# Most Probable Energy Distributions of Particles with Hierarchial Structures

Michael Romanovsky<EMAIL_ADDRESS>Science and Innovation Company, 24
Bolshaya Ordynka Street, Moscow, Russia
MIREA - Russian Technological University, 78 Vernadsky Avenue, Moscow, Russia

###### Abstract

Similarly to the derivation of the Gibbs-Boltzmann distribution for
structureless indistinguishable particles, we consider multi-particle systems
some of which are contained (or delimited) inside others (Problem 1), as well
as systems of particles delimited within other particles, which, in turn, are
delimited yet inside another kind of particles (Problem 2). Under the natural
assumptions concerning the conservation laws, such as the conservation of the
total number of particles, total energy, etc, the problem of the most probable
energy distributions is studied in a combinatorial formulation, with the
obtained distributions treated as externally observable. For Problem 1, the
particle distributions over maximal and minimal energies are also established
and shown to coincide with the ones found in the framework of the original
combinatorial treatment. The results for Problems 1 and 2 can be interpreted
as a sorting of lower-level particles based on their statistical mechanics
behavior observed in various experiments. An effective Pauli principle arises
in a non-contradictory manner in one-particle observations for the complete
probabilistic Problem 1 in the combinatorial formulation as well as in the
problem of distributions over maximal and minimal energies. Several of the
calculated distributions describe particles having no negative energy states.

most probable distributions, hierarchical structures, delimitation, effective
Pauli principle

###### pacs:

05.40.-a, 05.20.-y, 02.10.Ox, 05.10.Gg

##  I Introduction

It is well-known that, under the natural assumptions that the total number of
particles is finite and that there exists a mean value of the energy of a
particle, a system of indistinguishable identical particles with a discrete
energy spectrum sustains, as the most probable, an exponential probability
density law given by
$W({\epsilon}_{i})={(1/{T_{eff}})}e^{-{{\epsilon}_{i}}/{T_{eff}}}$ for the
particles being in i-th energy state ${{\epsilon}_{i}}$. Here $T_{eff}$ is a
normalization coefficient standing for a mean energy or an effective
temperature of the system [1, 2]. Let us designate the result as ”Problem 0”.
The thermodynamics characteristics and laws for such systems of structureless
particles are well-known.

Systems of greater complexity, consisting of indistinguishable particles
having inner structures, may be subject to modified thermodynamic relations,
as in the plasma case [3], and moreover exhibit new effects like negative
friction [4] (such particles are routinely described as active). Thereby, the
question arises naturally about the distributions of the particles over
energies in the most general case, where certain groups of particles are
delimited from others or some of the particles, called the higher-level ones
in what follows, contain others which are indistinguishable lower-level ones.

This problem has also been considered (for example, see [1, 2]. Under the
additional a priori assumption that the numbers of lower-level particles
within the higher-level ones vary, the most probable distribution is shaped by
a large canonical ensemble. It is not postulated here that the higher-level
particles must be identical. Let us designate the above as ”Problem 0‘”. The
consideration does not have to be limited to particles and energy
distributions a system internally carrying certain structures which affect its
external characteristics may be considered as alternative example. The role of
the energy distribution for the lower-level particles is played by a
distribution of the lower-level system over one of its characteristics.

It appears natural to switch to a more generalized formulation of the problem,
taking into account the observable indistinguishability of the higher-level
particles. Designate this formulation as ”Problem 1” and note that it implies
constructing the most probable distribution of lower-level particles over
energies. Multiplicity of such distributions should not be excluded. First,
those would be the presumed ”genuine” lower-level particle distributions over
energies within a single higher-level particle. It should be established at
this point whether they are observable and, if they are, what experiments it
takes to register them. Secondly, the question is which distributions are
observable and what sets them apart from the genuine ones. Problem 0 clearly
reduces to a limit of Problem 1 if the higher-level particle (under certain
conditions) contains a single lower-level one.

The formulation acquires additional complexity when the particles have
internal structures composed of other particles, which further contain yet
another type of species. The above may be appropriately termed ”Problem 2”.
The common pertinent illustration is represented by particles contained in
boxes (for example, see [1, 2]). The boxes being enclosed in other boxes, we
arrive at Problem 2. To an extent, atomic nuclei consisting of nucleons,
which, in their turn, are built of quarks, can be viewed as some illustration
of Problem 2.

The combinatorial problem about the number of combinations in the case where
identical objects are distributed over groups is suggestive of a solution to
Problem 1. We further demonstrate that considering the distribution of minimal
and maximal energies for lower-level particles in a higher-level sample yields
the same result. A solution to Problem 2 is spelled out in what follows.

##  II Problem 1. Combinatorial Formulation

(see [5]). Suppose $m_{i}$ is a set of higher-level particles such that
$m_{1}$ of these indistinguishable species contain $p_{1}$ lower-level
particles 111Replacing the expression ”higher-level particles” with the word
”strings”, and the expression ”lower-level particles” with the word
”mushrooms”, one arrives at the problem about drying mushrooms [6, 7] or at
the cyclic permutations problem [8]., $m_{2}$ similarly contain $p_{2}$, etc.
up to the $N$-th lower-level particle. Here $n$ is the overall number of
arrangements of the lower-level particles over the higher-level ones. For the
total number of lower-level particles $N$, we have

$N={{\sum}_{i=1}^{n}}{m_{i}}{p_{i}}$ (1)

with a distribution over a total of $M$ higher-level particles

$M={{\sum}_{i=1}^{n}}{m_{i}}$ (2)

The total number of permutations of the lower-level particles over all the
higher-level particles makes $W_{10}$ given by [6]

$W_{10}={N!\over{{{\prod}_{i=1}^{n}}{(p_{i}!)}^{m_{i}}}}$ (3)

If the higher-level particles being indistinguishable as mentioned above, we
have

$W_{11}={N!\over{{{\prod}_{i=1}^{n}}{(p_{i}!)}^{m_{i}}{{\prod}_{i=1}^{n}}{(m_{i}!)}}}$
(4)

Rather, the expression for $lnW_{11}$ is typically examined

$lnW_{11}=lnN!-{{\sum}_{i=1}^{n}}{m_{i}}{ln(p_{i}!)}-{{\sum}_{i=1}^{n}}{ln(m_{i}!)}$
(5)

Note that if $m_{i}{\equiv}1$, the formulation of the problem of calculating
the most probable distribution, as defined by conditions (1), (2), (5),
becomes analogous to the formulation of Problem 0222Allowing variations of $N$
in a large canonical ensemble problem, we can reduce it to Problem 1. The
process has never been implemented (for example, see [8])..

Suppose every lower-level particle may be in a state with some energy
${{\epsilon}_{i}}$. To find the most probable distribution over these energies
for arbitrary $m_{i}$, one should take into account that, in addition to Eqs.
(4) or (5) and the relations expressed by Eqs. (1) and (2), the condition must
be met that the total energy of the system is finite

$E={{\sum}_{i=1}^{n}}{m_{i}}{p_{i}}{{\epsilon}_{i}}$ (6)

In practice, this means that there exists a mean ”energy” per one lower-level
particle $<{\epsilon}>$.

Let us solve Problem 1. As the initial step, consider the variation of Eq.
(5), incorporating the simplest constraints imposed by Eqs. (1), (2), and (6).
To this end, the terms ${\alpha}N+{\beta}M+{\gamma}E$, with Lagrange
multipliers $\alpha$, $\beta$, and $\gamma$ should be added to the right-hand
side of (5):

$\displaystyle
lnW_{1}=lnN!-{{\sum}_{i=1}^{n}}{m_{i}}{ln(p_{i}!)}-{{\sum}_{i=1}^{n}}{ln(m_{i}!)}+$
$\displaystyle+{\alpha}{{\sum}_{i=1}^{n}}{m_{i}}{p_{i}}+{\beta}{{\sum}_{i=1}^{n}}{m_{i}}+{\gamma}{{\sum}_{i=1}^{n}}{m_{i}}{p_{i}}{{\epsilon}_{i}}$
(7)

We also use Stirling’s lowest approximation for $m!$ and $p!$, and perform the
variation in $\delta p_{i}$ and $\delta m_{i}$. The expression for the
variation of the most probable distribution $lnW_{1}$ becomes

$\displaystyle{\delta}lnW_{1}={{\sum}_{i=1}^{n}}[-({\delta}{m_{i}})p_{i}(lnp_{i}-1)-$
$\displaystyle-({\delta}p_{i})m_{i}lnp_{i}-({\delta}m_{i})lnm_{i}+$
$\displaystyle+{\beta}({\delta}m_{i})+{\alpha}({\delta}m_{i})p_{i}+$
$\displaystyle+{\alpha}m_{i}({\delta}p_{i})+{\gamma}({\delta}m_{i})p_{i}{{\epsilon}_{i}}+$
$\displaystyle+{\gamma}({\delta}p_{i})m_{i}{{\epsilon}_{i}}]=0$ (8)

We further collect the terms related to the independent variations $\delta
p_{i}$ and $\delta m_{i}$. The first equation of the set is obtained upon the
cancellation of the non-zero factors $\delta p_{i}$ and $m_{i}$

$lnp_{i}={\alpha}+{\gamma}{{\epsilon}_{i}}$ (9)

For the second variation, the cancelation of the non-zero quantity $\delta
m_{i}$ and the use of (9) show that

$lnm_{i}={\beta}+p_{i}$ (10)

Eqs. (9) and (10) provide a solution to Problem 1.

A solution to Eq. (9) corresponds exactly to the representation embodied in
the ordinary Boltzmann-Gibbs distribution (see the Introduction, [1]) and to
the solution of Problem 0. The difference from the solution to Problem 0 is
that the latter involves no variation in the number of the lower-level
particles. Easily, the solutions to Eqs. (9) and (10) are

$p_{i}=e^{{\alpha}+{\gamma}{{\epsilon}_{i}}}$ (11)

and

$m_{i}=e^{{\beta}+p_{i}}$ (12)

Thus, the genuine function of the most probable energy distribution of lower-
level particles is the same as in the most probable distribution in Problem 0
- it represents the probability of finding the lower-level particle in the
${{\epsilon}_{i}}$ energy state.

However, this function is not directly observable or measurable. Measuring
this distribution function would take an appliance with a capability to
measure $p_{i}$ vs. ${{\epsilon}_{i}}$ . The above task may not be performed
for a higher-level particle since the particles $m_{i}$ are indistinguishable.

The related question is which functions would be observable. To clarify the
issue, we rewrite the set of Eqs. (9) and (10), transforming the simplest
linear combinations of the equations to

$\displaystyle lnp_{i}\pm
lnm_{i}={\alpha}\pm{\beta}+{\gamma}{{\epsilon}_{i}}\pm p_{i}=$
$\displaystyle={\alpha}\pm{\beta}+{\gamma}{{\epsilon}_{i}}\pm
e^{{\alpha}+{\gamma}{{\epsilon}_{i}}}$ (13)

where Eq. (12) has been utilized. Consequently, the two functions $p_{i}m_{i}$
and ${{p_{i}}\over{m_{i}}}$ which solve Eqs. (9) and (10) must be considered.

Constants in Eqs. (9) and (10) must be defined for further analysis. Consider
$\beta$ in Problem 1. The quantity $m_{i}$ varies from a positive value
$e^{{\beta}+e^{\alpha}}$ for ${\epsilon}_{i}=0$ to $e^{\beta}$ at
${\epsilon}_{i}\rightarrow\infty$. For large ${\epsilon}_{i}$, with $\gamma$
being obviously negative,

$e^{e^{{\alpha}+{\gamma}{{\epsilon}_{i}}}}\simeq
1+e^{{\alpha}+{\gamma}{{\epsilon}_{i}}}$ (14)

From the physics standpoint, a large number of high-level particles for a high
energy of a lower-level particle contained therein should be an unlikely
arrangement, meaning that, as the first step towards treating Problem 1,
$\beta=0$ can be assumed for ${\epsilon}_{i}\rightarrow\infty$, and that
$m_{i}$ must tend to 1 for ${\epsilon}_{i}\rightarrow\infty$, i.e., as stated
above, $\beta=0$.

Obviously, the function $p_{i}m_{i}$ is potentially observable as it describes
the particle number distribution (1). This function differs substantially from
$p_{i}$ (see Fig. 1 and the corresponding caption where the depicted functions
1-7 are explained in the Caption):

Figure 1:  Normalized distribution density functions $p_{i}m_{i}$ (dashed
lines 2, 4, and 6) and ${{p_{i}}\over{m_{i}}}$ (dots-and-dashes 1, 3, 5) for
various values of the parameter $\alpha$. The values of the parameter are
$\alpha$=2.5 for curves 1 and 2, $\alpha$=1.5 for curves 3 and 4, and
$\alpha$=0.5 for curves 5 and 6. The curve
$p_{i}=|\gamma|e^{\gamma{{\epsilon}_{i}}}$ is also shown (continuous line 7).
It is assumed that $|\gamma|=1$. The scale is semi-logarithmic.

$p_{i}m_{i}=e^{{\alpha}+{\gamma}{{\epsilon}_{i}}}e^{e^{{\alpha}+{\gamma}{{\epsilon}_{i}}}}$
(15)

The function describing this distribution, normalized to unity, describes the
probability density

${(p_{i}m_{i})}_{norm}=|\gamma|{{e^{{\alpha}+{\gamma}{{\epsilon}_{i}}}e^{e^{{\alpha}+{\gamma}{{\epsilon}_{i}}}}}\over{e^{e^{\alpha}}-1}}$
(16)

which is shown as dashed curves in Fig. 1 for $\alpha$ = 0.5, 1.5 and 2.5. As
the value of $\alpha/|\gamma|$ increases, e.g. for a decreasing temperature
while $\alpha$ stays constant, the value of the function given by Eq. (16) for
its argument closed to zero increases sharply against what is seen for line 7,
while the asymptote for moderate and large values of ${\epsilon}_{i}$ drop off
with the ”same rate” (see Fig. 1). Thus, it appears that the probability
density $p_{i}m_{i}$ (or the normalized probability density
${(p_{i}m_{i})}_{norm}$ describes condensation of lower-level particles at low
energies, and the effect is manifest at low temperatures. The observability of
this function seems to materialize best in multi-particle effects such as
phase transitions, when ”all” of the (higher-level) particles are drawn into
the process. Here the situation is not dramatically sensitive to the value of
$\alpha$.

The second function which is a solution to Eq. (13), namely
${{p_{i}}\over{m_{i}}}$, is the distribution $p_{i}$ per one higher-level
particle from the $m_{i}$ set. It is also depicted in Fig. 1. The conditions
for its observability are clearly of a different kind than those for
$p_{i}m_{i}$. It looks that single-particle measurements of the energy spectra
(spectroscopic measurements, in a broad sense) are most relevant to the case
in terms of the physical meaning.

Consider the distribution of the lower-level particles over energies ”on the
average” per single higher-level particle ${{p_{i}}\over{m_{i}}}$:

${p_{i}\over
m_{i}}=e^{{\alpha}+{\gamma}{{\epsilon}_{i}}}e^{-e^{{\alpha}+{\gamma}{{\epsilon}_{i}}}}$
(17)

Obviously, here $|\gamma|$ plays the role of an ”inverse temperature” of the
system. Normalization of the distribution given by Eq. (17) to unity
translates into factor of $|\gamma|/(1-e^{-e^{\alpha}})$ in the equation, and
the quantity practically coincides with $|\gamma|$ for moderate or, even more
so, large $\alpha$. The function defined by Eq. (17) peaks for
${\epsilon}_{i}={\alpha}/|\gamma|$ and, since $\alpha$ is a Lagrange
multiplier for the total number of particles given by Eq. (1), the physical
meaning of $\alpha$ (or, rather, ${\alpha}/|\gamma|$) may be that it is
related to the chemical potential of the lower-level particles $\mu$, with
$\alpha=|\gamma|\mu$. It should be noted that the probability density given by
Eq. (17) coincides with the probability density for the Gumbel distribution
(type I) [9].

The distribution ${p_{i}\over m_{i}}$ given by Eq. (17), when normalized to
unity, is presented below for convenience of calculations

${({p_{i}\over
m_{i}})}_{norm}={|\gamma|\over{1-e^{-e^{\alpha}}}}{e^{{\alpha}+{\gamma}{{\epsilon}_{i}}}\over
e^{e^{{\alpha}+{\gamma}{{\epsilon}_{i}}}}}$ (18)

Fig. 2 shows the function defined by the above equation for several values of
$\alpha$ with different effective temperatures $1\over{|\gamma|}$.

Figure 2:  Probability distribution density functions ${({p_{i}\over
m_{i}})}_{norm}$, normalized to unity, vs. ${\epsilon}_{i}$ (18) for $\alpha$
= 0.5, 3, and 7. Here $|\gamma|$= 0.5, for the dashed curves 1-3, and
$|\gamma|$= 5 for the continuous curves 4-6. The scale is uniform.

As $\alpha$ increases (curves 1 - 3 respectively), the normalizing factor in
Eq. (18) gradually tends to $|\gamma|$. The convergence is practically
immediate for low effective temperatures (large $|\gamma|$) and the
distribution ${({p_{i}\over m_{i}})}_{norm}$ simply translates along the
horizontal axis, preserving its shape. It can be seen in Fig. 2 that the
maxima become narrow as $|\gamma|$ increases, i.e. as the effective
temperature goes down. A sharp maximum of the distribution corresponds to the
value of the chemical potential ${\epsilon}_{i}=\mu={\alpha}/|\gamma|$.
Apparently, the pattern implies that lower-level particles may bond with a
higher-level one (or get detached from it) only to occupy energy states close
$\mu$. The width of the distribution function decreases if the temperature
goes down, and for low temperatures the distribution may become so narrow that
no more than a single lower-level particle would be accommodated in such
state.

Therefore, under combinatorial consideration, a two-level hierarchic system of
particles, where the lower-level particles are delimited in the higher-level
ones, exhibits a split into two sorts of lower-level particles. Tentatively,
the two sorts should be observed in different types of experiments
specifically examining either collective or individual behavior. Particles of
the first sort may condensate into low-energy states. Particles of the second
sort should occupy states characterized by the binding energy (or the
dissociation energy). In a naively formulated approximation, the states are
characterized by the chemical potential of the lower-level particles within
the higher-level ones, and, due to the narrowness of the function of
probability density of the energy distributions, these happen to be the only
states open to the particles.

##  III Problem 1 in terms of distributions of maximal and minimal values.

Problem 1 can be formulated in other terms such that the solution process and
the results would, first, coincide with those demonstrated above, and,
secondly, serve to better clarify the entire outcome.

Assume that Problem 0 is generalized so that there exists a probability
density of a distribution over energies $p_{i}$ for lower-level particles
within every higher-level particle (or a distribution inside a box, or a
distribution of strings of mushrooms over lengths) from a set $m_{i}$.
Treating the set $m_{i}$ as a [statistical] sample, we seek a distribution
function (and then - the probability density) of the maximal value $p_{i}$
over all samples $m_{i}$. For the purpose, we will follow the procedure
outlined in [9].

Denote the variable of the function of probability density of the distribution
$p_{i}$ as ${|\gamma|}{{\epsilon}_{i}}=x$, with the quantity $x$ varying from
$0$ to $\infty$ 333Note that the shift of the energy ”ground state” of the
probability density $p_{i}$ from zero to a negative value $-{{\epsilon}_{0}}$
simply leads to a shift of the variable ${{\epsilon}_{i}}$, which would then
vary from $-{{\epsilon}_{0}}$ to $+\infty$. The probability densities obtained
for Problem 1 may now be rewritten as ${|\gamma|{(\pm e^{e^{\pm\alpha}}\mp
1)}^{-1}}{e^{\alpha+{\gamma}({{\epsilon}_{i}}+{{\epsilon}_{0}})}}{e^{\pm{e^{\alpha+{\gamma}({{\epsilon}_{i}}+{{\epsilon}_{0}})}}}}$
for ${(p_{i}m_{i})}_{norm}$ (the upper sign) and
$({{p_{i}}\over{m_{i}}})_{norm}$ (the lower sign) respectively. The quantities
$\alpha$ can have negative values which, compared to those considered in the
previous Section, should be shifted by $-{{\epsilon}_{0}}$ to obtain the same
probability densities. All the results laid out above would remain unchanged..
The distribution function $F(x)$ is, naturally, $F(x)=1-e^{(-x)}$. The
probability of encountering a maximal value in a sample $m_{i}$ is determined
by the expression [9]

$m_{i}[1-F({x_{im}})]=1$ (19)

where $x_{im}$ is the characteristic maximal value. It follows from Eq. (19)
that

$x_{im}={\ln}m_{i}\;and/or\;{{e^{x_{im}}}\over{m_{i}}}=1$ (20)

The distribution function ${F_{m}}(x)$ of the greatest value over the entire
$m_{i}$ sample is, obviously, the function $F(x)$ to the power $m_{i}$ [6].
Using the second of Eqs. (20) and a transition to a limit, we arrive at

${F_{m}}(x)=(1-{{e^{-(x-x_{im})}}\over{m_{i}}})^{m_{i}}\rightarrow{e^{-e^{-(x-x_{im})}}}$
(21)

while $m_{i}\rightarrow\infty$. Differentiating the limiting function of the
distribution as defined by Eq. (21), one finds the probability density
${p_{i}}\over{m_{i}}$, where $x_{im}=\alpha$, which corresponds to the
normalization of the Boltzmann-Gibbs distribution

${m_{i}}e^{-x}=e^{-(x-{\ln}m_{i})}$ (22)

by the number of higher-level particles $m_{i}$.

Next, consider the function $p_{i}m_{i}$ in line with an analogous scheme. In
contrast to $1\over{m_{i}}$, the function $m_{i}$ is not a distribution
function of any random variable. At the same time, it is known [9] that the
function of the probability density of the distribution of minimal values
$p_{i}$ in the sample $m_{i}$ (or the distribution over energies of the lower-
level particles within the higher-level ones) is ${m_{i}}e^{-m_{i}x}$ (we
retain the notation ${|\gamma|}{{\epsilon}_{i}}=x$).

Let us consider the case of zero values of $\alpha$. We develop a new function
of the probability density according to the following scheme: initially we
adopt the above distribution of the minimal value, as it can occur in any
specific case over the entire sample $m_{i}$ (within any of the higher-level
particles). Since it is irrelevant in which realization (within which higher-
level particle) this would happen, we ascribe to this probability density the
statistical weight of all combinatorial permutations with repetitions, i.e.
$1\over{(m_{i}!)}$. The term
${[{(m_{i}-1)}!]}^{-1}e^{-m_{i}x}=e^{-x}{[{(m_{i}-1)}!]}^{-1}e^{-(m_{i}-1)x}$
is to serve as the first term of the sum to be compiled, which will represent
the final value of a function of probability density.

The probability density of the minimal value in the remaining $m_{i}-1$
realizations (the remaining higher-level particles) makes
$({m_{i}}-1)e^{-(m_{i}-1)x}$. Similarly, the weight to be ascribed to this
function is $1\over{(m_{i}-1)!}$, the result being another term of the sum and
a contribution to the probability density function
${[{(m_{i}-2)}!]}^{-1}e^{-(m_{i}-1)x}=e^{-x}{[{(m_{i}-2)}!]}^{-1}e^{-(m_{i}-2)x}$.
The procedure should be repeated a total of $m_{i}-1$ times. With a limiting
transition

$e^{-x}{[{(m_{i}-k^{\prime})}!]}^{-1}e^{-(m_{i}-k^{\prime})x}\rightarrow
e^{-x}{(k!)}^{-1}e^{-kx}$ (23)

the probability density functions add up to the required probability density
function (see Fig. 3):

Figure 3:  The function $e^{-x}e^{e^{-x}}$ (the curve consisting of open
squares) compared to the sum of the functions
$e^{-x}{\sum}_{k=0}^{5}{{e^{-kx}}\over{k!}}$ (the upper bold dashed curve).
The six narrow dashed curves below represent the functions
$e^{-x}{{e^{-kx}}\over{k!}}$, with $k$ ranging from 0 to 5. The main plot,
uniform coordinates. The inset - same in semilogarithmic scale.

It is possible to do since we take different group of particles each time. The
final probability distribution density function is now the sum of partial
ones:

$f(x)={\lim}_{m_{i}\rightarrow\infty}e^{-x}{\sum}_{k=0}^{m_{i}-1}{{e^{-kx}\over{k!}}\rightarrow
e^{-x}e^{e^{-x}}}$ (24)

The function $p_{i}m_{i}$ exactly equals $f(x)$ for $\alpha=0$. Practically,
$f(x)$ represents the probability density of minimal values of $p_{i}$,
obtained in every realization of the $m_{i}$ sample (the energy of the lower-
level particle has a minimal value within a higher-level particle provided
that the values are also minimal in other realizations). Apparently, this is
direct evidence of the fact that $f(x)$ (or $p_{i}m_{i}$) describes a kind of
collective behavior of the higher-level particles as well as of the lower-
level particles within them.

There is significant difference between $f(x)$ and $p_{i}\over m_{i}$. The
latter describes the density of the probability distribution of a ”normal”
particle having a certain energy, which happens to be maximal over a set of
higher-level particles. The result, namely the maximal value distribution, is
the same in every higher-level particle (in every realization of the sample),
though it is clearly determined by the sample as a whole. For $f(x)$ (or,
generally, for the function $p_{i}m_{i}$) the result for every realization is
fundamentally different. This may mean that the system of higher-level
particles with the given lower-level particles comprises a certain collective
system, where the lower-level particles within a higher-level particle are
somehow affected by each others presence. Thus, delimitation in this case
additionally defines a certain (potentially collective) interaction between
species. The above is a stronger statement than a one describing ”normal”
particles with the function$p_{i}\over m_{i}$ would take, as here the
delimitation only leads to the observation of maximal values of the $p_{i}$
distribution.

A calculation of the probability density of lower-level particles over energy
for a sample of the higher-level particles produces exactly the same result as
the combinatorial calculation. From the standpoint of statistical mechanics,
only states with maximal energy or conditionally (see above) minimal energy of
the lower-level particles may emerge as the most probable (observable). An
effective Pauli principle arises in the former case.

##  IV Nature of the effective Pauli principle.

Several facts related to the above calculations are indicative for an
explanation of the effective Pauli principle nature. First, the function
$p_{i}\over m_{i}$ in the framework for maximal value describes the
probability density of the maximal value $p_{i}$ over the sample $m_{i}$ or,
which is the same in terms of particles, the probability density for
registering a lower-level particle in a maximal energy state over the set of
higher-level particles. This perspective affords only one lower-energy
particle in a definite energy state, which is a manifestation of the effective
Pauli principle.

Secondly, when the temperature goes down both types of calculations
establishing the probability density $p_{i}\over m_{i}$ point to a narrowing
of the distributions (Fig. 2). For low temperatures, the distribution can
become so narrow that now more than one lower-level particle would ”fit” into
it. A legitimate question in this connection is what states would be open to
the remaining lower-level particles.

Fig. 4 illustrates a model situation where an ensemble of higher-level
particles with various $\alpha$ (various chemical potentials for corresponding
groups of the lower-level particles) is available.

Figure 4: The functions $p_{i}\over m_{i}$ (dashed curves) with the values of
$\alpha$ from 0.5 to 5 (over 0.5), and a continuous curve, representing the
sum of these functions. The open squares show the Fermi functions
$1\over{exp{(|\gamma|\epsilon-\alpha)}+1}$ for $\alpha\simeq 5.3$. The inset
shows in semilogarithmic scale the values of the functions for large values of
their arguments, explicating the exponential asymptotic of the summary
function. $|\gamma|=1$.

In the case, the remaining lower-level particles are accommodated in the
energy states of the corresponding distributions (dashed curves in Fig. 4)
with identical maxima. If they are distributed densely, with
${\alpha}_{min}<\alpha<{\alpha}_{max}$ (${\alpha}_{min}$ may equal zero), the
summary probability density (dashed curve in Fig. 4) over energy appears to be
constant wherever $\epsilon<{{\alpha}_{max}\over{|\gamma|}}$, and then
decreases exponentially (see the inset in Fig. 4). Such summary distribution
would agree well with the Fermi distribution for
$\mu\simeq{\mu}_{max}={{\alpha}_{max}\over{|\gamma|}}$. Consequently, the
Fermi distribution density for all delimited particles (quantitatively, the
delimitation is defined by some energy - the Fermi energy which is close to
the chemical potential for the isolated set of particles) can be represented
by a sum of the probability densities of the distributions of groups of
particles, every one of which has its unique value of the chemical potential.
The above may also provide a key to the interpretation of the origin of the
effective Pauli principle.

For the same purpose, it is enough to note, without citing the complexities
which Fig. 4 reflects, that some energy of a lower-level particle would always
be registered in single-particle measurements of the state of the system of
species, and that for low temperatures the energy may be unique, as other
values do not fit under curves 4-6 in Fig. 2 due to the spacing between levels
of the discrete energy spectrum. So, one energy state would correspond to only
one particle, which may be interpreted as a case of the effective Pauli
principle.

At this point, the principle can be formulated as follows. Lower-level
particles delimited in higher-level ones have, as the most probable, a
probability density of a distribution over energies shaped as a narrow bell-
shaped curve. On top of the fact that only one particle ”fits” into the
distribution at low temperature, the function is also a distribution of lower-
level particles having a maximal energy within a higher-level particle.The
probability fo find two particles in the state with maximal energy is clearly
extremely small. Thus, only one delimited particle can be found in one
(highest) energy state - and namely these particles are observed.

##  V Problem 2

(see [5]). As it is noted in the Introduction, a formulation of the problem
can address the distribution over energies of lower-level particles, contained
in a set of intermediate-level particles, which, in their turn, are enclosed
in higher-level particles. The particles are indistinguishable on all the
three levels 444In terms of the generalized ”mushroom drying” problem, this
would mean a calculation of the lengths of mushroom strings which are
organized in bunches of different volumes. Not only the strings, but also the
bunches - not to mentions the mushrooms - are indistinguishable.. Denote the
number of lower-level particles contained in an intermediate-level particle as
$p_{i}$, the number of intermediate-level particles with this number of lower-
level particles as $m_{i}$, and the number of higher-level particles in which
intermediate-level particles in the amount of $m_{i}$ are packed as $t_{i}$.
The total number of permutations of lower-level particles over all
intermediate-level particles, and of those - over the higher-level particles
is

$W_{21}={N!\over{{{\prod}_{i=1}^{n}}{(p_{i}!)}^{m_{i}t_{i}}{{\prod}_{i=1}^{n}}{{(m_{i}!)}^{t_{i}}}{{\prod}_{i=1}^{n}}{(t_{i}!)}}}$
(25)

Here again $n$ is the full number of distinct arrangements of the lower-level
particles over the intermediate-level particles, and of the latter - over the
higher-level particles. Naturally, the total number of lower-level particles
is

$N={{\sum}_{i=1}^{n}}{m_{i}}{p_{i}}{t_{i}}$ (26)

the total number of intermediate-level particles is

$M={{\sum}_{i=1}^{n}}{m_{i}}{t_{i}}$ (27)

and the total number of higher-level particles is

$T={{\sum}_{i=1}^{n}}{t_{i}}$ (28)

The total energy is given by

$E={{\sum}_{i=1}^{n}}{m_{i}}{p_{i}}{t_{i}}{{\epsilon}_{i}}$ (29)

Following the general rule, consider the logarithm of the number of
permutations

$\displaystyle lnW_{21}=lnN!-{{\sum}_{i=1}^{n}}{m_{i}}{t_{i}}{ln(p_{i}!)}-$
$\displaystyle-{{\sum}_{i=1}^{n}}{t_{i}}{ln(m_{i}!)}-{{\sum}_{i=1}^{n}}{ln(t_{i}!)}$
(30)

Accordingly, the quantity given by Eq. (30) is subject to variation under the
constraints given by Eqs. (26) - (29) with Lagrange multipliers
$\alpha,\beta,\gamma,\Delta$, in analogy with Eq. (8):

$lnW_{2}=lnW_{21}+\alpha N+\beta M+\gamma E+\Delta T$ (31)

The variables for variation here are $p_{i},m_{i},t_{i}$. Variation over
$p_{i}$ results in an equation identical to Eq. (9) (leading to a solution
same as for Problem 0):

$lnp_{i}={\alpha}+{\gamma}{{\epsilon}_{i}}$ (32)

The variation of Eq. (31) over $m_{i}$ combined with Eq. (28) produces an
equation identical to Eq. (10):

$lnm_{i}={\beta}+p_{i}$ (33)

Finally, the variation of Eq. (31) over $t_{i}$, combined with Eqs. (32) and
(33), results in a third equation for Problem 2

$lnt_{i}={\Delta}+m_{i}$ (34)

Certain remarks are due concerning the quantities $\beta$ and $\Delta$. The
latter quantity features in all of the equations within an exponential factor,
being potentially necessary only in the case of a special normalization, while
it can be set to zero in the lowest approximation, though it would have been
natural to assume that $\Delta=-1$, so that $t_{i}$ would tend to $1$ for
${{\epsilon}_{i}}\rightarrow\infty$ (see above). The quantity $\beta$ in
Problem 2 can, for a start, be set equal to $\beta$ from Problem 1, which is
also zero. The case of $\beta$ not equal to zero will be treated further on.

We compile, as in the case of Problem 1, a simple linear combination of Eqs.
(32) - (34):

$\displaystyle lnp_{i}\pm lnm_{i}\pm
lnt_{i}=={\alpha}\pm{\beta}\pm\Delta+{\gamma}{{\epsilon}_{i}}\pm p_{i}\pm
m_{i}$

and make use of Eqs. (32) and (33). The result is

$\displaystyle lnp_{i}\pm lnm_{i}\pm lnt_{i}=$
$\displaystyle={\alpha}\pm{\beta}\pm\Delta+{\gamma}{{\epsilon}_{i}}\pm$
$\displaystyle\pm e^{{\alpha}+{\gamma}{{\epsilon}_{i}}}\pm
e^{\beta+e^{{\alpha}+{\gamma}{{\epsilon}_{i}}}}$ (35)

In full similarity to Problem 1, the solutions to Eq. (35) and the
distribution functions in the case considered are
$p_{i}m_{i}t_{i},{{p_{i}t_{i}}\over m_{i}},{p_{i}\over{m_{i}t_{i}}}$, as well
as ${{p_{i}m_{i}}\over t_{i}}$. The first of the functions describes
multiparticle measurements with the effect of particle condensation at low
energies, which is more intense than in problem 1 (see Fig. 5). The second
function describes the effect of condensation of the lower-level particles,
but it is less pronounced than in the case of the first of the functions, as
illustrated by the comparison of Curves 3 and 4 against Curves 1 and 2 in Fig.
5. The third one describes a certain function measured per intermediate-level
particle and having a sharp maximum. Here, however, the maximum does not
correspond to the value of $\alpha$ even for zero $\beta$, but exceeds it
approximately by $1$ for ${p_{i}\over{m_{i}t_{i}}}$ under the conditions of
Fig. 5 (this is equivalent to approximately one unit of temperature), as
demonstrated by Curves 5 and 6.

Figure 5:  Probability distribution density functions $p_{i}m_{i}t_{i}$,
normalized to unity, vs ${\epsilon}_{i}$ (Curve 1 for $\alpha$ = 1 and Curve 2
for $\alpha$ = 0.5), ${{p_{i}t_{i}}\over m_{i}}$ (Curve 3 for $\alpha$ = 1 and
Curve 4 for $\alpha$ = 0.5), ${p_{i}\over{m_{i}t_{i}}}$ (Curve 5 for $\alpha$
= 1 and Curve 6 for $\alpha$ = 0.5). Uniform scale, $|\gamma|$ is set to 1.

Finally, ${{p_{i}m_{i}}\over t_{i}}$ is the probability density for the
distribution ${e^{e^{\beta}}}\over t_{i}$ , if the energy, which is the
argument here, is extrapolated to cover negative values. The normalized
distribution of the probability density for the positive values of the
argument is

${({{p_{i}m_{i}}\over t_{i}})}_{norm}={|\gamma|\over{{1\over
e}-e^{-e^{e^{\alpha}}}}}{{e^{{\alpha}+{\gamma}{{\epsilon}_{i}}}e^{e^{{\alpha}+{\gamma}{{\epsilon}_{i}}}}}\over
e^{e^{e^{{\alpha}+{\gamma}{{\epsilon}_{i}}}}}}$ (36)

As in the case of Problem 1, the function defined by Eq. (36) is a result of
measuring an average value per one higher-level particle.

Fig. 6 illustrates the behavior of the normalized density of the probability
distribution given by Eq. (36) depending on the Lagrange multipliers $\alpha$
and $\beta$.

Figure 6:  Probability distribution density functions ${({{p_{i}m_{i}}\over
t_{i}})}_{norm}$ given by Eq. (36), normalized to unity, vs ${\epsilon}_{i}$.
Continuous curves 1, 4 and 7 for $\alpha$ = 0, $\beta$ = 1; $\alpha$ = 3,
$\beta$ = 1; $\alpha$ = 6, $\beta$ = 1 correspondingly. Dashed curves 2, 5 and
8 for $\alpha$ = 0, $\beta$ = 2; $\alpha$ = 3, $\beta$ = 2; $\alpha$ = 6,
$\beta$ = 2 correspondingly, point curves 3, 6 and 9 for $\alpha$ = 0, $\beta$
= 5; $\alpha$ = 3, $\beta$ = 5; $\alpha$ = 6, $\beta$ = 5 correspondingly. For
all curves, $|\gamma|=1$.

The maximum of ${{p_{i}m_{i}}\over t_{i}}$ for $\beta>1$ coincides with high
precision with $\alpha+\beta$, so that the quantity $\beta$ may be ”naively”
interpreted as the chemical potential on one lower-level particle within the
higher-level particle (for $\alpha$, the interpretation follows from Problem
1). It becomes obvious in the light of such reading that the full chemical
potential of a lower-level particle in the higher-level particle, in the
framework of Problem 2, comprises the two chemical potentials, namely, of the
lower-level particle within the intermediate-level one and of the
intermediate-level particle in the higher-level one. The fact can be seen
clearly in Fig. 6 as the maxima of the probability densities shown by curves
2, 3, 5, 6, 8, and 9 correspond to the sum $\alpha+\beta$.

For $0\leq\beta\leq 1$, the maximum of ${{p_{i}m_{i}}\over t_{i}}$ only
slightly exceeds $\alpha+\beta$, as it can be seen from curves 1, 4, and 7 in
Fig. 6. For other curves, the coincidence takes place with the precision up to
${e^{-2(\alpha+\beta)}\over 2}$. It also follows from Fig. 6 that the
different chemical potentials (proportional to $\alpha$ and $\beta$) yielding
identical sums define probability densities which are very similar (cures 3
and 5, as well as 6 and 8 in Fig. 6).

The calculations of the probability distribution functions of maximal and
minimal values that would serve to confirm and generalize the combinatorial
calculations are impossible to carry out in the case of Problem 2. Only one of
the four functions of probability density, namely ${{p_{i}m_{i}}\over t_{i}}$,
can be obtained from the probability distribution given by
${e^{e^{\beta}}}\over t_{i}$ , but this is not achieved by either finding the
maximal value distribution or by constructing the distribution of the
collective minimal value.

Thus, Problem 2, which describes a three-level hierarchic system, highlights
the splitting of the modes of behavior of the lower-level particles into four
types. Two of them describe the condensation of the particles into states with
low energies. One type of particles, described by the probability density
${{p_{i}m_{i}}\over t_{i}}$, is closed to the type of particles described by
the function ${{p_{i}}\over m_{i}}$ in Problem 1. In a naive representation,
this type (like the similar one in Problem 1), possibly corresponds to
”normal” bound particles.

It is unclear what the fourth probability density ${{p_{i}}\over{m_{i}t_{i}}}$
describes.

##  VI On the negligible probability to find negative-energy particles

Let us note that, if the argument $x$ is allowed to vary over the range from
$-\infty$ to $+\infty$, the function ${1\over m_{i}}=e^{-e^{-(x-x_{im})}}$
exactly becomes a probability distribution of some random quantity. Let us
keep the relation $x=|\gamma|{\epsilon}_{i}$. It is interesting that for a
definite, not too large value $x_{i}m\simeq\alpha=|\gamma|\mu\geq 6$, the
lower-level particles with negative energies are totally absent (the
probability of finding at least one particle makes a vanishing quantity $\sim
10^{-180}$, as illustrated by Fig. 7.

Figure 7:  The distribution function $1\over m_{i}$ vs energy ${\epsilon}_{i}$
for $\alpha=|\gamma|\mu$ equal to 0 for the point curve, 1 for the dashed
curve, 2 for the dot-dashed curve, and 6 (more exactly, $6.025$) for solid
curve. For $\alpha=|\gamma|\mu=6.025$, the function for zero value of its
argument would make $~{}10^{-180}$. The event of the appearance of a particle
with zero energy with the probability of $~{}1$ would have taken around
$10^{180}$ higher-level particles, which is likely more than is available
across the universe. Uniform coordinates, $|\gamma|$ is set to $1$. The inset
- the solid curve with $\alpha=6.025$, the semilogarithmic scale

Since the delimitation of lower-level particles within lower-level ones is
achieved due to an interaction of some nature, it implies non-zero values of
the chemical potential. Apparently, the interaction is responsible for the
absence of particles in negative energy states.

The following remark is due in connection with the functions arising in
Problem 2. The function

${e^{e^{\beta}}\over t_{i}}=e^{e^{\beta}}e^{-e^{{\beta}+{e^{\alpha-x}}}}$ (37)

is a distribution of some random variable provided that the argument
$x=|\gamma|{\epsilon}_{i}$ varies from $-\infty$ to $+\infty$ (Fig. 8). For
$\alpha=0$, this function, depicted as Curve 3, has the value for zero
argument $<10^{-180}$ already for $\beta\geq 3.4$. If the span of the argument
starts with zero (or some other value corresponding to the ground state of a
lower-level particle), differentiation easily produces the function
${p_{i}m_{i}}\over t_{i}$ (Fig. 6).

Figure 8:  Probability distributions given by Eq. (36) for various values of
$\alpha$ and $\beta$: curve 1, $\alpha$=0 $\beta$=-1, curve 2, $\alpha$=0
$\beta$=1, curve 3 $\alpha$=0 $\beta$=3.4, curve 4 $\alpha$=2 $\beta$=-1,
curve 5 $\alpha$=2 $\beta$=1, curve 6 $\alpha$=2 $\beta$=3.4. Uniform
coordinates, $|\gamma|$ is set to 1.

The function of probability density also describes lower-level particles,
which cannot exist in negative-energy states. At least one of the chemical
potentials not being zero is sufficient to warrant the statement (see above).

##  VII Discussion of results.

The results for Problems 1 and 2 can be interpreted as depended on the method
of observation. All observations in spectroscopic experiments (not necessarily
limited to optics, but generally dealing with the measurements of the energy
spectra) produce registrations of distributions pertinent to a higher-level
particle. The main result for Problem 1 is the finding of the maximum of the
probability density function for a distribution of particles over energies for
a one-particle observation. In the framework of the simplest descriptive
approach, the maximum coincides with the chemical potential of lower-level
particles within the higher-level particles, i.e. the energy of
binding/release of a lower-level particle to/from likewise species within a
higher-level particle. The simple treatment considers all particles with
energies below the chemical potential as bound.

As it has been already stated, for low effective temperatures the density of
the probability to find a lower-level particle in the state ${\epsilon}_{i}$
via a one-particle measurement exhibits an extremely narrow peak. This
translates into no more than one particle occupying the state at low (and yet
lower) temperature. The result can be understood as an effective Pauli
principle with one energy state taken by one particle of a certain sort,
corresponding to the minus sign in the left-hand side of Eq. (13). A
modification of Problem 1 giving rise to a set of functions ${p_{i}\over
m_{i}}$ with various $\alpha$ yields, as the functions add up, a function
close to the Fermi distribution. This is yet another potential realization of
the effective Pauli principle which emerges from the solution to Problem 1.

It is furthermore established that the function of the probability density
${p_{i}\over m_{i}}$ describes the distribution of the maximal value of
$p_{i}$ in $m_{i}$ sample, or, which is the same in terms of particles, the
probability density of finding a lower-level particle in a maximal energy
state over the entire higher-level sample. Since, obviously, only one lower-
level particle can occupy a maximal energy state, the situation amounts to a
direct realization of the Pauli principle.

The behavior of another observable function in Problem 1, namely $p_{i}m_{i}$,
is also governed by the effective temperature. Already for moderate values
$\alpha\simeq 6$ the share of particles subject to the normalized probability
density distribution given by Eq. Eq. (16) with energies beyond the
corresponding chemical potential becomes $\simeq 10^{-180}$, i.e. such (free)
particles are unlikely to exist in the universe. The formulation of the
problem about the probability density function for finding lower-level
particles with minimal energies brings about the same result. A nave
interpretation of the results of treating Problem 1 with two distribution
functions $p_{i}m_{i}$ and ${p_{i}\over m_{i}}$ is that we deal with particles
of two sorts - pseudobosons in the case of the first function (as it describes
condensation in low-energy states for low temperature) and pseudofermions (as,
for low temperatures, one energy state, seen to correspond to the maximal
energy, is occupied by only one particle).

The same interpretation of the outcome for Problem 2 reveals 4 sorts of
particles. Pseudobosons with an even more pronounced condensation effect are
described by the function $p_{i}m_{i}t_{i}$, the particles with energies
beyond the chemical potential being totally absent. Similar results are
sustained by the function ${p_{i}t_{i}}\over m_{i}$, but it reflects a weaker
condensation effect. In contrast, the higher-level particles described by the
function $p_{i}\over{m_{i}t_{i}}$ contain lower-level particles in energy
states beyond the chemical potential, while the function ${p_{i}m_{i}}\over
t_{i}$ appears to describe some ”normal” particles as the total chemical
potential of a lower-level particle within an intermediate-level one, and of
the latter - within a higher-level particle combine the chemical potentials of
the lower-level particle in an intermediate-level particle and of the
intermediate-level particle in a higher-level particle.

The most nave interpretation of the extremely low values of the of the
probability distributions $1\over m_{i}$ and ${e^{e^{\beta}}}\over t_{i}$ for
negative values of the normalized energy (in the energy range from $-\infty$
to $+\infty$), may be that the lower-level particles with negative energies
(or with energies below a certain negative threshold ${{\epsilon}_{0}}$ \- see
footnote 3) may not combine into higher-level particles, or that particles
with energies below the ground state cannot exist. 555It pulls to treat this
fact to absent of antiparticles. The reason behind this, i.e. the reason for
such low values of the distribution function for negative arguments, is the
delimitation of some of the particles within others, or an implicit assumption
of an interaction taking place.

##  VIII Acknowledgments.

The author acknowledges useful discussions of results with I.V. Kolokolov and
O.B. Shiryaev.

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