# Entropy Maximization, Cutoff Distribution, and Finite Stellar Masses

Authors: Ritesh Kumar Dubey; V. J. Menon; M. K. Pandey; D. N. Tripathi
Journal: Advances in Astronomy
 (2008)

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## 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 distributionfB  (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  f  (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 rK, cutoff energy  εK, and the huge quantum number K 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.


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## Body

## 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.Section2 below points out that since the MB function fB 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 ω 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 Sections2 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 ν. Also, a detailed treatment of our
variational conditions in Section 4 requires that ν! be replaced by Γ(ν+1) everywhere (even for ν≪1); 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 distributionfB  in equilibrium statistical mechanics. Particles are assumed to be moving in D spatial dimensions at temperature T under the influence of a mean field
potential energy W(r). The one-body
energy spectrum is divided into J cells, particles are distributed at random over these, those in the jth cell are regarded as
mutually identical, and the simple-minded Boltzmann-Shannon entropy functional  (1)SB=−k∑j=1Jgj{fjBln⁡fjB−fjB}+kα(∑j=1JgjfjB−N)−kβ(∑j=1JgjfjBεj−E) is set up. Here k is the Boltzmann constant,
gi  the cell degeneracy, N the total
number of particles, E the total energy, and α and β are Lagrange multipliers (see Appendix A for
precise definitions of various symbols). Next, one maximizes SB  with respect to fjB,α, and β to arrive at the MB solution (2)fB=eα−βε;νB=gfB:ε0≤ε≤∞, where the index j has been dropped in the quasicontinuum limit, ε0 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 W(r). 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 thelocal (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 ρ(r), one obtains the famous Boltzmann barometric
formula (3)ρB(r)∝exp⁡{−W(r)kT}:r0≤r≤∞.
The attractive
short-distance behaviour of W(r) cannot pose a real problem because the size r0 of the quantum ground level ε0 is finite [6]. But the long-distance
behaviour of W(r) is problematical as
regards astrophysics in D=3 dimensions. Indeed, for a dilute gaseous star
[2, page 114]
without the Poisson equation constraint, one finds asymptotically (4)W(r)~r→∞−0;ρB(r)~r→∞constant:Gaseous. Also, for the
isothermal Emden
sphere [3, 8] subject to the Poisson
equation constraint, one knows that (5)W(r)~r→∞ln⁡(r2a2);ρB(r)~r→∞r−2:Emden with a being the isothermal length scale.
Clearly, asr→∞,the
nil/slow decrease of ρB 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 〈r〉B of the system via (6)〈r〉B=∫0∞∂rr3ρB∫0∞∂rr2ρB=∞whichdiverges both for gaseous stars (4) and Emden
spheres (5).


#### 2.1.4. Mass

The total mass of the MB system is
calculated from(7)MB=4π∫0∞∂rr2ρB(r)=∞ 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 tor=∞ are allowed an open MB system, for example,
the dilute gaseous star goes on evaporating with time, producingas athereby a net outgoing fluxϕB of particles [2, page 114] at
every positive thermodynamic temperature T: (8)φB>0. Of course, the
isothermal sphere can be stable against evaporation [9] but its
mean field growing like ln(r2/a2) up to r=∞ 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 radiusRbox [7], or by
modifying the original distribution heuristically into non-Maxwellian form fLow such as (9)fLow∝exp⁡(−εkT)−1,∝exp⁡(−εkT)+εkT−1,   and so forth, holding  in the range ε<0 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 forfLow
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).


## 2.1. Preliminaries

This section
begins by quickly recalling a standard derivation of the famous
Maxwell-Boltzmann distributionfB  in equilibrium statistical mechanics. Particles are assumed to be moving in D spatial dimensions at temperature T under the influence of a mean field
potential energy W(r). The one-body
energy spectrum is divided into J cells, particles are distributed at random over these, those in the jth cell are regarded as
mutually identical, and the simple-minded Boltzmann-Shannon entropy functional  (1)SB=−k∑j=1Jgj{fjBln⁡fjB−fjB}+kα(∑j=1JgjfjB−N)−kβ(∑j=1JgjfjBεj−E) is set up. Here k is the Boltzmann constant,
gi  the cell degeneracy, N the total
number of particles, E the total energy, and α and β are Lagrange multipliers (see Appendix A for
precise definitions of various symbols). Next, one maximizes SB  with respect to fjB,α, and β to arrive at the MB solution (2)fB=eα−βε;νB=gfB:ε0≤ε≤∞, where the index j has been dropped in the quasicontinuum limit, ε0 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 W(r). 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 thelocal (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 ρ(r), one obtains the famous Boltzmann barometric
formula (3)ρB(r)∝exp⁡{−W(r)kT}:r0≤r≤∞.
The attractive
short-distance behaviour of W(r) cannot pose a real problem because the size r0 of the quantum ground level ε0 is finite [6]. But the long-distance
behaviour of W(r) is problematical as
regards astrophysics in D=3 dimensions. Indeed, for a dilute gaseous star
[2, page 114]
without the Poisson equation constraint, one finds asymptotically (4)W(r)~r→∞−0;ρB(r)~r→∞constant:Gaseous. Also, for the
isothermal Emden
sphere [3, 8] subject to the Poisson
equation constraint, one knows that (5)W(r)~r→∞ln⁡(r2a2);ρB(r)~r→∞r−2:Emden with a being the isothermal length scale.
Clearly, asr→∞,the
nil/slow decrease of ρB 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 〈r〉B of the system via (6)〈r〉B=∫0∞∂rr3ρB∫0∞∂rr2ρB=∞whichdiverges both for gaseous stars (4) and Emden
spheres (5).


### 2.1.4. Mass

The total mass of the MB system is
calculated from(7)MB=4π∫0∞∂rr2ρB(r)=∞ 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 tor=∞ are allowed an open MB system, for example,
the dilute gaseous star goes on evaporating with time, producingas athereby a net outgoing fluxϕB of particles [2, page 114] at
every positive thermodynamic temperature T: (8)φB>0. Of course, the
isothermal sphere can be stable against evaporation [9] but its
mean field growing like ln(r2/a2) up to r=∞ 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 radiusRbox [7], or by
modifying the original distribution heuristically into non-Maxwellian form fLow such as (9)fLow∝exp⁡(−εkT)−1,∝exp⁡(−εkT)+εkT−1,   and so forth, holding  in the range ε<0 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 forfLow
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).


## 2.1.1. Entropy

In the case of gravitational
systems, one always looks for thelocal (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 ρ(r), one obtains the famous Boltzmann barometric
formula (3)ρB(r)∝exp⁡{−W(r)kT}:r0≤r≤∞.
The attractive
short-distance behaviour of W(r) cannot pose a real problem because the size r0 of the quantum ground level ε0 is finite [6]. But the long-distance
behaviour of W(r) is problematical as
regards astrophysics in D=3 dimensions. Indeed, for a dilute gaseous star
[2, page 114]
without the Poisson equation constraint, one finds asymptotically (4)W(r)~r→∞−0;ρB(r)~r→∞constant:Gaseous. Also, for the
isothermal Emden
sphere [3, 8] subject to the Poisson
equation constraint, one knows that (5)W(r)~r→∞ln⁡(r2a2);ρB(r)~r→∞r−2:Emden with a being the isothermal length scale.
Clearly, asr→∞,the
nil/slow decrease of ρB 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 〈r〉B of the system via (6)〈r〉B=∫0∞∂rr3ρB∫0∞∂rr2ρB=∞whichdiverges both for gaseous stars (4) and Emden
spheres (5).


## 2.1.4. Mass

The total mass of the MB system is
calculated from(7)MB=4π∫0∞∂rr2ρB(r)=∞ 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 tor=∞ are allowed an open MB system, for example,
the dilute gaseous star goes on evaporating with time, producingas athereby a net outgoing fluxϕB of particles [2, page 114] at
every positive thermodynamic temperature T: (8)φB>0. Of course, the
isothermal sphere can be stable against evaporation [9] but its
mean field growing like ln(r2/a2) up to r=∞ 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 radiusRbox [7], or by
modifying the original distribution heuristically into non-Maxwellian form fLow such as (9)fLow∝exp⁡(−εkT)−1,∝exp⁡(−εkT)+εkT−1,   and so forth, holding  in the range ε<0 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 forfLow
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 SB 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 εK.
This is because a star moving in the mean potential field  W(r) will have a farthest turning at distance rK satisfying W(rK)=εK, where rK 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 quantumnumbers. The cutoff quantum number K and cutoff energy εK 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 jth cell has central energy εj,
width Δj,
degeneracy gj, occupation
number νj,
and occupation probability per state fj=νj/gj defined by
Appendix A. Next, treating the particle in the jth cell as
indistinguishable, a Gibbs combinatorial entropy functional S is constructed via (10)S=kln{∏j=1Jgjνjνj!}.Gamma Function Form
We deliberately rewrite (10) in the
equivalent form  (11)S=k∑j=1J{νjln⁡gj−ln⁡Γ(νj+1)}.   The replacement of
factorials by gammas has several algebraic advantages. (i) The equality νj!=Γ(νj+1) is exact at the integer values νj=0,1,2,3,…,∞.
(ii) The asymptotic behaviour, namely, (2π)1/2νν+1/2e−ν of both ν!   and Γ(ν+1) are the same as ν→∞.
(iii) Hence, by a theorem due to Carlson
[21, 22], Γ(ν+1) provides the most economical, essentially
unique continuation of ν!  to all continuous values throughout the range 0≤ν≤∞.
(iv) While setting up the variational conditions, later we shall need to replace
the derivative ∂lnΓ(νj+1)/∂νj evaluated at integer values by the digamma
function ψ(ν+1)  [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 lnΓ(ν+1) can be readily done even at small valuesν=0.2,0.4,  and so forth, giving results in good agreement with ψ(ν+1).Exact Variational Conditions
Next, we consider the following
objective functional to be maximized:(12)S*=S+kα(∑jνj−N)−kβ(∑jνjεj−E), where α and β are unknown Lagrange multipliers. Equating to
zero, the partial derivatives ∂S*/∂νj,∂S*/∂α,
and ∂S*/∂β lead to the following set of exact variational
conditions still using the discrete index
j
[12]:
(13a)ψ(νj+1)=ln⁡gj+α−βεj:1≤j≤J,(13b)∑jνj=N;∑jνjεj=E.Comments
Without making any assumption concerningthe largeness or smallness ofνj,  we can rewrite (13a) in the compact form(14a)ψ(νj+1)=exactln⁡νjB;νjB≡defgjeα−βεj, where the symbol νjB was already encountered earlier in (2). In
principle, (14a) can be solved for the desired cell occupation numbers νj in terms of α,β,εj.
Thereafter, the Lagrange multipliers α and β can be determined from the constraints (13b).
Equivalently, the chemical potential μ and thermodynamic temperature T may be introduced via (14b)kα=μT=−∂S*∂N;kβ=1T≡∂S*∂E.
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 ν versus ε 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 ε, we get (15)∂ψ(ν+1)∂ν⋅∂ν∂ε=1νB⋅∂νB∂ε. Clearly, the MPD occupancyν and MB occupancy νB are both peaked at a common energy εp which satisfies(16)ν˙(εp)=0;ν˙B(εp)=0,g˙(εp)g(εp)−β=0, where the dot stands
for derivative with respect to ε.
Typical algebraic estimates of εp for the soluble potential models will be
reported later in (28).Largeν Region
In the so-called head
region of the continuous distribution, the cells have large occupancyν≫1 so that the Stirling’s approximation ψ(ν+1)≈lnν+O(1/ν) holds in the fundamental equation (14a). Hence the MB solution is roughly retrieved,
namely,(17)ν≈νB≫1;f≈eα−βε:Large⁢⁢ν,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,ν≪1.
Then the digamma function possesses a Taylor
expansion (18)ψ(ν+1)≈ν≪1ψ(1)+ζ(2)ν+O(ν2):Tail, where ψ(1)=−0.577 is the negative of Euler’s constant and ζ(2)=π2/6 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
pointεK such that(19)νK=0;ψ(1)=ln⁡νKB≡ln⁡gK+α−βεK, where the suffix K refers to energy εK.(ii)
The said intersection happens linearly because, in its neighbourhood,
the occupancy(20)ν≈(εK−ε)ζ(2){β−g˙KgK}:Tail, where g˙K stands for ∂g/∂ε evaluated at εK.(iii)
Extension
of the graph ofν versus ε beyond the cutoff point is not allowed because
that would tend to make ν negative in (13a), that is,(21)νj≡0;f(ε)≡0:j≥K,ε≥εK,implying that the original occupied spectrum (A.2) has shrunk below J or εJ due to strict entropy maximization under
stable equilibrium. (The possibility K>J,εK>εJ would correspond to unstable equilibrium, that
is, continuous evaporation of the system.)  Schematic plots of fB and f versus ε are shown in Figure 1. Typical algebraic
estimates of the cutoff energy εK and quantum number K for the solvable models will be reported later in
(28).Figure 1
Schematic plots (not to scale) of the distribution functions in the MB
approach (cf. (2)) and MPD theory (cf.
(23)). (Plot
of the occupancy function would have shown a peak in both cases.)Compact Solution forν(ε)
Our equation (14a) is a transcendental
equation in ν and its precise analytical solution in closed
form is not known. Fortunately, there  exists an ansatz (22)ψ(ν+1)=ansatzln⁡{ν+θ};θ≡eψ(1)=0.562, which works
excellently throughout 0≤ν≤∞ 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(23)ν=νB−θ;f=fB−θg:ε1≤ε≤εK. It is interesting to
note that if gwere replaced by a
constant in (23), our MPD solution f 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 g(ε) is strongly energy-dependent.Figure 2
The exact digamma functionψ(ν+1) and three numerical approximations to the
same used in the context of (22), (17), and (18).
The values of three important constants are ψ(1)=−0.577,θ=eψ(1)=0.562,
and ς(2)=π2/6=1.646.Compact Number Condition
Combining the number constraint (13b) with the
general solution (23), we can define an effective number
Neff  through
(24a)Neff≡N+Kθ=∑j=1K(νj+θ)=∑j=1KνjB=∫ε0εK∂εΔ⋅geα−βε=eα⋅N∗.   Here we have
employed the quasicontinuum limit (A.4) and
introduced (24b)N∗≡∫ε0εK∂εw(ε)e−βε;eα=NeffN∗.This gives a formal expression for the Lagrange multiplier (or reduced
chemical potential)α provided that the underlying mean field W or
its reduced degeneracy function w=g/Δ is known.Compact Cutoff Condition
Lastly, we
convert the cutoff criterion (19) into (25a)θ≡eψ(1)=νKB=gKeα−βεK. Eliminating eα with the help of (24b), we find (25b)eβεK=NeffgKN∗θwhich yields a formal expression for the cutoff energyεK=ε(K) 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 Table1: (26)W(r)=−|W0|{1−r2R2}⋅step(R−r). 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 r>R,
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 r→0. (d) As
they stand, the depth W0 and range R 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 rK of the galaxy. We are now ready to apply the
MPD program to (26).Table 1
Properties of one-body semiclassical
spectrum labeled by the integerj for
four solvable potentials W(r)  in D dimensions. The well-depth W0≡−|W(0)| and the range R of the rectangular, linear, and oscillator wells are made finite
using the unit step function. The Coulomb well is left unrestricted over all r. For other notations, see
Appendix A. (We
do not tabulate the rigid box potential
model which has the upper end of the energy spectrum at εJ=E−(N−1)W0≈+∞.
Such a model is of little interest in cosmology In usual physical applications, one puts D=3.)
PotentialRectangularLinearOscillatorCoulombFormW(r)=W0step(R−r)W0{1−rR}⋅step(R−r)W0{1−r2R2}⋅step(R−r)Cr=ε1r1r:0≤r≤∞Highest
labelJ=(8m|W0|)1/2Rπℏ43⋅(2m|W0|)1/2Rπℏ(m|W0|2)1/2Rℏ∞Ionization
thresholdεJ=0000=ε∞Spectrumεj=W0{1−(jJ)2}W0{1−(jJ)2/3}W0{1−(jJ)}−mC22ℏ2j2=ε1j2Turning
pointrj=R(jJ)2/3R(jJ)1/2R2ℏ2j2m|C|=r1j2SeparationΔ=2|W0|jJ223⋅|W0|j−1/3J2/3|W0|J2|ε1|j3Geometrical
coefficientB=πD22D−2D2Γ2(D/2)(3π/4)DΓ(D)2D−1Γ(D/2)Γ(3D/2+1)1Γ(D+1)1DΓ(D/2+1/2)Accumulated
statesIjBjDBjDBjDBjDLevel
degeneracygj=DBjD−1DBjD−1DBjD−1DBjD−1The Tilde Nomenclature
First, we read off the symbolsJ,ε,Δ,B,
and g from the fourth column of Table 1. Next, for
algebraic convenience, the following dimensionless quantities are defined along
with the thermal de Broglie wavelength λ: (27)J≡J|βW0|1/2=(m|W0|/2)1/2R/ℏ(β|W0|)1/2=πRλ≫1,λ=πℏ(2βm)1/2;ε≡β(ε−W0)=jJ,α≡α−βW0,εK≡β(εK−W0)=KJ,j=J⁢ε,g=(DBJD−1)εD−1,νB=geα−βε=geα−ε,N≡eβW0N∗≡∫0Kdj(DBjD−1)e−j/J=DBJDγK,γK≡γ(D,εK).            A few remarks are in
order concerning these definitions. The Planck constant h or thermal de Broglie wavelength λ≡h/(2mkT)1/2 has appeared in the value of the symbol J and the inequality λ/R≪1 is essential for the validity of classical
motion (cf. (A.8)). The function ε measures the single-particle energy from the
ground level in terms of kT.
The symbols α and εK may be called the dimensionless chemical
potential and dimensionless cutoff energy, respectively, whose fixation using
MPD constraints is yet to be done. The integral N will play a crucial role below with γ 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 εp, peak height νpB, dimensionless chemical potential α,
and dimensionless cutoff  energy εK through (28)εp=D−1~1;νpB=gpeα−εp~NeffJγK,eα~NeffJDγK;eεKεKD−1~NeffJγKNeff≡N+Kθ;D>1,     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 εK=K/J has markedly different behaviours.Case 1  (well depth large compared tok times temperature).
For the truncated oscillator potential
(26), we recall the tilde notations (27)
and impose the following inequalities: (29)εK=KJ≫1;|βW0|=|W0|kT≫1,J~Rλ≫1;J≪Neff≪JD,J=(β|W0|)1/2J>K≫J≫1.     The physical meaning
of these restrictions is as follows. The inequalities J≫1,K≫1,J≫1  guarantee the validity of classical dynamics
in states of large quantum numbers, the
condition J>K ensures that the Kth  level lies below the ionization threshold for
stable MPD, the assumption εK≫1 in Case
1 implies that the actual cutoff energy εK is several kT above the ground level, |βW0|≫1 means that the well depth is large compared to kT,
the condition J≪Neff implies that the effective number Neff/J of particles grossly counted per cell is much
more than unity, and Neff≪JD means that the system is dilute or
nondegenerate (because the packing fraction Neff/JD~NeffλD/RD≪1, that is, the average number of particles
contained inside a D-dimensional sphere of radius λ is small compared to unity). Then the
incomplete gamma function γK→Γ(D) and consistent handling of (28) leads to the
estimates (30)εp~1;νpB~NeffJ≫1,α~ln⁡(NeffJD);εK~ln⁡(NeffJ),K~J⁢εK~(Rλ)ln⁡(NeffλR).     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 tok times temperature).
Again we recall the tilde notations (27) and
impose the orders of magnitude (31)εK=KJ~1;|W0kT|~1. Then the incomplete
gamma function γK~1 and (28) are found to admit the
self-consistent estimates (32)εp~1;νpB~NeffJ~1,Neff~N~K~J~Rλ≫1,α~ln⁡(1JD−1);εK~ln⁡(NeffJ)~1. The physics of (31)
and (32) is as follows. The statement |W0/kT|~1 applies to gravitational systems obeying
virialization, N~K tells that the total number of particles is of
the same order as the number of MPD cells, and νpB~1 signifies that the cell occupancies have
become comparable to unity with ℏ again playing a role through the symbol J.
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.


## 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 SB 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 εK.
This is because a star moving in the mean potential field  W(r) will have a farthest turning at distance rK satisfying W(rK)=εK, where rK 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 quantumnumbers. The cutoff quantum number K and cutoff energy εK 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 jth cell has central energy εj,
width Δj,
degeneracy gj, occupation
number νj,
and occupation probability per state fj=νj/gj defined by
Appendix A. Next, treating the particle in the jth cell as
indistinguishable, a Gibbs combinatorial entropy functional S is constructed via (10)S=kln{∏j=1Jgjνjνj!}.Gamma Function Form
We deliberately rewrite (10) in the
equivalent form  (11)S=k∑j=1J{νjln⁡gj−ln⁡Γ(νj+1)}.   The replacement of
factorials by gammas has several algebraic advantages. (i) The equality νj!=Γ(νj+1) is exact at the integer values νj=0,1,2,3,…,∞.
(ii) The asymptotic behaviour, namely, (2π)1/2νν+1/2e−ν of both ν!   and Γ(ν+1) are the same as ν→∞.
(iii) Hence, by a theorem due to Carlson
[21, 22], Γ(ν+1) provides the most economical, essentially
unique continuation of ν!  to all continuous values throughout the range 0≤ν≤∞.
(iv) While setting up the variational conditions, later we shall need to replace
the derivative ∂lnΓ(νj+1)/∂νj evaluated at integer values by the digamma
function ψ(ν+1)  [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 lnΓ(ν+1) can be readily done even at small valuesν=0.2,0.4,  and so forth, giving results in good agreement with ψ(ν+1).Exact Variational Conditions
Next, we consider the following
objective functional to be maximized:(12)S*=S+kα(∑jνj−N)−kβ(∑jνjεj−E), where α and β are unknown Lagrange multipliers. Equating to
zero, the partial derivatives ∂S*/∂νj,∂S*/∂α,
and ∂S*/∂β lead to the following set of exact variational
conditions still using the discrete index
j
[12]:
(13a)ψ(νj+1)=ln⁡gj+α−βεj:1≤j≤J,(13b)∑jνj=N;∑jνjεj=E.Comments
Without making any assumption concerningthe largeness or smallness ofνj,  we can rewrite (13a) in the compact form(14a)ψ(νj+1)=exactln⁡νjB;νjB≡defgjeα−βεj, where the symbol νjB was already encountered earlier in (2). In
principle, (14a) can be solved for the desired cell occupation numbers νj in terms of α,β,εj.
Thereafter, the Lagrange multipliers α and β can be determined from the constraints (13b).
Equivalently, the chemical potential μ and thermodynamic temperature T may be introduced via (14b)kα=μT=−∂S*∂N;kβ=1T≡∂S*∂E.
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 ν versus ε 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 ε, we get (15)∂ψ(ν+1)∂ν⋅∂ν∂ε=1νB⋅∂νB∂ε. Clearly, the MPD occupancyν and MB occupancy νB are both peaked at a common energy εp which satisfies(16)ν˙(εp)=0;ν˙B(εp)=0,g˙(εp)g(εp)−β=0, where the dot stands
for derivative with respect to ε.
Typical algebraic estimates of εp for the soluble potential models will be
reported later in (28).Largeν Region
In the so-called head
region of the continuous distribution, the cells have large occupancyν≫1 so that the Stirling’s approximation ψ(ν+1)≈lnν+O(1/ν) holds in the fundamental equation (14a). Hence the MB solution is roughly retrieved,
namely,(17)ν≈νB≫1;f≈eα−βε:Large⁢⁢ν,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,ν≪1.
Then the digamma function possesses a Taylor
expansion (18)ψ(ν+1)≈ν≪1ψ(1)+ζ(2)ν+O(ν2):Tail, where ψ(1)=−0.577 is the negative of Euler’s constant and ζ(2)=π2/6 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
pointεK such that(19)νK=0;ψ(1)=ln⁡νKB≡ln⁡gK+α−βεK, where the suffix K refers to energy εK.(ii)
The said intersection happens linearly because, in its neighbourhood,
the occupancy(20)ν≈(εK−ε)ζ(2){β−g˙KgK}:Tail, where g˙K stands for ∂g/∂ε evaluated at εK.(iii)
Extension
of the graph ofν versus ε beyond the cutoff point is not allowed because
that would tend to make ν negative in (13a), that is,(21)νj≡0;f(ε)≡0:j≥K,ε≥εK,implying that the original occupied spectrum (A.2) has shrunk below J or εJ due to strict entropy maximization under
stable equilibrium. (The possibility K>J,εK>εJ would correspond to unstable equilibrium, that
is, continuous evaporation of the system.)  Schematic plots of fB and f versus ε are shown in Figure 1. Typical algebraic
estimates of the cutoff energy εK and quantum number K for the solvable models will be reported later in
(28).Figure 1
Schematic plots (not to scale) of the distribution functions in the MB
approach (cf. (2)) and MPD theory (cf.
(23)). (Plot
of the occupancy function would have shown a peak in both cases.)Compact Solution forν(ε)
Our equation (14a) is a transcendental
equation in ν and its precise analytical solution in closed
form is not known. Fortunately, there  exists an ansatz (22)ψ(ν+1)=ansatzln⁡{ν+θ};θ≡eψ(1)=0.562, which works
excellently throughout 0≤ν≤∞ 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(23)ν=νB−θ;f=fB−θg:ε1≤ε≤εK. It is interesting to
note that if gwere replaced by a
constant in (23), our MPD solution f 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 g(ε) is strongly energy-dependent.Figure 2
The exact digamma functionψ(ν+1) and three numerical approximations to the
same used in the context of (22), (17), and (18).
The values of three important constants are ψ(1)=−0.577,θ=eψ(1)=0.562,
and ς(2)=π2/6=1.646.Compact Number Condition
Combining the number constraint (13b) with the
general solution (23), we can define an effective number
Neff  through
(24a)Neff≡N+Kθ=∑j=1K(νj+θ)=∑j=1KνjB=∫ε0εK∂εΔ⋅geα−βε=eα⋅N∗.   Here we have
employed the quasicontinuum limit (A.4) and
introduced (24b)N∗≡∫ε0εK∂εw(ε)e−βε;eα=NeffN∗.This gives a formal expression for the Lagrange multiplier (or reduced
chemical potential)α provided that the underlying mean field W or
its reduced degeneracy function w=g/Δ is known.Compact Cutoff Condition
Lastly, we
convert the cutoff criterion (19) into (25a)θ≡eψ(1)=νKB=gKeα−βεK. Eliminating eα with the help of (24b), we find (25b)eβεK=NeffgKN∗θwhich yields a formal expression for the cutoff energyεK=ε(K) 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 Table1: (26)W(r)=−|W0|{1−r2R2}⋅step(R−r). 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 r>R,
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 r→0. (d) As
they stand, the depth W0 and range R 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 rK of the galaxy. We are now ready to apply the
MPD program to (26).Table 1
Properties of one-body semiclassical
spectrum labeled by the integerj for
four solvable potentials W(r)  in D dimensions. The well-depth W0≡−|W(0)| and the range R of the rectangular, linear, and oscillator wells are made finite
using the unit step function. The Coulomb well is left unrestricted over all r. For other notations, see
Appendix A. (We
do not tabulate the rigid box potential
model which has the upper end of the energy spectrum at εJ=E−(N−1)W0≈+∞.
Such a model is of little interest in cosmology In usual physical applications, one puts D=3.)
PotentialRectangularLinearOscillatorCoulombFormW(r)=W0step(R−r)W0{1−rR}⋅step(R−r)W0{1−r2R2}⋅step(R−r)Cr=ε1r1r:0≤r≤∞Highest
labelJ=(8m|W0|)1/2Rπℏ43⋅(2m|W0|)1/2Rπℏ(m|W0|2)1/2Rℏ∞Ionization
thresholdεJ=0000=ε∞Spectrumεj=W0{1−(jJ)2}W0{1−(jJ)2/3}W0{1−(jJ)}−mC22ℏ2j2=ε1j2Turning
pointrj=R(jJ)2/3R(jJ)1/2R2ℏ2j2m|C|=r1j2SeparationΔ=2|W0|jJ223⋅|W0|j−1/3J2/3|W0|J2|ε1|j3Geometrical
coefficientB=πD22D−2D2Γ2(D/2)(3π/4)DΓ(D)2D−1Γ(D/2)Γ(3D/2+1)1Γ(D+1)1DΓ(D/2+1/2)Accumulated
statesIjBjDBjDBjDBjDLevel
degeneracygj=DBjD−1DBjD−1DBjD−1DBjD−1The Tilde Nomenclature
First, we read off the symbolsJ,ε,Δ,B,
and g from the fourth column of Table 1. Next, for
algebraic convenience, the following dimensionless quantities are defined along
with the thermal de Broglie wavelength λ: (27)J≡J|βW0|1/2=(m|W0|/2)1/2R/ℏ(β|W0|)1/2=πRλ≫1,λ=πℏ(2βm)1/2;ε≡β(ε−W0)=jJ,α≡α−βW0,εK≡β(εK−W0)=KJ,j=J⁢ε,g=(DBJD−1)εD−1,νB=geα−βε=geα−ε,N≡eβW0N∗≡∫0Kdj(DBjD−1)e−j/J=DBJDγK,γK≡γ(D,εK).            A few remarks are in
order concerning these definitions. The Planck constant h or thermal de Broglie wavelength λ≡h/(2mkT)1/2 has appeared in the value of the symbol J and the inequality λ/R≪1 is essential for the validity of classical
motion (cf. (A.8)). The function ε measures the single-particle energy from the
ground level in terms of kT.
The symbols α and εK may be called the dimensionless chemical
potential and dimensionless cutoff energy, respectively, whose fixation using
MPD constraints is yet to be done. The integral N will play a crucial role below with γ 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 εp, peak height νpB, dimensionless chemical potential α,
and dimensionless cutoff  energy εK through (28)εp=D−1~1;νpB=gpeα−εp~NeffJγK,eα~NeffJDγK;eεKεKD−1~NeffJγKNeff≡N+Kθ;D>1,     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 εK=K/J has markedly different behaviours.Case 1  (well depth large compared tok times temperature).
For the truncated oscillator potential
(26), we recall the tilde notations (27)
and impose the following inequalities: (29)εK=KJ≫1;|βW0|=|W0|kT≫1,J~Rλ≫1;J≪Neff≪JD,J=(β|W0|)1/2J>K≫J≫1.     The physical meaning
of these restrictions is as follows. The inequalities J≫1,K≫1,J≫1  guarantee the validity of classical dynamics
in states of large quantum numbers, the
condition J>K ensures that the Kth  level lies below the ionization threshold for
stable MPD, the assumption εK≫1 in Case
1 implies that the actual cutoff energy εK is several kT above the ground level, |βW0|≫1 means that the well depth is large compared to kT,
the condition J≪Neff implies that the effective number Neff/J of particles grossly counted per cell is much
more than unity, and Neff≪JD means that the system is dilute or
nondegenerate (because the packing fraction Neff/JD~NeffλD/RD≪1, that is, the average number of particles
contained inside a D-dimensional sphere of radius λ is small compared to unity). Then the
incomplete gamma function γK→Γ(D) and consistent handling of (28) leads to the
estimates (30)εp~1;νpB~NeffJ≫1,α~ln⁡(NeffJD);εK~ln⁡(NeffJ),K~J⁢εK~(Rλ)ln⁡(NeffλR).     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 tok times temperature).
Again we recall the tilde notations (27) and
impose the orders of magnitude (31)εK=KJ~1;|W0kT|~1. Then the incomplete
gamma function γK~1 and (28) are found to admit the
self-consistent estimates (32)εp~1;νpB~NeffJ~1,Neff~N~K~J~Rλ≫1,α~ln⁡(1JD−1);εK~ln⁡(NeffJ)~1. The physics of (31)
and (32) is as follows. The statement |W0/kT|~1 applies to gravitational systems obeying
virialization, N~K tells that the total number of particles is of
the same order as the number of MPD cells, and νpB~1 signifies that the cell occupancies have
become comparable to unity with ℏ again playing a role through the symbol J.
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 Section2
by employing the MPD solution obtained in Section 3.


### 4.1. Entropy

The Boltzmann-Shannon
entropySB of (1) is simple-minded, its maximization leads to
the MB solution νB 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 ν 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 ρ,
based on the transformation σ=ε−W(r),
we obtain (33)ρ(r)∝∫0σK∂σσD/2−1f(ε);σK≡εK−W(r), where ρ surprisingly vanishes if W(r)  equals εK.
This is explained by remembering that since no particle in MPD is allowed to
have an energy more than εK,there exists a largest classical turning
point atrK beyond which the density must become zero
identically, that is, (34)W(rK)=εK;ρ(r)≡0forr≥rK, in sharp contrast to
the MB density profiles (3)–(5). We can also
find the rate at which ρ(r) approaches zero as r tends to rK.
However,
(20) has already told us that f(ε)∝εK−ε∝σK−σ in the tail region. Hence, (33) yields the
leading behaviour (35)ρ(r)∝σK→0∫0σK∂σσD/2−1⋅σ∝W(r)→εK{W(rK)−W(r)}D/2+1. Since in a “good”
MPD solution εK and rK are finite, our result (35) tells that the
mass density obeys a (rK−r)D/2+1 law near the edge of the system.


### 4.3. Radius

Clearly, the distancerK in (34) is the upper bound on the size of our
galactic system and, for binding, we must have rK<rJ with rJ being the turning point just before the
continuum starts. (In the soluble models of
Table 1, this rJ was called R). Since the densityρ vanishes beyond rK the MPD integral defining the average size〈r〉 will also converge, that is,  (36)〈r〉<∞, 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,(37)M<∞, in sharp contrast to
the MB mass (7).


### 4.5. Nonevaporation

As is well
known if an attractive mean fieldW(r)  vanishes asymptotically, then the energy ε=0 is called the ionization threshold. Hence, our
galaxy will be stable against evaporation if the MPD cutoff energy εK happens to be negative at the given
thermodynamic temperature T. Consequently, for εK<0, there is no net outgoing particle flux, that
is, (38)φ=0, 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 εK.Poisson Equation Implications
So far in our treatment, the
detailed algebraic form of the mean fieldW(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 ρ in accordance with the Poisson equation in 3 dimensions (39)∇2W≡1r2∂∂r(r2∂W∂r)=4πGmρ;D=3, where G 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 densityρ(r) is sharply cutoff at rK,
by Gauss theorem, the exact potential energy and force at exterior points become (40)W(r)=−GMmr;F(r)=−GMmr2:r≥rK.(b)
At theedgerK itself, the potential energy becomes equal to the cutoff
energy, namely, (41)εK=W(rK)=−GMmrK.(c)
In theinterior region, the mean field may get smoothened so as to yield a
finite depth (42)W0≡W(0)~−kT by virtue of the
gravitational virial theorem.(d) Atinterior 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 W(r)=W0(1−r2/R2) if r≤rK with unknown phenomenological constants W0 and R.
The corresponding interior potential energy and force at the system edge rK then become (43)W(rK)=W0(1−rK2R2);F(rK)=2W0rKR2. Matching these to
the exterior values given by (40) at rK,
we identify (44)W0=3W(rK)2;R=3rK. Thus, W0 is deeper than W(rK) and R is larger than rK (although orders of magnitude are the same).Suggested Procedure for Cosmologists
Suppose a practical astronomical observation has been made on a
cluster ofN~105 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 sizerK and the known mass M of the cluster, the MPD cutoff energy εK is immediately given by (41) as εK=W(rK)=−GMm/rK.
Next, the oscillator well-depth W0 and the range parameter R for
motion inside the cluster are set up from (44) as W0=3W(rK)/2 and R=3rK.
Next, according to (32) applicable to cosmology, the MPD parameters have the
rough orders of magnitude (45)K~N~J~Rλ~(m|W(rK)|)1/2rKℏ~(GMm2rKℏ2)1/2 upon using the value
of J given by the first line of (27) under
virialization. Next, the cosmologist may treat (45) as providing a new mass
versus radius relationship M/rK~Gm4/ℏ2 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: (46)n(r)∝r→rK{W(rK)−W(r)}3/2+1∝MPD{1r−1rK}5/2 since the mean field W(r) becomes Newtonian near the
periphery. This can be cast into more convenient form by defining the variable x=1/r, choosing a normalization point
x1=1/r1, and
working with the modified function (47)n^2/5≡def{n(r)n(r1)}2/5=x−xKx1−xK which becomes unity
at x1 but
vanishes at xK=1/rK.
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 n^2/5 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).Figure 3
Normalized stellar
number density raised to 2/5 power, that is,n^2/5≡{n(r)/n(r1)}2/5 near the edge of the cluster M 15. The
experimental data points are adapted from King [14, Figure 2].
The theoretical
MPD prediction is computed from (47) with x1=0.202,xK=0.008.Step3 (comparison with King density).
Next, it is worthwhile to consider the functionn^1/2 and expand its MPD expression (47) around the
matching point x1 binomially in the form (48)n^1/2=MPD{1+δ}5/4=x→x11+54δ+O(δ2);δ≡x−x1x1−xK.  Dropping the δ2 term, the cosmologist retrieves the famous
formula proposed empirically by King, namely, (49)n^1/2=Kingx−xtx1−xt;xt=4xK+x15, whose square gives
the King’s profile [14, equation (2)] near the cluster’s periphery as (50)n(r)∝King{1r−1rt}2. 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 n^1/2 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 n^,
it is not possible to say whether MPD formula (47) or
King recipe (49) is
superior.Figure 4
Normalized stellar number density raised to1/2 power, that is, n^1/2 near the boundary of the M 15 cluster. The
experimental data points are read from King [14, Figure 2]. His model fit
is given by (49) with x1=0.202,xt=0.047.Step 4 (complete density profile).
Finally, the astronomer may like to have
an expression for the number densityn(r) valid throughout the range 0≤r≤rK.
In principle, our MPD distribution function f given by (23) yields the formal expression (51)n(r)=MPD∫∂Dp→hD{fB−θg} with the mean
potential W(r) 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 ω 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 α and K must be obtained numerically from the
transcendental conditions (24b) and (25b). We plan to accomplish this task in
a future communication.


## 4.1. Entropy

The Boltzmann-Shannon
entropySB of (1) is simple-minded, its maximization leads to
the MB solution νB 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 ν 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 ρ,
based on the transformation σ=ε−W(r),
we obtain (33)ρ(r)∝∫0σK∂σσD/2−1f(ε);σK≡εK−W(r), where ρ surprisingly vanishes if W(r)  equals εK.
This is explained by remembering that since no particle in MPD is allowed to
have an energy more than εK,there exists a largest classical turning
point atrK beyond which the density must become zero
identically, that is, (34)W(rK)=εK;ρ(r)≡0forr≥rK, in sharp contrast to
the MB density profiles (3)–(5). We can also
find the rate at which ρ(r) approaches zero as r tends to rK.
However,
(20) has already told us that f(ε)∝εK−ε∝σK−σ in the tail region. Hence, (33) yields the
leading behaviour (35)ρ(r)∝σK→0∫0σK∂σσD/2−1⋅σ∝W(r)→εK{W(rK)−W(r)}D/2+1. Since in a “good”
MPD solution εK and rK are finite, our result (35) tells that the
mass density obeys a (rK−r)D/2+1 law near the edge of the system.


## 4.3. Radius

Clearly, the distancerK in (34) is the upper bound on the size of our
galactic system and, for binding, we must have rK<rJ with rJ being the turning point just before the
continuum starts. (In the soluble models of
Table 1, this rJ was called R). Since the densityρ vanishes beyond rK the MPD integral defining the average size〈r〉 will also converge, that is,  (36)〈r〉<∞, 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,(37)M<∞, in sharp contrast to
the MB mass (7).


## 4.5. Nonevaporation

As is well
known if an attractive mean fieldW(r)  vanishes asymptotically, then the energy ε=0 is called the ionization threshold. Hence, our
galaxy will be stable against evaporation if the MPD cutoff energy εK happens to be negative at the given
thermodynamic temperature T. Consequently, for εK<0, there is no net outgoing particle flux, that
is, (38)φ=0, 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 εK.Poisson Equation Implications
So far in our treatment, the
detailed algebraic form of the mean fieldW(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 ρ in accordance with the Poisson equation in 3 dimensions (39)∇2W≡1r2∂∂r(r2∂W∂r)=4πGmρ;D=3, where G 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 densityρ(r) is sharply cutoff at rK,
by Gauss theorem, the exact potential energy and force at exterior points become (40)W(r)=−GMmr;F(r)=−GMmr2:r≥rK.(b)
At theedgerK itself, the potential energy becomes equal to the cutoff
energy, namely, (41)εK=W(rK)=−GMmrK.(c)
In theinterior region, the mean field may get smoothened so as to yield a
finite depth (42)W0≡W(0)~−kT by virtue of the
gravitational virial theorem.(d) Atinterior 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 W(r)=W0(1−r2/R2) if r≤rK with unknown phenomenological constants W0 and R.
The corresponding interior potential energy and force at the system edge rK then become (43)W(rK)=W0(1−rK2R2);F(rK)=2W0rKR2. Matching these to
the exterior values given by (40) at rK,
we identify (44)W0=3W(rK)2;R=3rK. Thus, W0 is deeper than W(rK) and R is larger than rK (although orders of magnitude are the same).Suggested Procedure for Cosmologists
Suppose a practical astronomical observation has been made on a
cluster ofN~105 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 sizerK and the known mass M of the cluster, the MPD cutoff energy εK is immediately given by (41) as εK=W(rK)=−GMm/rK.
Next, the oscillator well-depth W0 and the range parameter R for
motion inside the cluster are set up from (44) as W0=3W(rK)/2 and R=3rK.
Next, according to (32) applicable to cosmology, the MPD parameters have the
rough orders of magnitude (45)K~N~J~Rλ~(m|W(rK)|)1/2rKℏ~(GMm2rKℏ2)1/2 upon using the value
of J given by the first line of (27) under
virialization. Next, the cosmologist may treat (45) as providing a new mass
versus radius relationship M/rK~Gm4/ℏ2 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: (46)n(r)∝r→rK{W(rK)−W(r)}3/2+1∝MPD{1r−1rK}5/2 since the mean field W(r) becomes Newtonian near the
periphery. This can be cast into more convenient form by defining the variable x=1/r, choosing a normalization point
x1=1/r1, and
working with the modified function (47)n^2/5≡def{n(r)n(r1)}2/5=x−xKx1−xK which becomes unity
at x1 but
vanishes at xK=1/rK.
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 n^2/5 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).Figure 3
Normalized stellar
number density raised to 2/5 power, that is,n^2/5≡{n(r)/n(r1)}2/5 near the edge of the cluster M 15. The
experimental data points are adapted from King [14, Figure 2].
The theoretical
MPD prediction is computed from (47) with x1=0.202,xK=0.008.Step3 (comparison with King density).
Next, it is worthwhile to consider the functionn^1/2 and expand its MPD expression (47) around the
matching point x1 binomially in the form (48)n^1/2=MPD{1+δ}5/4=x→x11+54δ+O(δ2);δ≡x−x1x1−xK.  Dropping the δ2 term, the cosmologist retrieves the famous
formula proposed empirically by King, namely, (49)n^1/2=Kingx−xtx1−xt;xt=4xK+x15, whose square gives
the King’s profile [14, equation (2)] near the cluster’s periphery as (50)n(r)∝King{1r−1rt}2. 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 n^1/2 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 n^,
it is not possible to say whether MPD formula (47) or
King recipe (49) is
superior.Figure 4
Normalized stellar number density raised to1/2 power, that is, n^1/2 near the boundary of the M 15 cluster. The
experimental data points are read from King [14, Figure 2]. His model fit
is given by (49) with x1=0.202,xt=0.047.Step 4 (complete density profile).
Finally, the astronomer may like to have
an expression for the number densityn(r) valid throughout the range 0≤r≤rK.
In principle, our MPD distribution function f given by (23) yields the formal expression (51)n(r)=MPD∫∂Dp→hD{fB−θg} with the mean
potential W(r) 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 ω 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 α 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 Sections2–4 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.Table 2
Salient features of King-like lowered isothermal modelsfLow (cf. (9)) and our MPD distribution f (cf. (23)).
FeaturefLowfPhysical motivation
before equilibriumAs a general stellar cluster evolves
tides are generatedAs a general stellar cluster evolves
entropy is producedTheoretical basis
at equilibriumTidal force of the galaxyEntropy maximizationEquation obeyedSimple, approximate,
and nonunique in (9)Transcendental,
exact, and unique in (23)Role of Planck
constantNot needed since the treatment is
classical Newtonian.Needed explicitly
to find the cell degeneracygj,
the index J of (27), and the huge cutoff number K.Upper energy
cutoff in the spectrumImposed by hand atε=0 in (9).Predicted by
theory to occur atεK<0 in (19).Functional formFor example,e−βε−1 in
which the subtraction term 1 is
constant.fB−θ/g in which the subtraction term θ/g is strongly energy-dependent.Mass and radius of
the clusterFiniteFiniteNumber density
profile near the edgen(r)∝King(1/r−1/rt)2n(r)∝MPD(1/r−1/rt)5/2Quality of fit ton1/2 and n2/5Good in Figure4Good in Figure3The
essence of our cosmological discussion in Section4 is the following.
Suppose that an astronomer makes
observations on a (quantum mechanically nondegenerate) cluster having N stars,
total mass M, and radiusrK.
Then, its MPD solution will be characterized by the cutoff energy εK=−GMm/rK and cutoff quantum number K~N. 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
fB whose tail extends to ∞ in (2).
Such effect is likely to be weak on the MPD solution f whose tail gets truncated at εK in (23).(ii) One may argue that a sharp radius is also
known to arise in the method of self-consistentHartree 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 ofcollisionless 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 ZN in a box of large
volume V, perform the individual
momentum integrals at the outset over the infinite domain −∞≤p≤∞,
write the entropy S as the logarithm
of a Gibbs integral over the density of states, and minimize the Helmholtz free
energy E−TS 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 energyεe=−0 (i.e., arbitrarily close to zero but still
negative) will go far away and yet come back. Since the corresponding turning
point re 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 rK.Actually,
the above logic has the following very subtle fault. While doing pure dynamics,
it is enough to find trajectories and their turning pointsre;
but while doing statistical mechanics, it is essential also to calculate the
density profile ρe(r) and the related total mass Me.
Now, in direct analogy with (35) but with εe=−0, the density profile at large distance and its
associated Poisson equation become (in D=3 dimensions) (52)ρe(r)∝r→∞{−We(r)}5/2∝∇2We(r);∴We(r)∝r−4/3.   This result is physically unacceptable
because the gravitational potential due to a finite mass object must fall
asymptotically like 1/r. Hence a
logic based on εe=−0 will not work. In sharp contrast, if the globular cluster has finite
experimental massM, then it can be easily described by our MPD solution (34)
characterized by bounded rK and finite   εK<−0.(vi) Finally, a cosmologist may argue that since real gravitating systems
have a mass spectrum of stars, the assumption of particles with the same massm 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 (53)m=∑iNimi∑iNi, where the suffix i runs over different
species and there are Ni particles of the ith
type. This prescription should be
reasonable for those clusters where the mass dispersion is small (in units of
the solar mass).



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*Source: 870804-2009-01-26.xml*