1  

The maximum entropy distributions of collisionless particle velocity, speed, and energy for

statistical mechanics of self-gravitating collisionless flow (SG-CFD)

Zhijie Xu1,a

1. Computational Mathematics Group, Physical and Computational Sciences Directorate, Pacific

Northwest National Laboratory, Richland, WA 99352, USA

Abstract

The halo-mediated inverse mass cascade is a key feature of the intermediate statistically steady

state for self-gravitating collisionless flow (SG-CFD). How the inverse mass cascade maximizes

the system entropy and develops limiting velocity/energy distributions are fundamental questions

to answer. We present a statistical theory concerning the maximum entropy distributions of particle

velocity, speed, and energy for self-gravitating systems involving a power-law long-range

interaction with an arbitrary exponent n. For system with long-range interaction (-2<n<0), a broad

spectrum of halos and halo groups are necessary to form from inverse mass cascade to maximize

the system entropy. While particle velocity in each halo group is still Gaussian, the velocity

distribution of entire system can be non-Gaussian. With the virial equilibrium for local mechanical

equilibrium of halo groups, the maximum entropy principle is applied for statistical equilibrium of

global system to derive the limiting distributions. Halo mass function is not required in this

formulation, but it is a direct result of entropy maximization. The predicted velocity distribution

involves a shape parameter α that is dependent on the exponent n. The velocity distribution

approaches Laplacian with 0 and Gaussian with . For intermediate α, the maximum

entropy distribution naturally exhibits a Gaussian core at small velocity and exponential wings at

 a) Electronic mail: zhijie.xu@pnnl.gov; zhijiexu@hotmail.com

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large velocity. The total energy of collisionless particles at a given speed follows a parabolic

scaling for small speed (ε~v2) and a linear scaling (ε~v) for large speed. Results are compared

against a N-body simulation with good agreement.

Key word: Maximum Entropy, Collisionless, Dark Matter Halos, Velocity/Energy Distributions,

Long-range Interaction, Statistical Mechanics, Mass Cascade

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Contents Nomenclature ............................................................................................................................................... 4 

1. Introduction .............................................................................................................................................. 5 

2. The statistical theory for limiting distributions of SG‐CFD ........................................................................ 7 

2.1 Statement of the problem .................................................................................................................... 7 

2.2 Limiting probability distributions ..................................................................................................... 11 

2.3 The virial equilibrium and particle energy ........................................................................................ 13 

2.4 The maximum entropy principle and the velocity distribution X ...................................................... 15 

3. Distribution of particle velocity (X distribution) ..................................................................................... 19 

3.1 Statistical properties of X distribution ............................................................................................... 19 

3.2 Comparison with N-body simulation ................................................................................................ 21 

4. Distributions of particle speed and energy (Z and E distributions) ........................................................ 23 

5. Conclusion ............................................................................................................................................... 30 

 

4  

Nomenclature

Symbol S.I. Unit Physical Meaning

n Dimensionless Exponent of the particle-particle interaction potential. Specifically, n = -1 represents the usual gravitational interaction.

pm kg Mass of a collisionless particle

pv m s Velocity vector of a collisionless particle 20 2 2m s One-dimensional velocity dispersion of all particles in the system

N Dimensionless Number of collisionless particles in the system

hN Dimensionless Number of halos in the system

M kg Total mass of the entire system

pn dimensionless Number of collisionless particles in a halo

hm kg Mass of a halo

hr m Characteristic size of a halo

hv m s Halo velocity as the mean velocity of all particles in that halo 2vh   2 2m s   One-dimensional halo virial dispersion of a given halo 2v 2 2m s One-dimensional halo virial dispersion of a given halo group 2h 2 2m s One-dimensional halo velocity dispersion of a given halo group *hm kg Critical halo mass at which 2 * 2 *

h h v hm m

2 2 2m s One-dimensional velocity dispersion for all particles in a group of halos of the same size

0 2 2m s Background gravitational potential for all particles

g 2 2m s Average particle energy per unit mass for particles in a group of halos of the same size

v m s Particle energy distribution with respect to the particle speed v

X v s m Probability distribution of one-dimensional particle velocity

Z v s m Probability distribution of particle speed

E 2 2s m Probability distribution of particle energy (per unit mass)

2vH 2 2s m Probability distribution of particle virial dispersion 2

v

v 2 2m s Mean particle energy for all particles with a given speed v

1 , 2 Dimensionless Lagrangian multipliers S Dimensionless The entropy functional of the collisionless particle system

Dimensionless The exponent for scaling 2h vm

Dimensionless The shape parameter for X distribution

0v m s Typical velocity scale introduced for X distribution

G 3 2m kg s Gravitational constant

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1. Introduction

One of the most fundamental questions for self-gravitating system consisting of collisionless

particles concerns the final stationary state after relaxation. This problem was originally proposed

more than six decades ago, motivated by the paradox between the apparent universally stable self-

gravitating structures and the extremely long, unphysical, two-body relaxation time required to

form those structures. Ogorodnikov [1] and Lynden-Bell [2] were among the first to seek a fast

relaxation mechanism for an efficient phase-space mixing that drives the system toward the

equilibrium. The process of “violent relaxation” was originally introduced [2] to describe the fast

energy exchange between the rapid fluctuation of gravitational potential field and collisionless

particles moving through it. In the same paper, a new statistical mechanics subject to an exclusion

principle was also developed, where two parcels of phase space are precluded from superimposing

because of the collisionless nature. The theory predicts isothermal spheres as the equilibrium state

with maximum entropy.

However, prediction of that theory is not entirely satisfactory. The predicted isothermal spheres

have infinite mass even though prediction was made with apparent constraints of fixed finite

energy and mass. The classical computer simulations [3-6] for the structure formation reveal a

remarkably universal halo density profile that cannot be explained by that theory. Despite

significant progress made during the last several decades [7-11], the statistical mechanics of a

collisionless self-gravitating system remains a long-standing puzzle and not yet completely solved.

The difficultly can be partially attributed to the unshielded, long-range nature of gravitational

force, and the associated negative heat capacity and lack of equivalence between canonical and

microcanonical ensembles [12]. In contrast, the collisional molecular gases have short-range

interactions and plasmas systems have an effective short-range interaction due to the Debye

6  

shielding. Because of this fundamental difference, the conventional statistical mechanics for short-

range interaction systems cannot be directly applied to the long-range self-gravitating system [12].

Hence, it is necessary to develop new techniques to formulate theory of statistic mechanics that

can handle the long-range nature of gravitational interaction.

A key feature of self-gravitating collisionless flow (SG-CFD) is the inverse mass cascade as

the intermediate statistically steady state while evolving toward the final equilibrium [13]. Mass

cascade is local, two-way, and asymmetric in mass space and mediated by halos. Halos inherit/pass

their mass mostly from/to halos of similar size. The net mass transfer proceeds in a “bottom-up”

fashion. Halos pass their mass onto larger and larger halos until mass growth becomes dominant

over mass propagation. A continuous injection of mass (“free radicals”) at the smallest scale is

required to sustain the everlasting mass cascade such that the total halo mass 1 2hM a , where a

is the scale factor. Two distinct ranges can be identified in mass space, i.e. a propagation range

with a rate of mass transfer 1m a independent of halo mass hm  for *

h hm m and a deposition

range with cascaded mass consumed to grow halos for halo mass *h hm m , where *

hm is a

characteristic mass scale. As a typical non-equilibrium system, the SG-CFD exhibits an inverse

mass cascade with propagation range extending to larger and larger mass scales with time ( *hm

increases with a). The final (stationary) thermodynamic equilibrium may never be reached because

of this everlasting intermediate statistically steady state. In this sense, the inverse mass cascade

provides SG-CFD a mechanism to continuously maximize system entropy. The effects of mass

cascade on halo mass functions were previously discussed with new mass function formulated

based on the random-walk in halo mass space [13]. The effects of mass cascade on halo

momentum, energy, size, and internal structure were also presented [14]. However, how and why

7  

the halo-mediated inverse mass cascade maximizes the system entropy is still not clear, which is

the focus of this paper.

In this paper, a new statistical theory for self-gravitating collisionless systems is developed.

The theory starts from halo-based description of self-gravitating systems [15]. The maximum

entropy principle [16, 17] is applied to identify the limiting distributions of particle velocity, speed,

and energy. It is assumed the concept of entropy is still valid for describing the global statistical

equilibrium of self-gravitating system. The mechanical equilibrium and the long-range interaction

can be sufficiently taken care by the virial theorem for individual halo and halo groups. The paper

is organized as follows. Section 2 formulates statistical theory for the limiting distribution of

particle velocity from maximum entropy principle. Section 3 presents the statistical properties of

velocity distribution and comparison with N-body simulation, followed by the limiting

distributions of particle speed and energy in Section 4.

2. The statistical theory for limiting distributions of SG-CFD

2.1 Statement of the problem

The problem considered is rather idealized for complex nonlinear gravitational clustering of

collisionless particles. We consider a system of N particles interacting through a two-body

potential V r as a function of particle-particle distance r. Without loss of generality, the particle-

particle interaction can be a power-law with an arbitrary exponent n, i.e. nV r r . Particularly,

the case 1n represents the usual gravitational interaction. The spatial distribution of

collisionless particles at statistically steady state can be thought of being made up of distinct

clusters (halos) with a range of different size [15].

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Statistical description of the entire system requires full knowledge of the distribution of halo

mass, the distribution of particles within individual halos, and the spatial clustering of halos.

Besides the observational evidence to support the halo description, numerous numerical

simulations for non-linear evolution of self-gravitating collisionless system reveals the

gravitational clustering of an initially smooth particle distribution into a complex network of

sheets, filaments, and dense knots (halos) [18-20].

The halo description of the statistically steady state for self-gravitating collisionless system

(studied mostly for 1n ) is a direct result of long-range interaction and can be presumably

extended to the dynamics of collisionless particles with a potential other than the usual

gravitational ( 1n ) but still of a long-range nature. Figure 1 is a schematic plot of the halo

picture by sorting all halos according to their sizes from the smallest to largest. Each column in

Fig. 1 is a group of halos of the same size. The statistics can be defined on three different levels:

1) individual halos; 2) group of halos of same size (columns outlined in Fig. 1); and 3) global

system with all halos of all different sizes.

……

1pn 2pn 3pn 4pn ……

21v pn 2

2v pn 23v pn 2

4v pn ……

20h 2

0h 20h 2

0h ……

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Figure 1. A schematic plot of halos of different size. Halos are grouped and sorted according to the number of particles pn in halo with increasing size from left to right.

Every group of halos of the same size pn is characterized by a halo virial dispersion

2v pn as a function of halo size pn , while halo velocity dispersion 2 2

0h h is relatively

independent of halo size. In simulation, halos of same size pn might have different virial

dispersion, where 2v pn should be the average of virial dispersion of all halos in the same

group.

With the halo picture in mind, we can describe the entire system on four different levels:

1. On the particle level: every collisionless particle, characterized by a mass pm and a velocity

vector pv , should belong to one and only one particular (parent) halo. No free particles are allowed

in the halo-based description of entire system.

2. On the halo level: every halo is characterized by a halo size, i.e. the number of collisionless

particles in it ( pn ) or equivalently the halo mass ( hm ), a one-dimensional halo virial dispersion (

2vh ), and halo mean velocity ( hv ) (the mean velocity of all particles in that halo). The particle

velocity pv can be decomposed into

'p h p v v v , (1)

i.e. halo mean velocity hv and velocity fluctuation 'pv . The halo virial dispersion is defined as

2 ' ' 'var var varx y zvh p p pv v v , (2)

i.e. the variance of velocity fluctuation of all particles in the same halo. The viral velocity

dispersion can be related to the halo (local) temperature, where x, y, z denotes the three Cartesian

coordinates in Eq. (2). The halo mean velocity h

h pv v is the mean velocity of all particles in

the same halo, where h stands for the average over all particles in the same halo.

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3. On the group level for all halos of same size (the columns outlined in Fig. 1): halo group can

be characterized by the size of halos in that group ( hn or hm ), the halo virial dispersion ( 2v ), and

the halo velocity dispersion ( 2h ) that is defined as the dispersion (variance) of halo mean velocity

hv for all halos in the same group

2 var var varx y zh h h hv v v . (3)

The halo velocity dispersion 2h represents the temperature of a halo group due to the motion of

halos. The statistics defined on halo group level is the ensemble average for all halos of same size.

The virial dispersion ( 2v ) of a halo group is the average of 2

vh for all halos in the same group.

4. On the global level: entire system can be characterized by the total number of collisionless

particles N and a one-dimensional velocity dispersion 20 for all N particles in the system, which

is a measure of the total kinetic energy (or the global temperature) of the entire system.

The global statistics can be different from the local statistics of individual halos or halo groups.

To develop the new statistical theory, the following assumptions are made:

A1. On the halo level: Halos of same size pn can have the different virial dispersion 2vh and 

different mean velocity hv .

A2. On the group level: Virial theorem is assumed to be valid. Group of halos of the same size

pn  have a virial dispersion 2v . This is based on the virial theorem that

2 2 1 3 1 1 3 1n nhv vh h h p pnh

h

Gmm m n n

r

, (4)

where 3h p p hm n m r is the halo mass, hr is the size (virial radius) of halo, and is the exponent

for the scaling 2h vm

, where 1 1 3n . Symbol

h stands for the average over all

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halos in the same halo group. Gaussian velocity distribution (Maxwell-Boltzmann statistics) is

expected for all particles in the same group. Due to the independence between hv and 'pv  (From

Eq. (1)), the velocity dispersion of all particles in the same group can written as

2 2 2p v p h pn n n , (5)

with two separate contributions from halo virial dispersion 2v  and from halo velocity dispersion

2h , respectively, where both can be a function of halo size pn .

A3. On the system level: The concept of entropy is still valid for self-gravitating collisionless

system and the maximum entropy principle is valid to describe the statistical equilibrium.

2.2 Limiting probability distributions

For system described above, four limiting distributions can be identified:

1) X v : the distribution of one-dimensional particle velocity v;

2) Z v : the distribution of particle speed (the magnitude of velocity vector pv );

3) E : the distribution of particle energy including potential and kinetic energy;

4) 2vH : the distribution of particle virial dispersion 2

v , i.e. the fraction of particles with a

halo virial dispersion between 2 2 2,v v vd . Particles’ virial dispersion is the same as the

virial dispersion 2v  of the halo group they belong to.

Among these distributions, a relationship between the distributions X and H can be established

through an integral transformation (based on the assumption of A2),

2 22 2 2

0

1

2v

v vX v e H d

, (6)

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where the particle velocity distribution (the X distribution) is expressed as a weighted average of

Gaussian distribution of particle velocity in group of halos of same size. This average is weighted

by the fraction of particles ( 2 2v vH d ) with virial dispersion between 2 2 2[ , ]v v vd . The total

particle velocity dispersion 2 for all particles in the same group is given by Eq. (5). The H

distribution is related to the halo mass function that will be discussed in a separate paper and can

be obtained by the inverse transform of Eq. (6).

Similarly, the relationship between Z and H distributions can be written as:

2 22

2 2 230

2 vv v

vZ v e H d

, (7)

where the term on the right hand comes from the Maxwellian distribution of particle speed for all

particles from the same group. Obviously from Eqs. (6) and (7), the zeroth and second order

moments of X and Z distributions are:

2 2

0 01v vX v dv Z v dv H d

, (8)

2 2 2 2 2 200 0

1

3 v vX v v dv Z v v dv H d

. (9)

The mean halo virial dispersion (group temperature due to the motion of particles in each halo)

and the mean halo velocity dispersion (group temperature due to the motion of halos in that group)

for all particles in the system are defined as,

2 2 2 2

0v v v vH d

and 2 2 2 2

0h v h vH d

, (10)

where denotes averaging over all particles in entire system. From Eq. (5), it is readily to

confirm that the velocity dispersion of all particles,

2 2 2 20v h , (11)

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where the total particle kinetic energy is decomposed into contributions from the random motion

of particles in halos ( 2v ) and the random motion of halos ( 2

h ), respectively. How the total

kinetic energy 20 partitions between 2

v or 2h depends on the potential exponent n (Eq. (58)

). The energy equipartition requires a free exchange of energy among all available forms. However,

free energy exchange does not exist between the kinetic energy in 2v and the kinetic energy in

2h because two energies are defined on different scales, where 2

v is the dispersion of particle

velocity and 2h is the dispersion of halo velocity.

2.3 The virial equilibrium and particle energy

With four limiting distributions defined in Section 2.2, let us turn to the mechanical equilibrium

for halo groups (virial equilibrium). With assumption A2, the virial theorem for gravitational

potential with an exponent of n requires

2 0g g

KE n PE , (12)

where g

KE and g

PE are particle kinetic and potential energy, respectively, with subscript ‘g’

denoting an average over all particles from the same group. For particles in a halo group with a

total dispersion 2 , the specific particle kinetic and potential energy (per unit mass) are

23 2g

KE and 22 3g g

PE n KE n (13)

to satisfy the virial theorem (Eq. (12)).

The mean specific particle energy h for all particles in the same group can be written as

2 23 3

2h v g gKE PE

n

. (14)

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Particles in group of the smallest halos ( 0hm ) have the maximum energy

2 20

3 3 3 30 0

2 2h h h h hm mn n

, (15)

where 2 0v for the smallest halo group. The total energy of particles in all halos with a given

speed between [ , ]v v dv is:

2 22

2 2 2 230

12

2 vv h v v v

vN v dv e dv N H d

, (16)

where v v is the energy distribution with respect to the particle speed v and vN v dv is the

total energy of all particles with a speed between [ , ]v v dv . Term 1 is the total energy for all

particles in a halo group with a virial dispersion between 2 2 2[ , ]v v vd . For all particles in the

same halo group and with a given speed v, the instantaneous particle energy (kinetic and potential)

of each particle can be different and random. However, we expect that the average of the particle

energy of all particles with a given speed v from the same group can be approximated by 2h v

. Namely, the mean specific energy for all particles in the same group with a given speed v is given

by Eq. (14) and relatively independent of the particle speed v.

Term 2 is the fraction of particles with speed between [ , ]v v dv in that group due to the

Maxwellian distribution (assumption A2 and Eq. (7)). The integration is performed over all halo

groups with different virial dispersion 2v . The mean particle energy for all particles in entire

system is (from Eqs. (16) and (10)),

200

3 3

2h v v dvn

, (17)

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where parameters n and 20 fully determine the mean particle energy. It can be easily verified from

Eqs. (6), (14) and (16) that the particle energy distribution is related to the velocity distribution X,

263v v X v v

n

. (18)

The energy per particle v with given speed between [ , ]v v dv is the total energy v v dv

normalized by total number of particles Z v dv , i.e. the fraction of particles with a speed between

[ , ]v v dv . Hence,

2 63v v dv X v v

vZ v dv Z v n

. (19)

This particle energy v is not the instantaneous energy of a given particle. Instead, it is the mean

energy of all particles with a given speed v from all halo groups. Finally, a differential relation

between the X and Z distributions can be identified from Eqs. (6) and (7), where

2X

Z v vv

. (20)

Substitution of Eq. (20) into the Eq. (19) gives the particle energy v that is dependent only on

the X distribution,

3 3

2

X v vv

X v n

. (21)

2.4 The maximum entropy principle and the velocity distribution X

The principle of maximum entropy requires the velocity distribution with the largest entropy

and the least prior information. With assumption A3, the principle of maximum entropy is applied,

where the distribution X v should be a maximum entropy distribution under two constraints,

16  

1X v dv

, (22)

1X v v dv

. (23)

Here Eq. (22) is the normalization constraint for probability distribution and Eq. (23) is an energy

constraint requiring the mean particle energy to be a fixed constant 1 . The corresponding

entropy functional can be constructed as:

1 2 1ln 1S X v X v X v dv X v dv X v v dv

,(24)

where 1 and 2 are two Lagrangian multipliers introduced to enforce two constraints in Eqs.

(22) and (23). The entropy functional attains its maximum when the variation of the entropy

functional with respect to the distribution X vanishes such that

1 2ln 1 0S X v

X v vX

. (25)

The particle energy can be further expressed as (from Eq. (25))

12

1ln 1v X v

. (26)

By equating Eq. (26) with Eq. (21), a differential equation for distribution X v can be obtained,

2

1

3 21

2 1 ln

XvX

v nX

. (27)

The general solution of the X distribution from this first order differential equation is:

2 21 1 2

3 2exp 1 1

2

nX v C v

n

, (28)

where C is a constant of integration. To simplify the expression for X v , let’s equivalently

introduce three parameters , , and 0v to replace the original four parameters 1 , 2 , n , and C,

17  

1 1 , 2 C , and 0

2

2

3 2

nv

n

. (29)

The simplified expression for X distribution is:

220expX v v v . (30)

To satisfy the first constraint (Eq. (22)), we have:

0 12 1e v K . (31)

Finally, a family of distributions that maximize the system entropy can be obtained for one-

dimensional particle velocity (the X-distribution) that depends on two free parameters and 0v ,

220

0 1

1

2

v veX v

v K

, (32)

where yK x is a modified Bessel function of the second kind satisfying identity,

1 1

2x x xK K xK

. (33)

The velocity parameter 0v is introduced as a typical scale of velocity. The shape parameter

dominates the general shape of X distribution. The X distribution approaches a double-sided

Laplace distribution with 0 and a Gaussian distribution with , respectively. For an

intermediate shape parameter , we identify a Gaussian distribution

2

20 1 0

exp2 2

e vX v

v K v

for 0v v (34)

and a Laplace distribution

0 1 0

1exp

2

vX v

v K v

for 0v v . (35)

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It is shown that the X distribution naturally has a Gaussian core for small velocity v (with a

variance of 20v ) and exponential wings for large velocity v . This is a direct result of maximizing

the system entropy. Similar features are also observed from large scale N-body simulations [21].

Figure 2 plots the X distribution for four different shape parameters α=0, 1, 10, and , all with a

unit variance ( 2 20 1v ). With decreasing , the distribution becomes sharper with a narrower

peak and a broader skirt. The general shape of distribution can be further characterized by its

statistical properties that will be presented in Section 3.1.

Figure 2. The X distribution with a unit variance for four different shape parameters α. The X distribution approaches a Laplace distribution with 0 and a Gaussian distribution with , respectively. At intermediate α, the X distribution has a Gaussian core for small velocity v and exponential wings for large v.

The derivation of velocity distribution X requires the virial theorem for mechanical equilibrium

of halo groups and the maximum entropy principle for statistical equilibrium of global system.

-5 -4 -3 -2 -1 0 1 2 3 4 5v/

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

=0 (Laplace)=1.0=10= (Gaussian)

19  

Note that the halo mass function is not explicitly required to derive the maximum entropy

distribution X, which indicates that the mass function should be a direct result of entropy

maximization. Since the distribution X does not explicitly involve parameters characterizing the

system ( n and 20 ), additional connections may be identified between , 2

0v and n and 20 .

The Gaussian core of the velocity distribution X mostly comes from particles in small size

halos with small viral dispersion 2v , where the halo velocity dispersion is much larger than the

virial dispersion, i.e. 2 2h v . On the other side, the exponential wing of velocity distribution is

mostly due to the particles in large halos with 2 2v h . There exists a critical halo mass scale *

hm

where 2 * 2 *h h v hm m . The critical mass *

hm increases with the scale factor a (or decreases with

the redshift z). For small halos, the total velocity dispersion 2 2h with 2 0v . Therefore, it is

reasonable to assume that the variance of Gaussian core is comparable to the halo velocity

dispersion 20h  of small halos, i.e.

2 2 20 0hv ,   where 2 2

0 0h h hm . (36)

We will revisit this result in Section 4 using the particle energy result (Eqs. (15) and (55)).

3. Distribution of particle velocity (X distribution)

3.1 Statistical properties of X distribution

With velocity distribution explicitly derived in Eq. (32), some statistical properties of this

distribution can be easily obtained here and listed in Table 1. The moment-generating function is:

2

1 0

2

1 0

1

1

vtX

K v tMGF t X v e dv

K v t

. (37)

20  

The characteristic function is:

2

1 0

2

1 0

1

1

ivtX

K v tCF t X v e dv

K v t

. (38)

The mth order moment of the X distribution can be obtained as:

2

1 2

01

2 1 2m

mm mX

KmM m X v v dv v

K

. (39)

More specifically, the second order moment (variance) should be 20 ,

2 2 20 0

1

2X

KM n v

K

, (40)

which provides an additional relation between , 20v and 2

0 .  

  The mth order generalized kurtosis is defined as:

2

1 212

2 1

1 22

2

m

mXX m

X

KmM m KK m

K KM

. (41)

For example, 3XK is a skewness factor, a measure of the lopsidedness of the distribution and

4XK is a flatness factor indicating how far the velocity v departs from zero. The X distribution

with a smaller shape parameter α has a narrower peak and broader skirt, as shown in Fig. 2. Finally,

the Shannon entropy of the X distribution can be obtained explicitly as:

0

0 11

ln 1 ln 2X

KS X v X v dv v K

K

. (42)

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3.2 Comparison with N-body simulation

In this section, we will compare the theory with the particle velocity from a large-scale N-body

simulation carried out by the Virgo consortium. A comprehensive description of the simulation

data can be found in [22, 23]. The friends-of-friends algorithm (FOF) was used to identify all halos

from the simulation data that depends only on a dimensionless parameter b, which defines the

linking length 1 3b N V

, where V is the volume of the simulation box. Halos were identified

with a linking length parameter of 0.2b . All halos identified from the simulation data were

grouped into halo groups of different sizes according to halo mass hm (or particle number pn ).

We first verify the assumption A2 in Section 2.1 by computing the cumulative distribution

function of particle velocity for halo groups of different size at redshift z=0. For a Gaussian

distribution expected for particle velocity, the cumulative distribution should be an error function.

Figure 3 plots the cumulative function from N-body simulation (symbols) and the best fit of the

simulation data with error functions. The cumulative distribution of particle velocity (normalized

by 0 354.61u km s , the one-dimensional velocity dispersion of entire system at z=0) is computed

for halo groups of different sizes pn = 2, 10, 50, 100. Simulation data confirms the Gaussian

distribution of particle velocity for all particles in the same group, regardless of the halo size.

However, the particle velocity distribution for all halos (X distribution) can be non-Gaussian, as

derived in Section 2.4.

22  

Figure 3. The cumulative distribution of particle velocity (normalized by 0u ) in halo groups

of size hn = 2, 10, 50, 100. Gaussian velocity distributions are expected for particle velocity

of all particles in the same group. The corresponding cumulative distribution is expected to be error function. Symbols plot the original data from a large-scale N-body simulation and lines plot the best fit using an error function. Simulation data confirms the Gaussian distribution of particle velocity for all particles in the same group, while velocity of all particles in all halos can be non-Gaussian (the X distribution).

Figure 4 presents a comparison of theory (Eq. (32)) and simulation data for a one-dimensional

velocity distribution measured at a given scale r. We first identify pairs of particles with a given

separation r = 0.1Mpc/h at z=0. These pairs of particles likely resides in the same halo because of

the small separation r . The one-dimensional velocity Lu  is then computed as the projection of

particle velocity u  along the direction of separation r , i.e. Lu u r . Particle velocity is finally

normalized to have a unit variance and compared with the proposed X distribution. The best fit

leads to parameters 1.33 and 2 20 01 3v , where 2

0 var Lu .

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2v/u

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cum

ulat

ive

dist

ribu

tion

fun

ctio

nn

p=2fitted

np=10fitted

np=50fitted

np=100fitted

np=2

np=10

np=50

np=100

23  

Figure 4. The X distribution with a unit variance compared with one-dimensional velocity ( Lu normalized by Lstd u ) distribution from a N-body simulation. Vertical axis is in the

logarithmic scale (log10). The X distribution with parameters 1.33 and 2 20 01 3v

matches the simulated velocity distribution for small separation r , where both pair of particles likely reside in the same halo and different pairs can be from different halos. The Gaussian core and exponential wings can be clearly identified.

4. Distributions of particle speed and energy (Z and E distributions)

With velocity distribution (X distribution) explicitly derived in Eq. (32), the distribution of

particle speed for entire system (Z distribution) can be obtained from Eq. (20),

2202

3 221 0

0

1 v vv eZ v

K v v v

. (43)

Figure 5 plots the particle speed distribution (Z distribution) for different ( = 0, 1, 10, ∞) with

20 1 . The Z distribution approaches a Maxwell-Boltzmann distribution with .

Specifically, with increasing , Z distribution shifts towards large velocity with more particles

have an intermediate speed.

Pro

babi

lity

dis

trib

utio

n fu

ncti

on

24  

Figure 5. Distribution of particle speed (Z distribution) for different ( = 0.1, 1, 10, ∞). The Z distribution approaches a Maxwell-Boltzmann distribution with . With increasing α, the distribution shifts towards the large velocity with more fraction of particles having an intermediate speed and less fraction of particles having a small and large speed.

Some statistical properties of the Z distribution are also computed here and listed in Table 1. The

moment-generating function is:

0 1 2

00 1

22 3

2 !

m

mxtZ

m

v t KmMGF t Z x e dx

m K

. (44)

The mth order moment of Z distribution is:

1 2

1 2

01

2 3 2m

m mZ

KmM m v

K

, (45)

and the generalized kurtosis of different orders is:

2

1 212

2 1

2 3 22

32

m

mZZ m

Z

KmM m KK m

K KM

. (46)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5v/

0

0

0.1

0.2

0.3

0.4

0.5

0.6 =0=1.0=10=

25  

Finally, we have the particle energy v from Eqs. (19), (32) and (43),

2

2 20

0

3 21

2

vv v

n v

. (47)

In kinetic theory of gases, the energy of molecules is of a kinetic nature and proportional to 2v . 

While for collisionless particles in self-gravitating system, the particle total energy (including both

kinetic and potential) follows a parabolic scaling when 0v v  and a linear scaling when 0v v ,

2

20

3 21

2 2

vv v

n

for 0v v , (48)

and

0

3 21

2v v v

n

for 0v v . (49)

Figure 6 plots the dependence of normalized particle energy on the particle speed for five different

potential exponents n with a fixed 1.33 . Both parabolic and linear scaling are clearly shown

in Fig. 6 for small and large speeds, respectively. Dash line presents particle energy varying with

particle speed from a N-body simulation, where the one-dimensional velocity dispersion for all

particles in all halos is 0 395.18km s . The average energy (both kinetic and potential energy)

of all particles in all halos with a given speed v is computed for each speed v. The deviation at

large velocity might be due to the insufficient particles sampling large velocity. Simulation

matches an effective exponent 1.2n for virial theorem (not -1 because of nonzero halo surface

energy, see 1.3en for large halos Eq. (96) in [14]).

26  

Figure 6. The dependence of normalized particle energy v on particle speed v for five

different potential exponents n with a fixed shape parameter 1.33 . For small speed, the particle energy follows a parabolic law with particle speed ( 2v v Eq. (48)). While for

large speed, the particle energy follows a linear scaling with particle speed ( v v in

Eq. (49)) that is different from gas molecules. Particle energy from N-body simulation is plotted as the dash line. Simulation matches an effective exponent 1.2n for virial theorem (not -1 due to the mass cascade and nonzero halo surface energy [14]).

Finally, the mean particle energy is:

200

3 3

2h Z v v dvn

. (50)

The energy constraint in Eq. (23) can be obtained with Eqs. (47) and (32),

2

2 2 01 0 0 2

0

3 31

2

vX v v dv v

n

,  (51)

where 1 n with coefficient n depending on the exponent n. From Table 2, 1 2

(v)/

2 0

27  

for 0n and 1 for 2n (Gaussian distribution). By substituting Z distribution (Eq. (20))

into Eq. (50) and integrating by parts,

21 0

3 3

2

vX v v dv v

v n

 ,          (52)

such that the velocity scale 20v  is related to the difference between two energies. More work is still

required to better understand this relation.

The particle energy distribution (E distribution) can be found as (with Z v from Eq. (43) and

v from Eq. (47)),

2 2

21 0

2

3 2

Z v enE

d dv n K v

, (53)

where the dimensionless particle energy is defined as

20

2

3 2

n

n v

. (54)

With , there exists a maximum particle energy (or minimum in absolute value) from Eq. (54)

corresponding to particles in the smallest halo groups (Eq. (15)), where

2max 0

3 21

2v

n

.        (55)

For comparison, the energy distribution for a Maxwell-Boltzmann velocity distribution reads

20

2 20 0

12MBf e

. (56)

Figure 7 plots the energy distribution for three different potential exponents n=-1.5, -1.0, and -0.5

with a fixed 1.33 . For comparison, the energy distribution of Maxwell-Boltzmann statistics is

28  

also presented in the same plot. Compared to Maxwell-Boltzmann, more particles have low energy

and less particles have high energy for self-gravitating collisionless flow (SG-CFD) with n = -1.

Figure 7. The particle energy distribution for three different potential exponents n = -1.5, -1.0, and -0.5 with a fixed 1.33 . For comparison, the energy distribution of a Maxwell-Boltzmann velocity statistics is also presented in the same plot.

In principle, both halo virial dispersions ( 2v for a halo group) and the halo velocity dispersion

( 2h for a halo group) are functions of halo size. However, they can scale very differently with the

halo size, where 2 2h v for massive and hot halos and 2 2

v h for small halos. The halo virial

dispersion scales with the halo size as 2 1v hm ,  where 1 1 3n . To a first order

approximation, one may assume that 2h is relatively independent of the halo size. Therefore,

2 2h h hn is a constant.

The distributions derived in Sections 2 and 3 have two free parameters α and 0v , while the

system is fully characterized by the potential exponent n, particle velocity dispersion 20 , and the

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

| |/02

0

0.5

1

1.5

2

2.5

3

Pro

babi

lity

dis

trib

utio

nMaxwell-Boltzmannn=-1.5n=-1.0n=-0.5

29  

halo velocity dispersion 2h . Equation (40) provides a connection with 2

0 . Another connection

can be found by identifying the maximum particle energy v at 0v in Eq. (55), which is the

mean energy of all particles with a vanishing speed in all halos. The maximum particle energy

2h v among particles in all halos is 23 2 3 hn with 2 0v for particles in the smallest

halos from Eq. (15). Since most particles with small speed reside in small halos with 2 0v ,

2 20 hv , (57)

which is the same as we discussed in Section 2.4 (Eq. (36). With the help from Eq. (40) (the

variance of X distribution), we can write

2

120 2

h K

K

, (58)

from which it can be easily confirmed the two extremes 2 0h for 0 and 2 20h for

. A special case is that 2 2 20 2h v with 1.647 .

Non-bonded interactions can be generally classified into two categories: the short- and long-

range interactions [24]. A force is defined to be long-range if it decreases with the distance slower

than dr , where d is the dimension of the system. Therefore, the pair interaction potential is long-

range for 2n and short-range for 2n in a three-dimensional space with d=3.

For short-range force with 2n , we expect system is not halo-based with and X

distribution approaches a Gaussian for 2n . For long-range force with 2n , a halo-based

system is expected to maximize system entropy. With 0 , the X distribution approaches a

Laplace distribution for 0n . The shape parameter reflects the nature of force (short or long

range) and should be related to the potential exponent n. For systems with short-range interactions,

30  

Gaussian is the maximum entropy distribution where no halo structures are required. For systems

with long-range interactions, sub-systems (halo and halo groups) are required to spontaneously

form to maximize system entropy. While each sub-system still follows Gaussian distribution, the

entire system is non-Gaussian and follows a more general distribution (the X distribution). Table

2 lists the X distribution for different shape parameter and potential exponent n.

Table 2. The X distribution family and parameters for different potential exponents n

n 20v 2

h 2v X v Distribution

0 1 0 20 2 0 2

0 02

02

ve

Laplace

-1 3/2

2

12

2 0

hK

K

20 1

2

K

K

20 1

2

K

K

1 20

2

1K

K

220

0 12

v ve

v K

X

distribution

-2 3 0 20 0

2 202

02

ve

Gaussian

5. Conclusion

The limiting distributions of particle velocity, speed and energy at statistically steady state are

fundamental questions for self-gravitating collisionless fluid (SG-CFD). This paper presents a new

statistical theory to find the limiting distributions for self-gravitating systems involving a power-

law interaction with arbitrary exponent n. The halo-based description is a direct result of entropy

maximization and virial equilibrium for systems involving long-range interaction. The virial

theorem is applied for (local) mechanical equilibrium in halo groups. The maximum entropy

principle is applied for statistical equilibrium on the system level. The predicted velocity

distribution (the X distribution in Eq. (32)) naturally exhibits a Gaussian core at small velocity and

exponential wings at large velocity. Prediction is also compared with a N-body simulation with

31  

good agreement (Fig. 4). The speed (Z distribution in Eq. (43) and Fig. 5) and energy (E

distribution in Eq. (53) and Fig. 7) distributions of collisionless particles are also presented. The

standard kinetic energy is proportional to 2v , while the collisionless particles in halo-based system

have a total energy that follows a parabolic scaling when 0v v and a linear scaling when 0v v

(Eq. (47) and Fig. 6), where 0v is a typical velocity scale. The shape parameter of X distribution

reflects the nature of force (long or short range) and should be related to the potential exponent n.

For systems with short-range interaction, Gaussian is the maximum entropy distribution, and no

halo structures are required. For systems with long-range interaction, sub-systems (halo and halo

groups) are required to spontaneously form to maximize system entropy. While each sub-system

still follows Gaussian distribution, the entire system can be non-Gaussian and follows a more

general distribution (X distribution). Since particle velocity must follow the X distribution to

maximize the entropy of self-gravitating system, a broad spectrum of halos with different sizes

must be formed from inverse mass cascade to maximize the system entropy. That spectrum (mass

function) is given by the H distribution and is related to the X distribution via Eq. (6) .

32  

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34  

Table 1. Statistical properties of X and Z distributions

Distribution Name X Z

Support ( , ) [0, )

PDF

220

0 1

1

2

x ve

v K

2202

3 221 0

0

1 x vx e

K v x v

CDF

22022

0 0

1

11

2 ( )

x v

aJ x v x v e

Sign x K

220

1

J x v

K

Mean 0

3 20

1

8 Kv

K

Variance

2 20

1

Kv

K

2

3 22 202

1 1

83

KKv

K K

2nd Moment

2 20

1

Kv

K

2 20

1

3K

vK

Moments

2

1 2

01

2 1 2m

m mKm

vK

1 2

1 2

01

2 3 2m

m mKm

vaK

35  

Generalized Kurtosis

2

1 21

2 1

1 22m

mKmK

K K

2

1 21

2 1

2 3 22

3

m

mKmK

K K

Entropy 0

0 11

1 ln 2K

v KK

2 22

00 1

1 1

ln1ln 2

te t t dtKv K

K K

Moment-generating function

2

1 0

2

1 0

1

1

K v t

K v t

0 1 2

0 1

22 3

2 !

m

m

m

v t Km

m K

   or  ZMGF t  

Characteristic function

2

1 0

2

1 0

1

1

K v t

K v t

0 1 2

0 1

22 3

2 !

m

m

m

v it Km

m K

Maximum Entropy Constraint

2

02

0 1

1Kx

Ev K

where sJ x function defined as the integral: 0 

2 2x ts s

J x e t s dt , 0sJ s and 1sJ sK s 1 

20

2

0 1 01

2 201 0 1 0

1 1

2 1 2 1

v txtv te J K v t

P t X x e dxK v t K v t

2 

0

2xtZ

PMGF t Z x e dx P t t

t

3 

Euler Constant 0.5772 4 

36  

5