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: [email protected]; [email protected]
2
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
3
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
5
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].
8
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 ……
9
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.
10
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
11
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)
12
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)
13
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)
14
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)
15
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)
18
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)
21
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
Reference
1. Ogorodnikov, K.F., SvA, 1957. 1: p. 748. 2. Lyndenbell, D., Statistical Mechanics of Violent Relaxation in Stellar Systems. Monthly Notices of
the Royal Astronomical Society, 1967. 136(1): p. 101‐+. 3. Navarro, J.F., C.S. Frenk, and S.D.M. White, Simulations of X‐Ray‐Clusters. Monthly Notices of
the Royal Astronomical Society, 1995. 275(3): p. 720‐740. 4. Navarro, J.F., C.S. Frenk, and S.D.M. White, A universal density profile from hierarchical
clustering. Astrophysical Journal, 1997. 490(2): p. 493‐508. 5. Einasto, J. and U. Haud, Galactic Models with Massive Corona .1. Method. Astronomy &
Astrophysics, 1989. 223(1‐2): p. 89‐94. 6. Merritt, D., et al., Empirical models for dark matter halos. I. Nonparametric construction of
density profiles and comparison with parametric models. Astronomical Journal, 2006. 132(6): p. 2685‐2700.
7. Shu, F.H., Statistical‐Mechanics of Violent Relaxation. Astrophysical Journal, 1978. 225(1): p. 83‐94.
8. Tremaine, S., M. Henon, and D. Lyndenbell, H‐Functions and Mixing in Violent Relaxation. Monthly Notices of the Royal Astronomical Society, 1986. 219(2): p. 285‐297.
9. White, S.D.M. and R. Narayan, Maximum‐Entropy States and the Structure of Galaxies. Monthly Notices of the Royal Astronomical Society, 1987. 229(1): p. 103‐117.
10. Hjorth, J. and L.L.R. Williams, Statistical Mechanics of Collisionless Orbits. I. Origin of Central Cusps in Dark‐Matter Halos. Astrophysical Journal, 2010. 722(1): p. 851‐855.
11. Kull, A., R.A. Treumann, and H. Bohringer, Note on the statistical mechanics of violent relaxation of phase‐space elements of different densities. Astrophysical Journal, 1997. 484(1): p. 58‐62.
12. Padmanabhan, T., Statistical‐Mechanics of Gravitating Systems. Physics Reports‐Review Section of Physics Letters, 1990. 188(5): p. 285‐362.
13. Xu, Z., Inverse mass cascade of self‐gravitating collisionless flow and effects on halo mass functions. arXiv:2109.09985 [astro‐ph.CO], 2021.
14. Xu, Z., Inverse mass cascade of self‐gravitating collisionless flow and effects on halo deformation, energy, size, and density profiles. arXiv:2109.12244 [astro‐ph.CO], 2021.
15. Neyman, J. and E.L. Scott, A Theory of the Spatial Distribution of Galaxies. Astrophysical Journal, 1952. 116(1): p. 144‐163.
16. Jaynes, E.T., Information Theory and Statistical Mechanics. Physical Review, 1957. 106(4): p. 620‐630.
17. Jaynes, E.T., Information Theory and Statistical Mechanics .2. Physical Review, 1957. 108(2): p. 171‐190.
18. Jenkins, A., et al., The mass function of dark matter haloes. Monthly Notices of the Royal Astronomical Society, 2001. 321(2): p. 372‐384.
19. Colberg, J.M., et al., Linking cluster formation to large‐scale structure. Monthly Notices of the Royal Astronomical Society, 1999. 308(3): p. 593‐598.
20. Moore, B., et al., Cold collapse and the core catastrophe. Monthly Notices of the Royal Astronomical Society, 1999. 310(4): p. 1147‐1152.
21. Sheth, R.K. and A. Diaferio, Peculiar velocities of galaxies and clusters. Monthly Notices of the Royal Astronomical Society, 2001. 322(4): p. 901‐917.
22. C. S. Frenk, et al., Public Release of N‐body simulation and related data by the Virgo consortium. arXiv:astro‐ph/0007362v1 2000.
23. Jenkins, A., et al., Evolution of structure in cold dark matter universes. Astrophysical Journal, 1998. 499(1): p. 20.
33
24. Cheung, D.L.G., Structures and properties of liquid crystals and related molecules from computer simulation. 2002, Durham University.
34
Table 1. Statistical properties of X and Z distributions
Distribution Name X Z
Support ( , ) [0, )
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