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Advances in Astronomy

Volume 2008 (2008), Article ID 870804, 14 pages

http://dx.doi.org/10.1155/2008/870804

## Entropy Maximization, Cutoff Distribution, and Finite Stellar Masses

Department of Physics, Banaras Hindu University, Varanasi 221 005, India

Received 18 April 2008; Revised 16 July 2008; Accepted 26 August 2008

Academic Editor: Giovanni Carraro

Copyright © 2008 Ritesh Kumar Dubey et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Conventional equilibrium statistical mechanics of open gravitational systems is known to be problematical. We first recall that spherical stars/galaxies acquire unbounded radii, become infinitely massive, and evaporate away continuously if one uses the standard Maxwellian distribution (which maximizes the usual Boltzmann-Shannon entropy and hence has a tail extending to infinity). Next, we show that these troubles disappear automatically if we employ the exact most probable distribution (which maximizes the combinatorial entropy and hence possesses a sharp cutoff tail). Finally, if astronomical observation is carried out on a large galaxy, then the Poisson equation together with thermal de Broglie wavelength provides useful information about the cutoff radius , cutoff energy , and the huge quantum number up to which the cluster exists. Thereby, a refinement over the empirical lowered isothermal King models, is achieved. Numerically, we find that the most probable distribution (MPD) prediction fits well the number density profile near the outer edge of globular clusters.

#### 1. Introduction

It is well known that standard methods [1, 2] of equilibrium statistical mechanics run into conceptual difficulties when applied to gravitational bound systems [3–11]. Basically, these troubles arise due to the peculiar behaviour of the gravitational interaction (either the pair potential or the mean field) at short or long distances. The aim of the present paper is to focus attention on the triple problems, namely, unbounded radius, infinite mass, and continuous evaporation of every stellar/galactic system described by the conventional Maxwell-Boltzmann (hereinafter referred to as the MB) distribution.

Section
2 below points out that since the MB function maximizes only the simple-minded
Boltzmann-Shannon entropy, its tail becomes illogical in the energy cells of
small occupancy. The ensuing problems of the Maxwellian distribution cannot be really overcome by
using ad hoc prescriptions such as enclosing the system in a hypothetical box
[7] or modifying the Maxwellian form
empirically by invoking gravitational tidal cutoff
[4, 8]. Next, Section 3 presents a detailed derivation of our most
probable distribution (MPD) * f* by
taking hints from a preliminary investigation by Menon and Agrawal
[12] in
the molecular context and by Menon et al. [13] in the cosmological context.
Such *f* maximizes rigorously the more
sophisticated combinatorial entropy and the corresponding variational
conditions dictate that *f* must
possess a sharply truncated tail. Next, Section 4 demonstrates how our MPD idea
applied to cosmology resolves the aforesaid troubles of the MB formalism, and
how the Poisson equation brings additional features into our theory. We feel
that the MPD philosophy may have bright applicational prospects in fitting
cosmological data such as the classic study of stellar number densities in
globular clusters done by King [14, Figure 2] and the important measurements
performed by van Loon et al.
[15, Figure 6] showing velocity distribution on the
post-mail-sequence stars in Centauri. Finally, the paper ends by
presenting several concluding remarks in Section 5 where some other approaches
to the subject (namely, self-consistent Hartree calculations, incomplete
relaxation in low-density tail, canonical ensemble treatment of virialization, occurrence
of a stellar mass spectrum in real gravitating systems, etc.) are also
mentioned.

Some related aspects of algebraic interest are reported in two useful appendices. Careful study of Sections 2 and 3 will reveal that quantum mechanical discretization of the single-particle levels is very convenient for setting up the combinatorial entropy and in finding the cutoff number; hence for the sake of ready reference we collect in Appendix A several known formulae concerning semiclassical one-body spectrum as well as the energy cell occupation number Also, a detailed treatment of our variational conditions in Section 4 requires that be replaced by everywhere (even for ); hence Appendix B tells why derivatives of factorials or gamma functions can be readily taken even in the cells of small occupation numbers.

#### 2. Difficulties with the MB Distribution

##### 2.1. Preliminaries

This section
begins by quickly recalling a standard derivation of the famous
Maxwell-Boltzmann distribution in equilibrium statistical mechanics. Particles are assumed to be moving in spatial dimensions at temperature under the influence of a mean field
potential energy The one-body
energy spectrum is divided into cells, particles are distributed at random over these, those in the th cell are regarded as
mutually identical, and the simple-minded Boltzmann-Shannon entropy functional is set up. Here is the Boltzmann constant,
the cell degeneracy, the total
number of particles, the total energy, and and are Lagrange multipliers (see Appendix A for
precise definitions of various symbols). Next, one maximizes with respect to and to arrive at the MB solution where the index *j* has been dropped in the quasicontinuum limit, is the ground level, and the upper end of the
simple-particle energy spectrum has been extended to both for confining as well as nonconfining
potentials Although (2) has
been widely applied [1, 2] to gases/liquids kept in
the laboratory, yet its application to open astronomical systems leads to the
following serious conceptual puzzles.

###### 2.1.1. Entropy

In the case of gravitational
systems, one always looks for the *local* (not global) maxima of the entropy functional. The MB solution
(2) does this
job *exactly* for the
Boltzmann-Shannon entropy defined by (1), but only *approximately* for the more sophisticated combinatorial entropy
defined by (10) later. It will be shown in Section 3 that *the tail of the MB solution becomes illogical in the energy cells of
small occupation numbers.*

###### 2.1.2. Density

If (2)
is inserted back into the general expression (A.7) of
Appendix A for the mass
density one obtains the famous Boltzmann barometric
formula
The attractive
short-distance behaviour of cannot pose a real problem because the size of the *quantum* ground level is finite [6]. But the long-distance
behaviour of is problematical as
regards astrophysics in dimensions. Indeed, for a dilute gaseous star
[2, page 114]
without the Poisson equation constraint, one finds asymptotically Also, for the
isothermal Emden
sphere [3, 8] subject to the Poisson
equation constraint, one knows that with *a* being the isothermal length scale.
Clearly, *as the
nil/slow decrease of in (4) and (5) and the logarithmic increase of
W in (5) are unphysical.*

###### 2.1.3. Radius

From the MB density (3), one computes the
mean size of the system via *which**diverges both for gaseous stars (4) and Emden
spheres (5)*.

###### 2.1.4. Mass

The total mass of the MB system is
calculated from which also diverges
for the two cases mentioned above. Thus, *in
the Boltzmann-Shannon view, the most likely state of an isotropic stellar
system has infinite mass.*

###### 2.1.5. Evaporation

Since all regions of the phase space up to are allowed an open MB system, for example,
the dilute gaseous star goes on evaporating with time, *producing**as a**
thereby a net outgoing flux of particles [2, page 114] at
every positive thermodynamic temperature T*: Of course, the
isothermal sphere can be stable against evaporation [9] but its
mean field growing like up to is unphysical.

*King-like Lowered Isothermal Models*

In the conventional
literature, the above difficulties are usually circumvented by enclosing the
system within a hypothetical box of some radius [7], or by
modifying the original distribution heuristically into non-Maxwellian form such as and so forth, holding in the range and vanishing elsewhere
[4, 8, 16–19]. In particular, King [4] and
Wilson [19]
appealed to the tidal force field of the galaxy for physically setting its
outer boundary and assumed the velocity distribution of stars to be cut off at
the local escape velocity. Lowered isothermal prescriptions such as (9) are
often employed by astronomers to fit data.

*Physical Motivation for *

If one takes a stellar cluster in an original Liouville
collisionless state then the cluster will start evolving in space-time through trajectory
mixing and stellar encounters which are most frequent in the core region.
Mathematically, the complicated dynamics of such a nonequilibrium system is
governed by the coupled Fokker-Planck and Poisson equations
[17]. Physically, this evolution will involve momentum/mass/heat flow, tide
generation, and entropy production. At equilibrium, the macroscopic flows will
stop, tides will stabilize, and the entropy would become maximum. Naturally,
von Hoerner [20] and
King [14] realized that a finite boundary to the star
cluster is set up by the tidal force of the galaxy, that is, the cutoff tail in the essentially *classical* stellar systems can be
ascribed to the physical outcome of the boundary conditions and/or constraints
(independent of the Plank constant).

#### 3. Our Most Probable Distribution (MPD)

##### 3.1. Preliminaries

We adopt the view
that the above-mentioned King models can be refined further by utilizing the
following facts. (i) At equilibrium, the entropy of a multiparticle thermal
system should become a *(local) maximum.* Of course, the Boltzmann-Shannon definition of in (1) will not serve the purpose due to the
difficulties of the Maxwellian; we shall show in (11) and (12) below that a
more suitable candidate is the so-called combinatorial entropy *S* that counts the number of microstates
in energy cells corresponding to specified total particle number *N* and total energy
*E.* (ii) The resulting most probable distribution *f* should develop a tail which is *automatically truncated* at a finite
energy
This is because a star moving in the mean potential field will have a farthest turning at distance satisfying where may now be identified with the classical King
radius of the galaxy. (iii) By Bohr’s
correspondence principle, classical motion is the limiting case of quantum
motion in states of very *large quantum**numbers*. The cutoff quantum number *K* and cutoff energy should be determinable from the variational
constraint equations of our MPD theory provided that
*h* is
brought into the picture explicitly. (iv) Our MPD solution for *f* should be able to provide a
theoretical justification (or better characterization) of the lowered
isothermal Maxwellian models (9). Now we shall demonstrate how such a task is
accomplished in practice.

*Gibbs Combinatorial Entropy*

We follow the basic theme of
Huang [1, page
182], and a preliminary investigation by Menon and Agrawal
[12] as well as
by Menon et al. [13]. The single-particle spectrum is divided into
*J* cells into which the particles are
distributed at random such that the th cell has central energy
width
degeneracy occupation
number
and occupation probability per state defined by
Appendix A. Next, treating the particle in the th cell as
indistinguishable, a Gibbs combinatorial entropy functional *S* is constructed via

*Gamma Function Form*

We deliberately rewrite (10) in the
equivalent form The replacement of
factorials by gammas has several algebraic advantages. (i) The equality is exact at the *integer* values
(ii) The *asymptotic* behaviour, namely, of both and are the same as
(iii) Hence, by a theorem due to Carlson
[21, 22], provides the most economical, essentially
unique continuation of to all *continuous* values throughout the range
(iv) While setting up the variational conditions, later we shall need to replace
the derivative evaluated at integer values by the digamma
function [23] computed at
general continuous values. This problem of *integer
programming* is handled in Appendix B by using an efficient
finite-difference package for all natural numbers up to 4. (v) Appendix B also
shows that the
numerical differentiation of can be readily done even at *small values* and so forth, giving results in good agreement with

*Exact Variational Conditions*

Next, we consider the following
objective functional to be maximized: where and are unknown Lagrange multipliers. Equating to
zero, the partial derivatives
and lead to the following set of exact variational
conditions still using the *discrete* index
[12]:

*Comments**Without making any assumption concerning**the largeness or smallness of we can rewrite (13a) in the compact form* where the symbol was already encountered earlier in (2). In
principle, (14a) can be solved for the desired cell occupation numbers in terms of
Thereafter, the Lagrange multipliers and can be determined from the constraints (13b).
Equivalently, the chemical potential and thermodynamic temperature *T* may be introduced via
Finally, if the
total number *J* of levels is very
large, we are permitted to take the quasilimit (A.4) leading to a *continuous* distribution for versus by dropping the index *j* and converting sums into integrals. Let us derive several
interesting properties of our most probable distribution (MPD) defined by
(13a), (13b)
and (14a), (14b) with the suffix *j* omitted.

*Location of The Peak*

Differentiating (14a) with respect to we get Clearly, *the MPD occupancy and MB occupancy are both peaked at a common energy which satisfies* where the dot stands
for derivative with respect to
Typical algebraic estimates of for the soluble potential models will be
reported later in (28).

*Large Region*

In the so-called head
region of the continuous distribution, the cells have large occupancy so that the Stirling’s approximation holds in the fundamental equation (14a). *Hence the MB solution is roughly retrieved,
namely,**but it must be violated in the cells where the occupancy becomes
comparable to, or less than, unity.*

*The Tail Region*

On the other extreme lies the tail region of the
continuous distribution where the cell occupancy becomes small, that is,
Then the digamma function possesses a Taylor
expansion where is the negative of Euler’s constant and is a Riemann zeta value. Substitution of the
expansion (18) into the fundamental equation (13a) leads to the following three
surprising yet important observations.

(i)*The tail of the distribution
intersects the energy axis at a cutoff
point such that* where the suffix *K* refers to energy (ii)
The said intersection happens linearly because, in its neighbourhood,
the occupancy where stands for evaluated at (iii)*Extension
of the graph of versus beyond the cutoff point is not allowed because
that would tend to make negative in (13a), that is,**implying that the original occupied spectrum (A.2) has shrunk below J or due to strict entropy maximization under
stable equilibrium.* (The possibility would correspond to unstable equilibrium, that
is, continuous evaporation of the system.) Schematic plots of and versus are shown in Figure 1. Typical algebraic
estimates of the cutoff energy and quantum number *K* for the solvable models will be reported later in
(28).

*Compact Solution for *

Our equation (14a) is a transcendental
equation in and its precise analytical solution in closed
form is not known. Fortunately, there exists an ansatz which works
excellently throughout as shown graphically in Figure 2. *Combining (14a) and
(22), we obtain a very
compact, quite accurate, MPD solution valid in all energy cells of relevance as* It is interesting to
note that if *were* replaced by a
constant in (23), our MPD solution would agree with the first line of (9) *implying a sort of justification for the
lowered isothermal Maxwellian models*. Actually, our numerically accurate
solution (23) should be regarded as a better characterization since the
degeneracy function is strongly energy-dependent.

*Compact Number Condition*

Combining the number constraint (13b) with the
general solution (23), we can define an effective number
through
Here we have
employed the quasicontinuum limit (A.4) and
introduced *This gives a formal expression for the Lagrange multiplier (or reduced
chemical potential) provided that the underlying mean field W or
its reduced degeneracy function is known.*

*Compact Cutoff Condition*

Lastly, we
convert the cutoff criterion (19) into Eliminating with the help of (24b), we find *which yields a formal expression for the cutoff energy whose functional dependence can be inverted to
specify also the number K of levels.* The sharply cutoff tail of (21) will
play a crucial role in the cosmological application to be discussed later in
Section 4.

*Illustration for the (Truncated) Oscillator Well*

The above
methodology may be illustrated in the case of the truncated harmonic oscillator
potential listed in Table 1: Before going ahead
with the algebra, the following important remarks should be kept in mind.
(a) If the well was untruncated, that is,
the step function in (26) was absent then all particles would remain truly
confined, the Boltzmann mass density (3) would vanish asymptotically, and the
MB distribution would not be problematical. (b)
However, if the well is truncated by the use of the step function in
(26), then the potential vanishes for
particles can be ejected into the continuum, the MB distribution becomes
problematical, and the MPD philosophy becomes very useful. (c) Near the origin
the oscillator potential is rather flat, that is, smoothly varying so that it
can approximately mimic the realistic mean field in the core region of
astronomical galaxies. In sharp contrast, the truncated-linear and Coulomb-like
potentials of Table 1 cannot do so since these vary rather quickly as (d) As
they stand, the depth and range are only illustrative parameters introduced in
(26). However, when we come to cosmological applications in
Section 4
(especially the Poisson equation), it will be found that these parameters are
directly related to the physical mass *M* and observed radius of the galaxy. We are now ready to apply the
MPD program to (26).

*The Tilde Nomenclature*

First, we read off the symbols
and from the fourth column of Table 1. Next, for
algebraic convenience, the following dimensionless quantities are defined along
with the thermal de Broglie wavelength : A few remarks are in
order concerning these definitions. The Planck constant or thermal de Broglie wavelength has appeared in the value of the symbol and the inequality is essential for the validity of classical
motion (cf. (A.8)). The function measures the single-particle energy from the
ground level in terms of
The symbols and may be called the dimensionless chemical
potential and dimensionless cutoff energy, respectively, whose fixation using
MPD constraints is yet to be done. The integral will play a crucial role below with being the incomplete gamma function.

*Use of MPD Conditions*

Remembering the tilde quantities, we
can readily evaluate the conditions (16),
(24b), and (25b). This yields the
peak location peak height dimensionless chemical potential
and dimensionless cutoff energy through where the wavy
symbol ~
implies the order of magnitude, and the multiplicative factors of order
unity have been suppressed. We still have to show that the formal equations
(28) do admit valid, that is, self-consistent MPD solutions under suitable
restrictions. For this purpose, we consider below two cases in which the
parameter has markedly different behaviours.

*Case 1 (well depth large compared to times temperature). *
For the truncated oscillator potential
(26), we recall the tilde notations (27)
and impose the following inequalities: The physical meaning
of these restrictions is as follows. The inequalities guarantee the validity of classical dynamics
in states of large quantum numbers, the
condition ensures that the th level lies below the ionization threshold for
stable MPD, the assumption in Case
1 implies that the actual cutoff energy is several above the ground level, means that the well depth is large compared to
the condition implies that the effective number of particles grossly counted per cell is much
more than unity, and means that the system is dilute or
nondegenerate (because the packing fraction that is, the average number of particles
contained inside a -dimensional sphere of radius is small compared to unity). Then the
incomplete gamma function and consistent handling of (28) leads to the
estimates The present case
should apply to usual gases/liquids contained in the laboratory and we have
independently verified that the functional forms of (29) and (30) are very
rugged, that is, they hold for all the soluble models reported in
Table 1.

*Case 2 (well depth comparable to times temperature). *
Again we recall the tilde notations (27) and
impose the orders of magnitude Then the incomplete
gamma function and (28) are found to admit the
self-consistent estimates The physics of (31)
and (32) is as follows. The statement applies to gravitational systems obeying
virialization, tells that the total number of particles is of
the same order as the number of MPD cells, and signifies that the cell occupancies have
become comparable to unity with again playing a role through the symbol
The present case should correspond to open astronomical systems and the
ruggedness of the results (32) can be verified also for the other solvable
models in Table 1.

#### 4. Conceptual Application of MPD to Cosmology

We are now ready to resolve the conceptual difficulties of the MB distribution mentioned already in Section 2 by employing the MPD solution obtained in Section 3.

##### 4.1. Entropy

The Boltzmann-Shannon
entropy of (1) is simple-minded, its maximization leads to
the MB solution in (2) with untruncated tail, and its
generalization to quantum statistics is difficult. In sharp contrast, the *combinatorial entropy S of (11) is
sophisticated, its maximization leads to our MPD solution in (23) with a truncated tail, and its
generalization to quantum statistics as straightforward.*

##### 4.2. Density

If the MPD information (21) is
inserted back into the general expression (A.7) for the local mass density
based on the transformation
we obtain where surprisingly vanishes if *W(r)* equals
This is explained by remembering that since no particle in MPD is allowed to
have an energy more than *there exists a largest classical turning
point at beyond which the density must become zero
identically,* that is, in sharp contrast to
the MB density profiles (3)–(5). We can also
find the rate at which approaches zero as tends to
However,
(20) has already told us that in the tail region. Hence, (33) yields the
leading behaviour Since in a “good”
MPD solution * and are finite, our result (35) tells that the
mass density obeys a law near the edge of the system.*

##### 4.3. Radius

Clearly, the distance in (34) is the upper bound on the size of our
galactic system and, for binding, we must have with being the turning point just before the
continuum starts. (In the soluble models of
Table 1, this was called ). *Since the density vanishes beyond the MPD integral defining the average size** will also converge,* that is, in sharp contrast to
the MB mean radius (6).

##### 4.4. Mass

By the same token, the MPD integral defining the total mass of the stellar system also exists, that is, in sharp contrast to the MB mass (7).

##### 4.5. Nonevaporation

As is well known if an attractive mean field vanishes asymptotically, then the energy is called the ionization threshold. Hence, our galaxy will be stable against evaporation if the MPD cutoff energy happens to be negative at the given thermodynamic temperature Consequently, for there is no net outgoing particle flux, that is, in sharp contrast to the MB result (8).

*Comment*

Of course, *a galaxy which is observed experimentally to
evaporate is not in true equilibrium.* Then simplifying restrictions like
(31) may not hold, that is, the cutoff conditions (19) and (32) will admit a
positive root for

*Poisson Equation Implications*

So far in our treatment, the
detailed algebraic form of the mean field *W(r)* was not required explicitly for self-gravitating systems. Actually this is a
tough problem theoretically/numerically because one must solve the coupled
equations for the distribution function *f* and mass density in accordance with the Poisson equation in *3* dimensions where is the
gravitational constant. Our limited aim in the present paper will, however, be
served by noting the following features.

##### 4.6. Features

(a) Since the density is sharply cutoff at
by Gauss theorem, the exact potential energy and force at *exterior* points become

(b)
At the *edge itself,* the potential energy becomes equal to the cutoff
energy, namely,

(c)
In the *interior* region, the mean field may get smoothened so as to yield a
finite depth by virtue of the
gravitational virial theorem.

(d) At *interior* points, the exact profile of the mean field is not known a priori since it has
to be, in general, computed numerically by solving (39). However, for the
purposes of illustration, we can represent it by an oscillator form if with unknown phenomenological constants and *R*.
The corresponding interior potential energy and force at the system edge then become Matching these to
the exterior values given by (40) at
we identify Thus, is deeper than and *R* is larger than (although orders of magnitude are the same).

*Suggested Procedure for Cosmologists*

Suppose a practical astronomical observation has been made on a
cluster of stars.
For utilizing our MPD theory with respect to his collected data, the
cosmologist should proceed through the following steps.

*Step1 (characterization parameters). * From the observed size and the known mass *M* of the cluster, the MPD cutoff energy is immediately given by (41) as
Next, the oscillator well-depth and the range parameter *R* for
motion inside the cluster are set up from (44) as and
Next, according to (32) applicable to cosmology, the MPD parameters have the
rough orders of magnitude upon using the value
of given by the first line of (27) under
virialization. Next, the cosmologist may treat (45) as providing a new mass
versus radius relationship for nondegenerate clusters whose experimental
status is, however, not yet studied. Finally, for an accurate interlink among
all MPD parameters, the astronomer may like to solve the transcendental
equations (27) numerically.

*Step2 (MPD density near the edge). *
Next, the astronomer may look at (35)
which gives the leading behaviour of the stellar number density near the
cluster’s boundary: since the mean field *W(r)* becomes Newtonian near the
periphery. This can be cast into more convenient form by defining the variable choosing a normalization point
and
working with the modified function which becomes unity
at but
vanishes at
To test the validity of (47), the astronomer may, for example, concentrate on
the star counts made on photographs of the cluster M 15
(see Figure 2 of King [14]) taken with the 48-inch Schimdt camera in the Palomer
observatory. The results of are plotted in Figure 3. Clearly, there is
quite good agreement between experimental observation and MPD prediction,
although a slight curvature in the data trend may imply the presence of
additional weak nonlinear terms on the RHS of (47).

*Step3 (comparison with King density). *
Next, it is worthwhile to consider the function and expand its MPD expression (47) around the
matching point binomially in the form Dropping the term, the cosmologist retrieves the famous
formula proposed empirically by King, namely, whose square gives
the King’s profile [14, equation (2)] near the cluster’s periphery as It is well known
that the phenomenological proposal (49) has been extensively used in the past
by astronomers. For example, in context of M15 cluster
Figure 4 shows the plot
of near
the cluster’s boundary. Clearly, the agreement between experimental observation
and King’s parametrization is good, ignoring the slight curvature in the data
trend. Incidentally, the qualities of fit seen in Figures 3
and 4 are quite
comparable implying that, with the present accuracy of measuring
it is not possible to say whether MPD formula (47) or
King recipe (49) is
superior.

*Step 4 (complete density profile). *
Finally, the astronomer may like to have
an expression for the number density valid throughout the range
In principle, our MPD distribution function *f* given by (23) yields the formal expression with the mean
potential being approximately
harmonic oscillator in the interior and Newtonian near the edge. Unfortunately,
analytical evaluation of the phase space integral (51) is somewhat tedious and
will be dealt with in a future communication. However, the cosmologist should
note that the integral (51) is the algebraic difference of two terms which is
very satisfying because *the empirical
full density profile written by King [14, equation (14)] also contains a difference of
two terms.*

*Step 5 (velocity distribution of stars). *
It is a
standard astronomical practice to measure the local radial velocity
distributions (along with other properties) of stars in a globular cluster, for
example, see the extensive photometric study made by van Loon et al.
[15] on the
post-main-sequence stars in Centauri (NGC 5139). The cosmologist may ask
how well our MPD distribution function *f* given by (23) fits the observed data.
Unfortunately, a straightforward answer to this question is difficult because
exact values of the unknown parameters and *K* must be obtained numerically from the
transcendental conditions (24b) and (25b). We plan to accomplish this task in
a future communication.

#### 5. Concluding Remarks

The main results of the present work
appear in the abstract along with Sections 2–4 and are often
emphasized by italics. It is hoped that astronomers will benefit from the
algebraic properties of MPD derived in Section 3, its cosmological implications
mentioned in Section 4, numerical plots of number density profiles in Figures 3
and 4, and a pointwise comparison between the King model and MPD philosophy
made in Table 2. Clearly, both types of theories can be applied to cosmology
although our *f* may be regarded as providing a better
characterization from the conceptual viewpoint.

The
essence of our cosmological discussion in Section 4 is the following.
*Suppose that an astronomer makes
observations on a (quantum mechanically nondegenerate) cluster having N stars,
total mass M, and radius
Then, its MPD solution will be characterized by the cutoff energy and cutoff quantum number * Before ending the paper, we mention below briefly several important points not
discussed explicitly in the earlier sections.

(i) In the mean field description of a
multiparticle system, *fluctuations* arising
from short-range pair correlations are usually ignored. The effect of
fluctuations is likely to be stronger on the MB solution
whose tail extends to in (2).
Such effect is likely to be weak on the MPD solution *f* whose tail gets truncated at in (23).

(ii) One may argue that a sharp radius is also
known to arise in the method of self-consistent *Hartree* fields applied to gravitational systems
[17]. We stress, however, that the Hartree method is done through a numerical
algorithm because the coupled equation for the mean field and distribution
function must be solved iteratively on a computer. Therefore, our analytical
maximum-entropy treatment of Section 3 still retains its novelty.

(iii) One may also
argue that it is not meaningful to demand thermodynamic equilibration in the
peripheral region of the galaxy because, due to low densities, relaxation may
remain incomplete there. However, it
must be kept in mind that since gravitational forces are of long range, the
mechanism of *collisionless relaxation* [9] still
operates. Therefore, our assumption of
equilibration even in the tail region may remain justified.

(iv) Next, mention must be made of some
recent investigations [10, 11] carried out
on the question of gravitational galactic clustering, their virialization, and
peculiar velocity distribution superposed over the local Hubble flow. These
authors start from the *N* body
cosmological canonical partition function in a box of large
volume perform the individual
momentum integrals at the outset over the infinite domain
write the entropy *S* as the logarithm
of a Gibbs integral over the density of states, and minimize the Helmholtz free
energy with respect to the internal energy *E*. Of course, these investigations are
very *different* from our work because
we do not need an enclosing box, momentum integrations over infinite domain are
never performed, the entropy functional is combinatorial, and maximization is
done with respect to the cell occupation numbers.

(v) Next,
suppose that one considers a time span long compared to the two-body relaxation
time in a globular stellar cluster. One may argue that a star having energy (i.e., arbitrarily close to zero but still
negative) will go far away and yet come back. Since the corresponding turning
point may be arbitrarily big, one expects a very *small* (but not zero) possibility of the
star’s existence even at a very large radius. This logic apparently contradicts
the MPD result (34) which had claimed that there is no density outside a finite
distance

Actually,
the above logic has the following very subtle fault. While doing pure dynamics,
it is enough to find trajectories and their turning points
but while doing statistical mechanics, it is essential also to calculate the
density profile and the related total mass
Now, in direct analogy with (35) but with the density profile at large distance and its
associated Poisson equation become (in dimensions) This result is physically unacceptable
because the gravitational potential due to a finite mass object must fall
asymptotically like Hence a
logic based on will not work. In sharp contrast, *if the globular cluster has finite
experimental mass then it can be easily described by our MPD solution (34)
characterized by bounded and finite *

(vi) Finally, a cosmologist may argue that since real gravitating systems
have a mass spectrum of stars, the assumption of particles with the same mass *m* in MPD may not be justified. We wish
to point that some workers have attempted to apply hydrodynamical equations to
globular clusters employing a phase space density involving the continuous mass
[24] as an extra variable. Some other workers have analyzed
phenomenologically the mergence
of clusters such as Praesepe [25] employing four mass bins.
Although, in principle, a multicomponent combinatorial entropy will now replace
(10), yet the corresponding variational conditions
(13a) and (13b) will be hard to handle analytically
because different chemical potentials and different cutoff energies may have to
be assigned to various components present in the system. An easy approximation
will be to still use the MPD formalism of Section 3 based on the single
particle average mass where the suffix runs over different
species and there are particles of the th
type. This prescription should be
reasonable for those clusters where the mass dispersion is small (in units of
the solar mass).

#### Appendices

#### A. One-Body Description Recapitulated

##### A.1. Preliminaries

This section will summarize our notations along with several known formulae dealing with the semiclassical single-particle spectrum/distribution without invoking entropy constraints. Some of these formulae will be used explicitly in Sections 3–5 of the text.

##### A.2. Assumptions and Notations

Consider the
nonrelativistic localized motion of a particle in spatial dimensions under
the influence of a smooth attractive central field. *Classically,* the symbols respectively, denote
the mass, distance, absolute momentum, potential energy, applied force, and
mechanical energy of this particle. *Quantum* mechanically invokes the Planck constant and solves the Schrödinger equation for
determining the energy spectrum where is the ground level and the highest
bound level supported. Of course, solution of the Schrödinger equation
for the exact eigenvalues, eigenfunctions, and their degeneracy is generally
tedious.

##### A.3. Sommerfeld Quantization

Perhaps the easiest semiclassical link between the
descriptions (A.1) and (A.2) is provided by Sommerfeld’s criterion
[26] which says that the phase integral or action variable over a
complete oscillation should be an integer multiple of Then a discrete level
in (A.2) corresponds to the classical turning point
local momentum variable principal quantum number
and level spacing given by Since the presence
of zero point energy is of little consequence here, hence the more
sophisticated WKB quantization [27] will not be needed for our
purpose. Also, if the number *J* of
supported levels is very large compared to unity, then the *quasicontinuum limit* can be taken by writing

*Gibbs’ Prescription*

Further
information is obtained by imagining a spherical region of range *R* and remembering that several quantum
states of different orbital angular momenta and magnetic projections may be
nearly degenerate at a given energy level. Then, the *D*-dimensional solid angle
total volume *V* of the region, useful
geometrical factor *A*, Gibbs’ phase space
element
accumulated number of quantum states below
local number of states
per unit energy interval, and the degeneracy
of the th level itself are read off from Here is the
single-particle Hamiltonian, the quasicontinuum limit (A.4) is understood, is the gamma function, step the unit step
function, and the delta function.

*Solvable Potential Models*

The methodology described by (A.3)–(A.5) is best
illustrated in the case of 4 soluble models, namely, the rectangular, truncated
linear, truncated harmonic oscillator, and Coulomb wells. The results are
summarized in Table 1 and the following features are worth noticing.

(i)
In the case of the rectangular, linear,
and oscillator wells, the range *R* represents the distance beyond which the particle goes into the continuum. By
the same token, the highest level *J* is fixed through the requirement that (ii)
For the Coulomb well, however, since the
bound orbits can have any size, one sets By the same token, the ionization threshold
appears at (iii)
In every model of Table 1, the
semiclassical energy * ε* increases
monotonically with the principal quantum number

*j*, but the trend of the level spacing Δ is not uniform.(iv) In every soluble model, the level degeneracy where the geometrical factor is of order unity. Hence it is reasonable to expect that, for a more general attractive central field in dimensions, at least for large (v) Table 1 does not explicitly treat the infinite rectangular well, that is,

*rigid box*in which particles of any momentum would remain confined. Then, the highest kinematically allowed level would have using the many-body notation of (A.6) below. Of course, the rigid box model is irrelevant in cosmology.

*Multiparticle, Statistical System*

In the present paper, we shall *not* consider pure Bose/Fermi many-body systems where the strict quantum mechanical
identity of all particles is crucial. Ours is the so-called *Boltzmann* system where the one-body
spectrum is obtained from the Schrödinger equation/semiclassical quantization
but strict identity among all particles is not imposed except within the same
energy cell. The mean field of (A.1)
may be either externally applied or internally generated. Assuming spherical
symmetry, independent-particle motion, and ignoring short-range pair
correlations, we let the symbols respectively, denote the specified number of particles, total energy, global
average number density, global mean thermodynamic temperature, Boltzmann
constant, inverse temperature parameter, and thermal de Broglie wavelength. The
one-body phase space may be imagined to be composed of the differential
elements (cf. (A.5)) or of *J* energy cells of successive widths which are arranged in the sequence (A.2). Then
a useful transformation
single-particle energy
one-body distribution function
cell occupancy
local number density local mass
density
total number *N*, and total energy *E* are read off from Two crucial comments
are in order at this stage. (i) The
functional form of is left unspecified at the moment. (ii) Convincing justifications are still needed
for retaining in our mechanical as well as statistical
expressions (A.3)–(A.7) especially
when application to classical galaxies of enormous sizes is being envisaged.

*Importance of Planck Constant*

(a) By Bohr’s correspondence
principle, the motion of a quantum Schrödinger/Sommerfeld particle tends to become
classical in the states of large principal quantum numbers. In the notation of
(A.4), this requires where *J* of
Table 1 contains explicitly.
(b) Strict Bose/Fermi statistical
systems tend to obey classical statistics at low density and high temperature
if in the notation of (A.6). This requires that
the linear size of the system be large compared to the thermal de Broglie
wavelength, that is, *Hence
the -dependent dual inequalities**tell very precisely when a multiparticle system can be called* “classical.” Such a precision would be
lacking if were dropped at the outset in cosmological
applications. (c) While the Sommerfeld quantum number *j* in (A.3) is very suitable for labelling
the distinct energy levels, the Gibbs degeneracy *g* (derived from the phase-space element ) in
Table 1 is equally convenient to count the precise number of states in any
cell. (d) The *precise knowledge of a cutoff quantum number and energy will be shown
to be crucial to find the rigorous most probable distribution in Section 3
which job cannot be done in the cosmological context of Section 4 if is dropped at the outset (in the classical
phase space element )*.

#### B. Extension from Integer to Continuous

In
this appendix, we carefully examine the *numerical* justification of some algebraic manipulations done on the combinatorial
entropy *S* of (10),
(11).

*Factorials versus Gammas*

As
is well known, identically equals at all nonnegative *integers* as seen from the second line of the following
brief table. Its third line records the corresponding values of the natural
logarithm to be used as the input in
Table 3.

##### B.1. Numerical Differentiation

Next,
we address the subtle question of computing where the suffix
“num” stands for “numerically” and the inequality implies that has become a continuous variable over a test
range [0, 4]. This is a problem of *integer
programming* and we tackle it by adopting the following procedure.

(i) First, a finite-difference table was prepared using the above-mentioned data on (ii) Next, at several chosen integral/fractional values of (B.1) was computed employing an efficient package based on Markoff’s version of Newton’s interpolation differentiated [23, page 883]. (iii) Finally, comparison was made with the standard values of the digamma function [23, pages 258, 267, 272] obtained from the “exact” definition

##### B.2. Results

The accompanying table shows that of (B.1) and of (B.2) agree within 1% to 5% at the input
integer values We also see that their mutual agreement is
good at the small fractional values of Therefore, *taking
derivatives of at all continuous values of (including )* is mathematically justified in (13a) and
(14a)
(see Table 4).

#### Acknowledgment

The authors thank the Council of Scientific and Industrial Research (CSIR), New Delhi, India, for the financial support.

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