disorder of the system. It is now evident how to solve the problem; in making inferences on the basis of partial information we must use that probability distribution which has maximum entropy subject to whatever is known. To maximize (15)

ð19Þ ð20Þ

elf ðxi Þ

ð21Þ

i

will be called the partition function. This may be generalized to any number of functions fr ðxÞ. Given the averages: X hfr ðxÞi ¼ pi fr ðxi Þ ð22Þ i

form the partition function: X Zðk1 ; . . . ; km Þ ¼ expf½k1 f1 ðxi Þ þ    þ km fm ðxi Þg i

i

where K is a positive constant. In other words, the Shannon’s theorem states that, if pi are a set of mutually exclusive probabilities, then the function Qðp1 ; p2 ; . . . ; pn Þ is a unique function, which, when maximized, gives the most likely distribution of the pi for a given set of constraints. Since this is just the expression for entropy as found in statistical mechanics, it will be called the entropy of the probability distribution pi . Henceforth, we can consider the terms ‘entropy’ and ‘uncertainty’ as synonymous. In fact, the correspondence with the law of entropy increase is striking and leads directly to the definition of the Gibbs entropy as X S ¼ k pi ln pi ð16Þ

ð18Þ

ð23Þ Then the maximum-entropy probability distribution is given by: pi ¼ expf½k0 þ k1 f1 ðxi Þ þ    þ km fm ðxi Þg

ð24Þ

in which the constants are determined from: hfr ðxÞi ¼ 

o ln Z okr

k0 ¼ ln Z

ð25Þ ð26Þ

5. Lynden-Bell In his seminal paper, Lynden-Bell (1967) argued that the violently changing gravitational field of a newly formed galaxy leads to a redistribution of energies between stars and provides a mechanism analogous to a relaxation in a gas. The logical space for a treatment of this kind is the phase-space (l-space) in which the phase-space volume is conserved and the following restrictions rules: (1) The total number of elements of phase (see later) which have any given phase density, f ð~ x;~ v; tÞ, is the same as it was initially, see, Eq. (1). (2) The total energy is conserved. (3) As a corollary of item (1) no two elements of phase can overlap in phase-space so that the phase-space density would be different in the region of overlap. This last assumption is equivalent to admit an exclusion principle for the phase-elements, and the fact that the phase-elements are distinguishable leads to introduce a

D. Bindoni, L. Secco / New Astronomy Reviews 52 (2008) 1–18

fourth type of statistics besides the classical one of Maxwell–Boltzmann and the two quantistic statistics, as follows:

No exclusion Exclusion

Indistinguishable particles

Distinguishable particles

I Einstein–Bose III Fermi–Dirac

II Maxwell–Boltzmann IV Lynden-Bell

We will present now a simplified version of the main steps in the derivation of Lynden-Bell’s statistics, as well reviewed by Efthymiopoulos et al. (2006). He considers a compact l-space (i.e., the escapes are negligible), and implements a coarse-graining process by dividing the l-space in an enormous number of I macrocells of equal volume (the squares labelled by index i ¼ 1; 2; . . . ; I in Fig. 1). He further divides each macrocell into a very large number, m, of microcells each of the same volume that may or may not be occupied by elements of the Liouville phase flow of the stars moving in l-space. In Fig. 1 these phase-elements (that is the microcells filled by particles) are shown by dark squares within each macrocell. He adopts the equal a priori probability assumption, namely he assumes that each element of phase flow has equal a priori probability to be found in any of I macrocells of Fig. 1 (complete mixing). As the system evolves in time, each phase element travels in phase-space by respecting this assumption. We should note that, because of phase mixing, the form of the phase-elements also changes in time. However, this deformation does not change the volume amount of an element. We can thus proceed in counting the number of phase-elements in each macrocell by keeping the simple schematic picture of Fig. 1. He denotes ni the occupation number of the ith macrocell, i.e., the number of fluid elements inside this macrocell at any fixed time t. The set of numbers fni g ¼ ðn1 ; n2 ; . . . ; nI Þ, called a macrostate, can thus be viewed as a discretized realization

5

of the coarse-grained distribution function of the system at the time t. In other words, that means the F ð~ x;~ v; tÞ of Eq. (2) is defined as a discrete function on the ith macrocell in this way: Z 1 ni n x;~ v; tÞ ¼ 6 f d3~ ð27Þ x d3~ v¼ 6 F i ð~ 6 D li D li D li where n is the number of particles inside a phase-element. He calculates the number W ðfni gÞ of all possible microscopic configurations that correspond to a given macrostate, and defines a Boltzmann entropy, S ¼ ln W P,I for this particular macrostate. If we express by N ¼ i¼1 ni the total number of phase elements and by m the (constant) number of microcells within each macrocell, the combinatorial calculation of W readily yields: W ðfni gÞ ¼

I Y N! m! n1 !n2 !    nI ! i¼1 ðm  ni Þ!

ð28Þ

He finally seeks to determine a statistical equilibrium state as the most probable macrostate, i.e., the one which maximizes S under the constraints imposed by all preserved quantities P of the phase flow. Besides mass conservation N ¼ Ii¼1 ni , is also P assumed total energy conservation of the system E ¼ Ii¼1 ni i (where i is the average energy of particles in the ith macrocell), and moreover the conservation of other quantities as the total angular momentum (if spherical symmetry is preserved during the collapse) or any other ‘third integral’ of motion may be taken into account. In the simplest case of mass and energy conservation, we maximize S by including the mass and energy constraints as Lagrange multipliers k1 , k2 in the maximization process, namely: d ln W  k1 dN  k2 dE ¼ 0

ð29Þ

Applying Stirlings formula for large numbers ln N !  N ln N  N , Eq. (27) becomes: gni g Fi ¼ jS¼max ¼ ð30Þ expðk1 þ k2 i Þ þ 1 m

Fig. 1. Lynden-Bell’s l-space partition inside a macrostate. The scheme is taken from Efthymiopoulos et al. (2006).

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D. Bindoni, L. Secco / New Astronomy Reviews 52 (2008) 1–18

Fig. 2. Lynden-Bell’s l-space partition inside a macrostate in case of J groups of phase-elements. (For interpretation to colours in this figure, the reader is referred to the web version of this paper.)

where the value of the phase-space density inside each moving phase-space element, g ¼ d n6 l ¼ f ð~ x;~ v; tÞ, is taken as constant (in Fig. 1, g turns to be proportional to the ‘darkness’ of phase-element). Eq. (30) is Lynden-Bell formula for the value F i of the coarse-grained distribution function within the ith macrocell at statistical equilibrium. Following the conventions of thermodynamics, we interpret k2 as an inverse temperature constant k2  b / 1=T and k1 in terms of an effective ‘chemical potential’ l ¼ k1 =b (or ‘Fermi energy’). We thus rewrite Eq. (30) in a familiar form reminiscent of Fermi–Dirac statistics Fi ¼

g g exp½bði  lÞ ¼ exp½bði  lÞ þ 1 1 þ exp½bði  lÞ

ð31Þ

The effective chemical potential in Eq. (31) has the same dimensions of the energy per unit mass, i . At any rate, in the so-called non-degenerate limit F i  g (i.e., exp½bði  lÞ  1), Eq. (31) tends to a Maxwell– Boltzmann distribution (that is, the final state approaches the isothermal model) in the following way: F i ¼ g exp½bði  lÞ ¼ A expðbi Þ

ð32Þ

where A ¼ g expðblÞ. The above exposition of Lynden-Bell theory is simplified in many aspects. In particular: the expression given for the constraint of the total energy is not precise. One should calculate the energy self-consistently by the gravitational interaction of the masses contained in each phase-element. However, the final result turns out to be the same once this more precise calculation is performed; all phase-elements in the above derivation are assumed to have the same value of the phase-space density g, i.e., the same ‘darkness’ in Fig. 1. A more general distribution function was derived by Lynden-Bell when the phase-elements of Fig. 1 can be grouped into J groups of distinct darkness j ¼ 1; . . . ; J (see, Fig. 2).

The final formula, derived also by the standard combinatorial calculation, reads: Fi ¼

J X j¼1

Fj ¼

J X j¼1

gj

exp½bj ði  lj Þ PJ 1 þ j¼1 exp½bj ði  lj Þ

ð33Þ

that is, it depends on a set of J pairs of Lagrange multipliers bj , lj . This more realistic formula links the initial conditions of system formation, parametrized by the values of gj (conserved during the relaxation) to the final distribution function. In the non-degenerate limit, Eq. (33) represents a superposition of nearly Boltzmann distributions, which means that each group of phase-elements is characterized by its own Maxwellian distribution of velocities which yields a different velocity dispersion in each group, related to the value of gj . In fact, if F j  gj then each factor exp½bj ði  lj Þ has to be small and we obtain the nondegenerate approximation: Fi ¼

J X

Aj expðbj i Þ

ð34Þ

j¼1 j lj Þ is determined from the condition Rwhere6 Aj ¼ gj expðb F j d l ¼ M j (d6 l ¼ d3~ xd3~ v and the integration is over g gM all phase-space) and bj ¼ b gj ¼ b P jg M where b is in turn j j

j

determined from the total energy condition. The result shows the correct coarse-grained distribution function to be a superposition of Maxwellian components whose velocity dispersion are inversely proportional to the phase-space density of the component at star mixture ½hv2 ij / 1=gj . Indeed for the velocity distribution we obtain F i exp½bj v2i  which means bj / hv12 i / gj . This poses the following problem: how to j express the overall distribution of velocities in the galaxy by a single Maxwellian function. (see, e.g., Shu (1978) and the debate of Shu (1987); Madsen (1987)).

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5.1. Incomplete relaxation

6.1. Shu’s approach

In real systems the relaxation is not complete (LyndenBell, 1967). In fact, when applied to a spherically symmetric stellar system, Eq. (34) substituted into Poisson’s equation leads to the Lane–Emden equation for an isothermal sphere. It is well known that unbounded isothermal spheres of extended spatial structure have infinite masses (e.g., Chandrasekhar, 1939), whereas isolated isothermal spheres of finite mass have zero spatial structure. Therefore, unfortunately, the resulting equilibrium state of the violent relaxation has infinite total mass. Thus real stellar systems tend towards the equilibrium state during violent relaxation but can not attain it: the gravitational potential variations die away before the relaxation process is complete. The suggested cure by Lynden-Bell for this is to introduce a cut-off in phase-space so that complete violent relaxation takes place in a limited region only. While physically reasonable, the need for such a cure raises doubts about the usefulness of the theories and introduces the quest for a theory of incomplete relaxation (Madsen, 1987). Nevertheless a statistical treatment of incomplete violent relaxation is still lacking. We will come back on this crucial point in Section 12, where it is underlined as TCV provides for a natural cut-off induced from the dark matter halo on the stellar component.

The velocity dispersion problem has been first considered by Shu (1978) with a method based on the particulate nature of the system. Shu divides the six-dimensional phase-space (l-space) into an enormous number of fixed microcells of quite arbitrary volume g. Each microcell is occupied by 0 or 1 particle (star of given mass, DM particle, black hole, etc.), in fact, the microcell volume g is chosen small enough that each microcell may be host of one star at most but large enough that two stars at adjacent microcells will not suffer a two-body encounter which would cause they travel, in a dynamical crossing time (the time scale associated with violent relaxation, Lynden-Bell, 1967) to a microcell different from the one into which they would have gone in absence of such an encounter. Shu makes a coarse-grained mesh of macrocells by grouping microcells into larger units of volume xi ¼ mi g, where mi is the number of microcells in the ith macrocell (its dimension may now change by passing from one macrocell to another). The set fxi g, i ¼ 1; . . . ; I, constitutes a coarse mesh of l-space, and the set fni g defines a given macrostate, where ni is the occupation number, i.e., the number of particles in xi (see, Fig. 3 for details). Similarity and differences with Lynden-Bell’s approach are manifest: fni g defines for both the macrostate but the meaning is different. In Lynden-Bell ni is the number of phase-elements, in Shu the number of particles. Since we deal with isolated system (or with system confined inside reflecting walls), the set of occupation numbers fni g satisfy the macroscopic constraints: XI mni ¼ Nm ð35Þ i¼1   XI 1 2 1 vi j þ U i ¼ E mni j~ ð36Þ i¼1 2 2

6. Shu’s criticism On reconsidering the foundations of Lynden-Bell’s statistical mechanical of violent relaxation in collisionless stellar system, Shu (1978) argues that Lynden-Bell’s formulation in terms of a continuum description introduces unnecessary complications. He considers a more conventional formulation in terms of particles. In fact, he stresses the point that stars are truly particles and not infinitesimals part of a continuum. In his opinion it must be possible to formulate a statistical description on the basis of a particulate description. He then finds the exclusion principle discovered by Lynden-Bell to be quantitatively important only at phase densities where two-body encounters are no longer negligible. Since the dynamical basis for the exclusion principle vanishes in such cases, the conclusion is that Lynden-Bell’s statistics always reduces in practice to Maxwell–Boltzmann statistics when applied to stellar systems. The last loose end underlined by Shu concerns LyndenBell’s conclusion that the general cases of his new statistics leads to ‘a superposition of Maxwellian components whose velocity dispersion are inversely proportional to the phasespace density of the star component’. This difficulties may vanish in the particulate description for a collisionless stellar system as long as stars of different masses are initially well mixed in phase-space.

where Ui  Uð~ xi ; tÞ  

I X l¼1;i6¼l

Gmnl j~ xi ~ xl j

ð37Þ

is the macroscopic gravitational field, m is the mass of the single species of particle, ~ vi the velocity corresponding to the center of macrocell i. The number W ðfni gÞ of all possible microscopic configurations in l-space that correspond to a given macrostate fni g defines a finite volume in 6N-dimensional C-space:    N! m1 !    mI ! W ðfni gÞ ¼ ½m3N gN  n1 !    nI ! ðm1  n1 Þ!    ðmI  nI Þ! ð38Þ The macrostate occupying the largest volume W under the constraints of conservation of mass, M, and energy, E, is the most probable state in which the system ends up, according to classical statistical mechanics. Entropy is measured by, ln W , so maximizing entropy under conservation of M and E corresponds to varying the following quantity

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D. Bindoni, L. Secco / New Astronomy Reviews 52 (2008) 1–18

Fig. 3. Shu’s l-space partition inside a macrostate.

ln W ðfni gÞ  a

I X i¼1

mni  b

I X i¼1

mni

  1 2 1 j~ vi j þ Ui 2 2

ð39Þ

with respect to ni in order to find its extremum. The parameters a and b are Lagrange multipliers introduced to remove the constraints on the independence of the ni variations. The most probable state according to the statistical mechanical procedure described above, including the assumption of complete mixing in phase-space, turns out to be: mi ni ¼ ð40Þ 1 þ exp½bmði  lÞ 2

vi j þ Ui is the energy per unit mass in cell i where i ¼ 12 j~ and the chemical potential l  a=b. Shu called this distribution the Lynden-Bell distribution for one type of particle. Disregarding to a normalization it resembles the Fermi– Dirac distribution, as might be expected due to the Pauli exclusion principle, but it is logically different since the particles are distinguishable.

In the nondegenerate limit ðni  mi Þ the Lynden-Bell distribution is simply the Maxwell–Boltzmann distribution, ni  mi exp½bmði  lÞ

ð41Þ

Up to now we confined the discussion to N stars of a single mass m. Generalizing to a collisionless stellar system with N j particles of mass mj the conserved quantities are now: XI

mj nij ¼ N j mj   I XJ X 1 2 1 j~ v m ¼E j nij i j þ Ui j¼1 2 2 i¼1 i¼1

ð42Þ ð43Þ

Here the macrostate (see, Fig. 4) is defined by the collection fnij g, where i runs from 1 to I (the total number of macrocells) and j runs from 1 to J (the number of particles types, where J is much less than N ðJ  N Þ, the total number of particles). The general most probable state, assuming that all type of particles obey the exclusion principle (that is at most one

Fig. 4. Shu’s l-space partition inside a macrostate in case of J groups of phase-elements. (For interpretation to colours in this figure, the reader is referred to the web version of this paper.)

D. Bindoni, L. Secco / New Astronomy Reviews 52 (2008) 1–18

particle is allowed in each microcell) and assuming complete mixing in phase-space, is expfbmj ½i  lj g P nij ¼ mi 1 þ j expfbmj ½i  lj g

ð44Þ

where lj is the chemical potential of the jth particle type. As we can see, in the non-degenerate limit ðnij  mi Þ the most probable distribution is a sum of Maxwellians with the same inverse ‘temperature’ b, nij ¼ mi expfbmj ½i  lj g

ð45Þ

But the mean kinetic energy of one particle of type j in the isothermal fluid is 1 3 mj hv2 ij ¼ kT kb1 ð46Þ 2 2 then the square of the velocity dispersion has to be inversely proportional to mass ½hv2 ij / 1=mj . This implies a dramatic consequence: the mass segregation, that is the heaviest objects are more centrally concentrated in respect to the lightest ones which go toward the border (see, Section 7). Shu argued that mass segregation can be avoided if the initial condition is well mixed, i.e., the mass composition of each macrocell is the same. For a collisionless stellar system, stars of all masses have nearly identical motions if they have nearly identical l-space locations. So the simplest assumption he made is that the mass function was initially well mixed. In Shu’s opinion the motion equation assures that the mass distribution function has always the same form throughout l-space, of this kind: nij ¼

Nj ni N

ð47Þ

wherePni is the total number of stars in the ith macrocell: J ni ¼ j¼1 nij . In other words, that means: owing to the fixed ratio of the total number of stars with mass mj over the total number N of the particles, the number of stars of given mass mj within the ith macrocell simply scales down proportionally to the total number of stars within the same macrocell, i.e., the relative composition of each macrocell is the same. Defining the coarse-grained mass density distribution function, F, as

We may write the coarse-grained entropy as k X Fi ln Fi D6 l S¼ m i

9

ð50Þ

which, generally speaking, is the Gibbs entropy. Eq. (50) becomes the Boltzmann entropy as soon as k is the Boltzmann constant. Maximizing W subject to the constraints (42), (43) and (47) is now equivalent to maximizing S subject to the constraints: X Fi D6 l ¼ constant ð51Þ i X 1 2 1  j~ vj þ U Fi D6 l ¼ constant ð52Þ i 2 2 The latter mathematical task is easy and the result is a single Maxwellian, Fi ¼ A expði =r20 Þ

ð53Þ

where A and r0 are constants and  is the energy per unit mass of a star: 1 2 vi j þ U i i ¼ j~ ð54Þ 2 It should be noted that the assumption that the mass distribution function is well mixed in l-space, Eq. (47), translates the collisionless problem with J mass species to an equivalent N-body problem with a single mass m. In particular, the equilibrium distribution function (53) is a single Maxwellian with a uniform velocity dispersion r0 , the same for all the mass species. In this way the mass segregation is completely avoided. 6.2. Incomplete relaxation Finally, also Shu as Lynden-Bell (1967) explicitly recognized in his work the primary difficulties (e.g., the predictions of systems with infinite masses) associated with the assumption of complete relaxation. Violent relaxation, being restricted in action both in space and time, must necessarily be incomplete. His own approach was to consider additional macroscopic constraints (e.g., Shu, 1969) and another time cut-off procedures have to be applied to the statistical mechanical results to obtain total masses which are not formally divergent.

xD3~ Fð~ x;~ v; tÞD3~ v xD3~ v centered on ð~ x;~ vÞ ¼ mass of stars at time t within D3~ ð48Þ

We consider a volume V in a system with a sample of particles of mass M and a sample of particles of minor mass m. If there is equipartition of the energy it means that

ð49Þ

1 1 Ekinetic ¼ MV 2 ¼ mv2 ð55Þ 2 2 where v2 is the typical velocity of the particles of mass m and V 2 is the typical velocity of the particles of mass M. So, it is possible to assign, in the same way as in a gas of molecules in equilibrium, a proper temperature T, such as

we recover it as Fi ð~ xi ;~ vi ; tÞxi ¼

J X

mj nij ¼ mni

7. Mass segregation

j¼1

which is the analogous of Lynden-Bell’s coarse-grained number density F i , Eq. (27), P as soon as we remember that mn now: g ¼ 1=g, gmi ¼ xi , m ¼ j nj i ij is the average mass of the macrocell and ð~ xi ;~ vi Þ is the center of the ith macrocell.

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D. Bindoni, L. Secco / New Astronomy Reviews 52 (2008) 1–18

3 ð56Þ Ekinetic ¼ kT 2 At the time t ¼ t, we take into account the subsample of the m particles; since the negligible cross section, it will appear practically decoupled from the sample of the M’s. If the sample of the m’s is in virial equilibrium (see, Section 3) in the V volume, such as 2T m ¼ Xm

ð57Þ

with 8 P1 2 > < T m ¼ j 2 mjvj P  > : Xm ¼ mj / j

ð58Þ

j

we wonder if, in the same volume, could the particles with mass M be in equilibrium too, i.e., if it is 2T M ¼ XM

ð59Þ

with 8 P1 > M iv2i M i /i : XM ¼

ð60Þ

Fig. 5. Mass segregation: stars of mass M goes to the center and stars of mass m goes toward the border.

from which we have / ¼  Mm v2 that gives m m Xm ¼ m/ ¼  mv2 ¼ 2T m M M and since m < M it results

ð67Þ

jXm j < 2T m

ð68Þ

Another time the virial equilibrium is violated and the mass segregation arises because the smaller masses go toward the border (see, Fig. 5).

i

Since the stars move in the field generated by the entire galaxy, if it was be chosen a sufficiently small volume, we have j ¼ /  i ’ constant ¼ / / ð61Þ as the particles M and m fill on average the same place. Moreover, since V 2i ’ constant ¼ V 2

v2j ’ constant ¼ v2

ð62Þ

we can remove from the expressions of T and X the sign of sum. Therefore, knowing that ( 1 mv2 ¼  12 m/ 2 ð63Þ 1 mv2 ¼  12 MV 2 2 we wonder if also the subsample of the Ms is in virial equilibrium, i.e., 12 MV 2 ¼  12 M/. From (63) we infer / ¼  Mm V 2 that gives M M ð64Þ XM ¼ M/ ¼  MV 2 ¼ 2T M m m and because of the hypothesis M > m it results jXM j > 2T M

ð65Þ

so the M particles are not in virial equilibrium in the V volume. Moreover, on the strength of a kinetic energy too lower in respect to the potential energy in that position, the particles of mass M, being not supported by T, collapse to the center giving place to the mass segregation. The construction can be inverted; if per hypothesis the M masses are in equilibrium, so ( 1 MV 2 ¼  12 M/ 2 ð66Þ 1 MV 2 ¼  12 mv2 2

8. Kull, Treumann and Bo¨hringer’s criticism In the present section, we deal with another issue which is related to the main crucial point of two previous approaches of violent relaxation: continuum and particulate. As we have seen, since the first investigation by Lynden-Bell (1967), one of the most important flaws of the theory has been considered to be the fact that despite the collisionless nature of violent relaxation, the statistical mechanics approach predicts a thermalized final state (Madsen, 1987; Shu, 1987), i.e., in which each phase-space elements of different densities has a square velocity dispersion inversely proportional to their density (in Lynden-Bell: hv2 ij / 1=gj ; in Shu hv2 ij / 1=mj ). The consequence, tendentially present also in Lynden-Bell (1967), is the mass segregation. To solve this problem, Shu (1978) applied to his particle approach very stringent assumption on the initial mean distribution function. In their work of 1997, Kull, Treumann and Bo¨hringer reexamine the statistical mechanics of violent relaxation in terms of phase-space elements of different densities. Starting from the consideration that Lynden-Bell’s statistical approach uses objects which are phase-elements of constant volume but different mass, they conclude that the mass independence of the final result is not fully represented. Obviously, considering phase-space elements of different volume but constant mass they remove the presence of different velocity dispersions linked to different mass elements. 8.1. Kull, Treumann and Bo¨hringer’s approach The authors (hereafter, KTB) attempt to derive the coarse-grained phase-space distribution, F ð~ x;~ v; tÞ, of a sys-

D. Bindoni, L. Secco / New Astronomy Reviews 52 (2008) 1–18

tem final state consisting of phase-space elements of J different phase-space densities, gj , subject to violent relaxation mechanism. The densities gj are obtained from the initial fine-grained phase-space density, f ð~ x;~ v; tÞ . To determine the final state, they apply the same maximum entropy principle used by Lynden-Bell (1967): as usual the state that maximizes the entropy under the constraint of conserved total energy and mass is considered the most probable final state attained by the system. To apply statistical mechanics, the l-space is divided into a large number of microcells. gj is the phase-space density of a phase-space element (i.e., a microcell occupied by mass of j-type) and gj is its volume, so that the mass associated with the phase element is gj gj . A microstate may then be described by the set of gj occupied microcells each of them has density gj . With respect to the continuum limit, the volumes gj of the phase-space elements are arbitrary and have no direct physical significance. Instead, physical significance is attributed to the mass gj gj (or the corresponding mass differences) of the phase-space elements in l-space. In this respect, it becomes natural to incorporate the universal mass independence of final motion, after the violent relaxation, into the statistical mechanics picture by introducing a constraint on the volumes gj . Indeed, the authors take as constant the phase-element mass, it means: gj gj ¼ m ¼ const:

ð69Þ

Therefore, in contrast to Lynden-Bell (1967), where the phase-elements have different mass but constant volume, the phase-elements considered here differ in volume and have constant mass. According to Eq. (69), the volume gj of a phase-element of density gj and mass m, is given by gj ¼ m=gj

ð70Þ

11

To simplify the discussion without introducing further restrictions, KTB assume that the different phase-space volumes gj are all multiples of some smallest elementary volume g, so that the volume gj occupied by a phaseelement of density gj becomes, g j ¼ cj g

ð71Þ

where cj is the factor by which gj is larger than the elementary volume, g. In other words, the microcell gj is made up of cj microcells of elementary volume g. In conclusion, the l-space is assumed to be divided into a large number of elementary microcells each one of volume g each. At the macroscopic level, these microcells are grouped into macrocells containing a large number, m, of microcells. The corresponding volume of the macrocell is mg. Suppose that there is a macrostate in which the ith macrocell contains nij phase-elements of different densities gj (see, Fig. 6). Because of the collisionless nature of the interaction in the violent relaxation process described by the Vlasov equation, there is no cohabitation of microcells which form a phase-element (Lynden-Bell, 1967). Under the assumption that the phase-space is resolved to the scale of the volume of the smallest microcell, g, TKB find the number of ways of assigning nij microcells of volume gj without cohabitation to the ith macrocell, P and then the total number of ways of assigning the j nij phase-space elements to the ith macrocell, wðnij Þ. The total number of microstates W ðfnij gÞ corresponding to a given set of occupation numbers fnij g is found by multiplying wðnij Þ and taking into account the number of ways of splitting the total of N distinguishable elements into groups nij . This yields: Y N! Y m! P Q ð72Þ W ðfnij gÞ ¼ i ðm  j cj nij Þ! nij ! j i

Fig. 6. KTB’s l-space partition inside a macrostate. (For interpretation to colours in this figure, the reader is referred to the web version of this paper.)

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D. Bindoni, L. Secco / New Astronomy Reviews 52 (2008) 1–18

The macroscopic constraints to which the system is submitted are the J constraints related to conservation of the total number N j (or total mass M j ) of phase-space elements of densities gj ; that is X gj gj nij ¼ gj gj N j ¼ M j ð73Þ i

The energy constraint reads:   XX 1 2 1 vi j þ Ui ¼ E gj gj nij j~ 2 2 j i where the gravitational potential is defined by: X X Ggj gj nl Ui ¼ Uðxi Þ ¼  jxi  xl j j l¼1;l6¼i

ð74Þ

ð75Þ

The most probable state is found by the standard procedure of maximizing, ln W , subject to the constraints of constant total energy and constant masses. Introducing the Lagrangian multipliers aj and b, which are related respectively to the constraints (73) and (74), the expression to be maximized is XX ln½W ðfnij gÞ  aj gj gj nij j

b

XX j

i

j

In conclusion, in the non-degenerate limit, the final state of the violent relaxation process defined by equation (81) is a superposition of Maxwellians characterized by a common velocity dispersion that is equivalent to an equipartition of energy per unit mass without regarding to the mass species. As a consequence, there is no mass segregation in systems which have undergone a violent relaxation, as one expects from the typical energy variation per unit mass due to this mechanism. That is e_  U_

ð82Þ

e being the energy per unit star mass. This is in agreement with the common velocity dispersion of the different components observed, for instance, in clusters of galaxies (see, e.g., Lubin and Bahcall, 1993). 9. Nakamura’s criticism

i

  1 2 1 vi j þ U i gj gj nij j~ 2 2

ð76Þ

Defining lj ¼ aj =b and taking into account Eq. (70), the most probable occupation numbers fnij g become nij ¼ P

same temperature into the same velocity dispersion r2o ¼ b1 . The coarse-grained phase-space distribution in the non-degenerate limit is X F i ð~ v;~ xÞ ¼ cj gj expfbm½i ð~ v;~ xÞ  lj g ð81Þ

m exp½bmði  lj Þ c j j exp½bmði  lj Þ þ 1

ð77Þ

2

where i ¼ 12 j~ vi j þ Ui stands for the total energy per unit mass related to the ith macrocell. The coarse-grained phase-space distribution F is defined as the sum of the J phase-space distributions where: cj nij gj F j ð~ ð78Þ x;~ vÞ  F j ð~ xi ;~ vi Þ ¼ m which is the analogous of Eq. (27) by considering that inside a phase-element there are cj elementary microcells. It is thus given as X X cj nij gj ð79Þ F i ð~ x;~ vÞ  F i ð~ xi ;~ vi Þ ¼ F j ð~ xi ;~ vi Þ ¼ m j j Substituting nij from Eq. (79), the coarse-grained phasespace distribution F becomes finally X cj gj expfbm½ð~ v;~ xÞ  lj g P ð80Þ F i ð~ x;~ vÞ ¼ v;~ xÞ  lj g þ 1 j cj expfbm½ð~ j 2

where i ð~ v;~ xÞ  i ð~ vi ;~ xi Þ ¼ 12 j~ vi j þ Ui is the total energy per unit mass. In the non-degenerate limit ðF j  gj ; 8jÞ the coarsegrained phase-space distribution (80) becomes a sum of Maxwellians. But the Maxwellians are all characterized by the same temperature b1 . The fact that all the phaseelements have the same mass, m, allows to translate the

The reason which motivate Nakamura’s review work (2000) is essentially the attempt to solve the velocity dispersion problem risen in the seminal work of Lynden-Bell. In the Nakamura’s opinion in fact, neither Shu (1978) nor Kull et al. (1997), succeeded in a satisfactory way in the purpose. As we have seen, in the Shu’s approach, based on the particulate nature of the system, the velocity dispersion depends on the mass of particle species, i.e., the equilibrium state exhibits mass segregation. Shu claims that mass segregation can be avoided if the initial condition is well mixed, i.e., the mass composition of each macrocell is the same, see, Eq. (73). This mass distribution function in a macrocell remains unchanged throughout the time evolution. But, as Nakamura remarks, if this assertion is true, trajectories of two neighboring particles with different mass must be so close that the two particles are always in the same cell throughout the relaxation process. On the other hand, trajectories of two neighboring particles with the same mass might be far apart at the equilibrium state because Shu’s theory allows permutation of any two particles in the final equilibrium state. Therefore, the initial condition assumed by Shu (1978) may be overrestrictive. Further investigation will be required to clarify the problem of mass segregation in this particulate approach. In his review analysis Nakamura takes also into account as Kull et al. (1997) have shown that a single Gaussian distribution is obtained when we use microcells with equal mass instead of divide the phase-space into microcells with equal phase-space volume like in Lynden-Bell’s theory. Though the theory by Kull et al. seems to have successfully solved the problem, in Nakamura’s opinion, it still contains two basic defects: one conceptual and one method-

D. Bindoni, L. Secco / New Astronomy Reviews 52 (2008) 1–18

ological. The conceptual defect is about the basis of the equal mass microcells. Why do we have to use microcells with equal mass, not equal volume? Kull et al. argue that mass has more physical significance than phase-space volume, however, this statement is subjective. One may consider, for instance, that energy is more fundamental and may use microcells with equal energy. Then one would end up with a distribution completely different from the Gaussian distribution. The methodological defect is the size and shape of microcells. In Lynden-Bell (1967) theory the microcells can be identical hypercubes in six-dimensional phase-space; it is easy to fill the whole phase-space with these hypercubes with any combination. On the contrary, microcells with equal mass inevitably have different shapes and sizes, thus only limited combinations are allowed to fill the phasespace without gaps (see, e.g., the white squares in Fig. 6). For instance, the phase-space density at v ! 1 (v, velocity) is infinitesimally small, thus a cell with equal mass must be infinitely large there. The purpose of the Nakamura work is to introduce a statistical theory of a collisionless system based on the ‘maximum information entropy principle’ introduced by Jaynes (1957a,b, 1983). The guiding line is that: ‘the probability distribution over microscopic states which has maximum entropy subject to whatever is known, provides the most unbiased representation of our knowledge of the state of the system’ (Jaynes, 1957b). Nakamura applies this principle to the relaxation of the coarse-grained distribution function to show that the equilibrium state is a single Gaussian distribution in the non-degenerate limit. In the case of ordinary collisional gases, the entropy must be calculated from the probability of particle existence, i.e., the probability to find a particle at a certain position in the phase-space. It is possible to use the same kind of probability for a collisionless system because the essential structure of the problem must be the same. It will be shown (Section 9.2) that Lynden-Bell’s statistics are equivalent to calculate the entropy from the probability of particle transition, i.e., the probability that a particle at a certain location at the initial time moves to another location, at infinite time. This is the reason, in Nakamura’s thought, why Lynden-Bell’s statistic gives the wrong answer. 9.1. Nakamura’s approach Nakamura divides the l-space into a set of i small cells of equal volume Dl ¼ D3~ xD3~ v. The box-averaged distribution F i ðtÞ is defined as Z F i ðtÞ ¼ f ð~ x;~ v; tÞd3~ x d3~ v ð83Þ Dli

where f ð~ x;~ v; tÞ is the true distribution (the fine-grained distribution) and Dli ¼ Dl indicates the volume integration over the ith cell. The box-averaged distribution F i represents the probability of finding a particle in the ith cell.

13

Fig. 7. Examples of initial (top) and equilibrium (down) distribution: F n ðt0 Þ ¼ P 1;n þ P 2;n þ P 3;n and F n ð1Þ ¼ P i;1 þ P i;2 þ P i;3 , respectively.

What author wishes to do is to calculate the box-averaged distribution in the limit of t ! 1, starting with a given initial distribution at t ¼ t0 . Here, the phase-space volume is a conserved quantity in addition to the energy and mass in ordinary statistical physics. Thus, he must specify the initial distribution to know the phase-space volume corresponding to it. Then, he introduces the joint probability P i;n as a probability to find a particle in the nth cell at t ¼ t0 and find the same particle in the ith cell at t ¼ 1. The initial and equilibrium distribution P P is calculated from P i;n as F n ðt0 Þ ¼ i P i;n and F i ð1Þ ¼ n P i;n (see, e.g., the examples in Fig. 7). The initial state of the fine-grained distribution f ð~ x;~ v; t0 Þ is assumed to be so smooth that we can regard it as a constant within a cell, i.e., f ð~ x;~ v; t0 Þ ¼ F n ðt0 Þ=Dl. The maximum entropy principle gives the inference that probability P i;n is the one that maximizes the following information entropy X X S¼ F i ð1Þ ln F i ð1Þ ¼  P i;n ln P i;n ð84Þ i

i;n

under the constraints of energy and phase-space volume conservation. We introduce now the quantities: ri;n ¼

P i;n F n ðt0 Þ

ð85Þ

where F n ðt0 Þ may be known or not. In this second case, F n ðt0 Þ, works simply as a mathematical normalization factor for the joint probability P i;n . In the first case it changes the meaning of the two probabilities P i;n and ri;n (see, later). Then the equilibrium distribution can be expressed in the following way: X F1 ri;n F 0n ð86Þ i ¼ n

where we write F n ðt0 Þ ¼ F 0n and F i ð1Þ ¼ F 1 i . The goal of this Section is to maximize the entropy S putting Eq. (86) into Eq. (84) under the conservation of energy and phase-space volume. From the above equation the energy conservation may be written as

14

X

D. Bindoni, L. Secco / New Astronomy Reviews 52 (2008) 1–18

ri;n F 0n

i;n

  1 2 j~ vi j þ Ui ¼ constant 2

ð87Þ

where vi and Ui is the velocity and potential per mass at the ith cell. Noting that the fine-grained distribution is uniform within a cell, we understand that Dlri;n represents the fraction of the phase-space volume of the nth cell that will go to the ith cell at t ! 1. The phase-space volume conservation is then expressed as X Dlri;n ¼ Dl ð88Þ n

There is one more condition: X ri;n ¼ 1 i

ð89Þ

to ensure i;n P i;n ¼ 1. Using Lagrange’s method to find the maximum of the entropy S in Eq. (84) under the constraints of equations (87)–(89), the function to maximize is X X 1 2 vi j þ Ui Þ  ri;n F 0n ln ri;n F 0n  b ri;n F 0n ð j~ 2 i;n i;n X X  ki ri;n  dn ri;n ð90Þ i

where b, ki and dn are the Lagrange’s undetermined coefficients related to the constraints. The result is: 1=F 0n

ri;n ¼ An Bi

expðbi Þ

ð91Þ

Bi ¼ expðki Þ, and with An ¼ expðdn =F 0n  1Þ, i ¼ v2i =2 þ Ui . The distribution at t ! 1 can be calculated from the above equation combined with Eq. (86) as X 1=F 0 An Bi n F 0n expðbi Þ ð92Þ F1 i ¼ n

The parameters An , Bi , and b must be determined in order to satisfy equations (87)–(89). Due to the difficulty to obtain explicit expressions for these parameters, thus Nakamura examines, in what follows, one limiting case, the non-degenerated limit. There are cells in which the probability distribution is negligibly small ðF 0n 0Þ at the initial time. We assume these empty cells have a very small probability F and calculate the limit of F ! 0. Introducing a new parameter 1=F B0i ¼ Bi , the ri;n may be re-written as ( 0F =F 0n A1 expðbi Þ ðF n0 6¼ F Þ ri;n ¼ lim An Bi expðbi Þ ¼ F !0 A0 B0i expðbi Þ ðF n0 ¼ F Þ ð93Þ

The equilibrium distribution of Eq. (92) becomes: 0 1 X X n @ A1 F 0 þ A0 B0i F A F1 i ¼ expðbi Þ lim F !0

¼ A1 expðbi Þ

F n0 6¼F

X

F n0 6¼F

F n0

2

F1 vi j Þ  ðbUi Þ i A1 ½expðbj~

ð95Þ

where the Gaussian distribution of the velocity is characterized by a unique mean square velocity dispersion b1 . Before closing let us examine the condition under which the above calculation is valid. The ri;n of Eq. (93) when F n0 ¼ 0 may be cast in this way: X jri;n F n ¼0 ¼ M 0 A0 B0i expðbi Þ ¼ 1  M 1 A1 expðbi Þ n 0

P

n

This result means the equilibrium state is a single Gaussian distribution that is proportional to expðbi Þ. Indeed we obtain:

F n0 6¼F

ð94Þ

ð96Þ F n0

where M 0 and M 1 are the number of cells with ¼ 0 and F n0 6¼ 0, respectively. Since r has to be a positive number and the expðbi Þ ranges inside 1–0, then the following inequality has to hold: M 1 A1 < 1 Therefore the condition is X X 1 X 0 bi F1 F 0n < A1 F 0n < F n ¼ hF 0n i i ¼ A1 e M 1 n n n

ð97Þ

ð98Þ

where h. . .i denotes the average over non-zero cells. What means this result? (1) The mean number of particles per unit volume inside the n-cell at t ¼ 0 has to be equal to the number of particles per unit volume inside the i-cell at =1. (2) Owing to item (1), that means there is not overlap inside each cell, then the exclusion principle does not play a role in this case. In other words we are in the non-degenerate limit. 0 (3) But if no-overlap is synonymous of F 1 i ¼ hF n i without regarding to the indices ði; nÞ and to time ð1; 0Þ, that from one side is what one expects to occur (see, Lynden-Bell’s items (1) and (3)) by choosing the phase-space volume conservation, from the other side it proves, in Nakamura words, ‘as in this limit the conservation of phase-space volume does not play a role and the equilibrium state becomes the same as the one obtained without it’. Even if this statement may only intuitively be accepted, in our opinion it appears not clearly demonstrated. (4) If the assumption of Nakamura related to a smooth fine-grained distribution is taken into account, the 0 item (3) becomes exactly F 1 i ¼ F n owing to: X Dlri;n ¼ Dl n X F n ðt0 Þ ri;n ¼ Dl n f ðx; v; t Þ 0 X F n ðt0 Þ P i;n ¼ Dl n f ðx; v; t Þ F ðt Þ 0 n 0 X P i;n ¼ Dl n f ðx; v; t Þ 0

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X 1 P i;n ¼ Dl f ðx; v; t0 Þ n F i ð1Þ ¼ Dl f ðx; v; t0 Þ F i ð1Þ F n ðt0 Þ ¼ Dl ¼ f ðx; v; t0 Þ f ðx; v; t0 Þ F i ð1Þ ¼ F n ðt0 Þ

ð99Þ

It should be noted that this strong condition is a consequence of the Nakamura assumption. Even if he claims the formulation of his statistics is completely general (see, Section 4 of his paper), we maintain some reserves on this crucial point. 9.2. To get Lynden-Bell statistics In the previous section, the equilibrium distribution is obtained by maximizing the entropy defined in Eq. (84). It is possible to think about another definition of entropy, i.e., entropy calculated from the probability of particle transition: ri;n ¼ P i;n =F n ðt0 Þ, which means the probability that a particle in the nth cell at t ¼ t0 goes to the ith cell at t ! 1. It should be noted that the meaning of particles transition probability ri;n becomes different in respect to the joint probability P i;n because now the term F n ðt0 Þ gains a deep meaning transforming the joint probability into a conditional probability. It means a probability after we know the fact that the particle was in nth cell. Because of the P probability for total ways of transition is expressed as P ¼ n ri;n thus the entropy may be defined as ! ! X Y Y X 0 S ¼ ri;n ln ri;n ¼ M ri;n ln ri;n i;n

n

n

i;n

ð100Þ where M is the total number of cells. Maximizing S0 under the constraints of (87)–(89), we obtain ri;n ¼ An Bi expðbF 0n i Þ

ð101Þ

where An and Bi are coefficients determined by the constraints. When An is known, we can express Bi as Bi ¼ P n

1 An expðbF 0n i Þ

ð102Þ

The equilibrium distribution calculated from Eq. (86) becomes: F1 i ¼

X X An F 0n expðbF 0n i Þ F 0n expðbF 0n i  ln Þ P P ¼ 0 1þ expðbF 0n i  ln Þ n An expðbF n i Þ n F 0 6¼0 n

F 0n 6¼0

ð103Þ

where ln is defined as An expðln Þ ¼ P 0 A expðbF 0 n i Þ F ¼0 n n

ð104Þ

15

From the comparison of Eq. (103) with Eq. (94) it is now manifest the great difference to use joint probability instead of conditional probability. The terms F 0n are outside of the exponentials in the first case, whereas they are rigidly located inside the exponentials in the second case. The above expression is identical to the result of Lynden-Bell (1967) given in his Appendix I. Therefore, we understand that Lynden-Bell’s theory is equivalent to applying the maximum entropy principle on the probability of particle transition, ri;n . Then the question is : which is true? We can determine it when we omit the constraint of phase-space volume conservation. Indeed Eq. (103) differs from a single Gaussian owing to the presence of, F 0n . But F 0n are the terms which link the joint probability, P i;n , to the transition probability ri;n . To use P i;n is equivalent to not know exactly the particle distribution F 0n which in turn is equivalent, in Nakamura’s opinion, to disregard the volume conservation (item (3)). But the same does not occur for the Lynden-Bell’s approach based on the ri;n . Indeed, in the Nakamura’s word: ‘Without this constraint (volume conservation) the theory must agree with the statistical physics of an ordinary gas, and the equilibrium distribution must be a Gaussian distribution. It can be done in our theory by omitting equation (87), and the result gives a single Gaussian distribution. In Lynden-Bell’s theory, one can remove the phase-space volume conservation to calculate the number of combinations; the result fails to give a single Gaussian distribution’. 10. Inconsistency in theories of violent relaxation In a recent work Arad and Lynden-Bell (2005) re-examine how well the fundamental assumptions of statistical mechanics apply in violent relaxation mechanism. They concentrate on the well-known Lynden-Bell theory and the more recent Nakamura’s theory. However, they do not try to answer the question on about which of these theories is more correct, as, in their opinion, this is still an open question1 but instead they highlight an inconsistency present in both the theories. The inconsistency arises from the non-transitive nature of these theories: a system that undergoes a violent relaxation, relaxes and then, under an addition of energy, undergoes again a new violent relaxation and would settle in an equilibrium state that is different from the one that is predicted the system had gone directly from the initial to the final state. In fact, the main purpose of their paper is to demonstrate that theories predict a definite statistical equilibrium, do not predict the same final state when the system undergoes two violent relaxation sessions separated in time, as they do when the two sessions are treated as one. This may be called an inconsistency or at best a lack of transitivity.

1 They do not consider Nakamura’s argument to be a proof for his correctness over the Lynden-Bell’s theory.

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In order to do this analysis they re-derive the Nakamura’s theory, which is based on the information-theory approach, using a combinatorial approach that enables them to compare it to Lynden-Bell’s theory. This derivation is possible realizing the phase-space density distribution using N  1 elements of equal mass m. As in Lynden-Bell theory, authors assume that initially the system is made of a discrete set of density levels g1 ; g2 ; . . . occupying phase-space volumes V 1 ; V 2 ; . . .. Then the overall number of elements that realize a phase-space density gJ is N J ¼ V J gJ =m. Once done it, authors test the transitivity of the theories and demonstrated that the statistical-mechanical theories of violent relaxation by Lynden-Bell and Nakamura give both a negative conclusion. This non-transitivity is a result of the phase mixing that occurs when the system relaxes. As the fine-grained phase-space density filaments become thinner and thinner, the system is better described in terms of the coarse-grained phase-space density. Any further relaxation of the system should be therefore considered in terms of the coarse-grained phase-space density which, as we have seen, would yield results different from the predictions based on the initial fine-grained phase-space density. This is a worrying aspect of these theories, as it is easy to imagine a scenario where part of the system mixes, then it fluctuates, and then mixes once again. The predictions of the theory, based on the fine-grained density, will then give us a wrong result. Arad and Lynden-Bell believe that these difficulties and ambiguities in exactly how to apply the statistical mechanics to collisionless particles which obey at Boltzmann’s equation, teach us an important lesson. The non-transitivity that they have shown is a sign that a kinetic description of violent relaxation is probably incomplete, as the equilibrium is dependent on the evolutionary path of the system. Instead, what is probably needed is a dynamical approach to the problem. Indeed, most of the above difficulties are circumvented if, instead of aiming to derive a universal most probable state, we reduce our aim to that of finding an appropriate and useful evolution equation for the coarse-grained distribution function. 11. The phase-space structure of dark matter halos Recent developments has been performed in order to get new insight into the phase-space structure of dark matter halos. According to Dekel and Arad (2004), dark-matter halos are the basic entities in which luminous galaxies form and live. Their gravitational potentials have a crucial role in determining the galaxy properties. While many of the systematic features of halo structure and kinematics have been revealed by N-body simulations, the origin of these features is still not understood, despite the fact that they are governed by simple Newtonian gravity. The halo density profile qðrÞ is a typical example. It is found in the cold dark matter (CDM) simulations to have

a robust non-power-law shape (originally Navarro et al., 1997; Power et al., 2003; Hayashi et al., 2004 and references therein), with a log slope of 3 at large radii, varying gradually toward 1 or even flatter at small radii. The slope shows only a weak sensitivity to the cosmological model and to the initial fluctuation power spectrum (e.g., Colı´n et al., 2004; Navarro et al., 2004), indicating that its origin is due to a robust relaxation process rather than specific initial conditions. In particular, violent relaxation (Lynden-Bell, 1967) may be involved in shaping up the density profile, but we have no idea why this profile has to assume the specific NFW shape. An interesting attempt to address the origin of the halo profile has been made by Taylor and Navarro (2001), who measured a poor-man phase-space density profile by fTN ðrÞ ¼ qðrÞ=r3 ðrÞ, and found that it displays an approximate power-law behavior, fTN / ra with a ¼ 1:87, over more than two decades in r. Using the Jeans equation, they showed that this power law permits a whole family of density profiles, and that a limiting case of this family is a profile similar to NFW, but with an asymptotic inner slope of 0.75 as r ! 0. This scale-free behavior of fTN ðrÞ is intriguing, and it motivates further studies of halo structure by means of phase-space density (i.e., Arad et al., 2004). Other studies have subsequently confirmed that q=r3 is a power-law in radius, but estimates of the exponent differ somewhat from the Taylor–Navarro value: a ¼ 1:95, 1.90 or 1:94 according to Rasia et al. (2004), Ascasibar et al. (2004) and Dehnen and McLaughlin (2005), respectively. Hartwick (2007) by starting from the ‘pseudo’ phasespace density QðrÞ ¼ q=r3 , presents a halo model which posses a constant Q core followed by a radial CDM-like power-law decrease in Q. The space density profile derived from this model has a constant density core and falls off rapidly beyond. Modelling dark matter halos with constant density cores is not new but it is usually done by parametrizing the density profile (see, e.g., Burkert, 1995 and references therein). Here the goal is to follow the effects of a finite primordial ‘pseudo’ phase-space density upper limit. Thus the constant density core results from a solution of the Jeans equation with a parametrized phase-space density profile. In fact, simple analytical arguments suggest that the effects of a primordial phase-space density bound should be seen in present structures even after many mergings (e.g., Dalcanton and Hogan, 2001). In the absence of cosmological simulations which include such a primordial bound, the author rely on the good agreement of predictions from his simple model with observations to argue that standard CDM simulations and hence the NFW profile may not be giving a complete picture. Finally, in a very recent paper, Vogelsberger et al. (2007) present a new and completely general technique for calculating the fine-grained phase-space structure of dark matter Galactic halo. Rather than improving simulations simply by increasing the number of particles, they

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attach additional information to each particle, namely a phase-space distortion tensor which allows them to follow the evolution of the fine-grained phase-space distribution in the immediate neighbourhood of the particle. They introduce the geodesic deviation equation (GDE) as a general tool for calculating the evolution of this distortion along any particle trajectory. The projection from phase-space to configuration-space yields the density of the particular CDM stream that particle is embedded in. This technique appears to be general and powerful in order to analyze phase-space structures embedded in different potentials. More sophisticated cosmological simulations should lead to a fuller understanding of the dark matter halo structures in the phase-space. 12. Conclusion We have analyzed the thermodynamics of violent relaxation in collisionless systems from the point of view of the DF in l-space. Different coarse-grained DFs proposed by different authors have been taken into account. It appears that for a collisionless gas of a unique mass specie there is not significant discrepancies among the different approaches which converge to the same DF at the end of relaxation process. The main problem is to avoid the non observed mass segregation in the case of multi-species composition. On this topic the results are very different and are depending on the shape and size one chooses for l-space tiles. Our run on the literature about this argument is started from the seminal paper of Lynden-Bell (1967) and is closed by that of the same author (Arad and Lynden-Bell, 2005). According to this last criticism, we can deduce the following focal points: the real relaxation is probably more complicated in respect to the ideal treatments until now considered. In them there is an inconsistency or at best a lack of transitivity; the fate of the system at the end of the violent relaxation is not definitively assigned; anyways it seems that if we take off the phase-space volume conservation, a single Gaussian would characterized the proper DF of a collisionless gas with a mass mixture. A number of authors have attempted to use entropy arguments to draw conclusions about the end state of collisionless relaxation (Tremaine et al., 1986; White and Narayan, 1987; Stiavelli, 1987; Stiavelli and Bertin, 1987; Spergel and Hernquist, 1992; Soaker, 1996). In addition to the difficulty just mentioned, these studies must deal with the fact that the equilibrium state of a stellar system depends strongly on the initial conditions. If the correct final state is to be singled out by an entropy maximization, the dependence of the final state on the initial state must somehow be translated into a

17

set of constraints. Inferring the final state then becomes equivalent to specifying these constraints. There is currently no clear understanding of how this can be done, and it is not even certain that the exercise would be physically enlightening (Merritt, 1999). It should be underlined that all the considerations of the present review concern a single component of collisionless ingredients. But a real galaxy has its baryonic component embedded inside a dark matter halo. Then the problem we are dealing with is to take into account a two-component system. In our knowledge, an equivalent treatment as that before considered does not exist even if some efforts have been done (see, e.g., Stiavelli and Sparke, 1991). The incompleteness of the treatments from one side and the incompleteness of the real relaxation on the other side, which requires the introduction of further constraints, open the door to the TCV approach. Indeed a new condition inside it appears, that is the maximization of Clausius’ virial energy (CV), to characterize the first virial equilibrium state. The special configuration that arises spontaneously in a real two-component system, introduces a truncation of the stellar component in the coordinate space, due to the presence of a scale length induced from the dark halo, as long as virial equilibrium holds. That should solve the problem of a cut-off in phase-space as suggested also by Lynden-Bell (1967) (the velocity of escape could provide the other truncation in the velocity space). Of this truncated phase volume one really needs in order to find the most probable phase distribution function for a stellar system in a stationary state, as Ogorodnikov (1965) has highlighted since 1965 and King (1962) has modelled since 1962. Moreover, the presence inside the TCV of a widest allowed configuration for the baryonic matter, due to the existence of a minimum value for the kinetic energy able to substain the structure, would justify also the fact that the stellar component in the ellipticals is less relaxed in respect to the collisionless dark halo in which it is embedded. As White and Narayan (1987) have pointed out, by studying single power-law stellar structures, the ellipticals seem indeed to have stopped their violent relaxation process before its end and before the collisionless dark matter structures. Really, the NFW density profile (Navarro et al., 1996) has the external slope b ¼ 3 instead of the Hernquist profile (in agreement with the de Vaucouleurs light profile) which requires b ¼ 4.

Acknowledgements We thank Volker Mu¨ller and Jan Peter Mu¨cket of the AIP (Potsdam) for their helpful suggestions and comments to this work and Massimo Stiavelli for useful discussions. Our thanks go also to the Drs. C.Efthymiopoulos, N.Voglis, and C.Kalapotharakos who allowed us to use some fragments of their 2006 paper in the present review.

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