The multiplicity Ω ( E , V , N ) of an N-particle, mono-atomic, ideal gas with energy E and volume V is the product of the number of cells of volume δ s 3 available in position space (that equals V / δ s 3 ) and the number of cells available in momentum space [4]. The volume of momentum space through which the system may move is the p-volume ( 2 e π / 3 N ) 3 N / 2 3 N R 3 N − 1 δ p of a one-cell-wide shell of radius R = 2 m E . As a consequence, the number of permissible momentum-space cells in this shell is (e is the basis of natural logarithms)

V p-shell = ( 2 e π / 3 N ) 3 N / 2 3 N ( 2 m E / 3 N ) ( 3 N − 1 ) / 2 ( δ p / δ p 3 N ) , (1)

so that [4]

Ω ( E , V , N ) = ( V / δ V ) N ( 2 e π ) 3 N / 2 3 N ( 2 m E / 3 N ) ( 3 N − 1 ) / 2 ( δ p / δ p 3 N ) . (2)

After some lengthy considerations, the author of [4] [pp. 56-57] [Eqs. (3.8)-(3.15)] rephrases the above relations as

Ω ( E / N , V ,1 ) = V ( E / N ) 3 / 2 ( 4 e π m / 3 h 2 ) 3 / 2 , (3)

an equation to be discussed below.

It has been known since at least 1925 that quantum mechanics includes, among its tenets, the following tree conditions: 1) phase space cells have a size determined by Planck’s constant h, 2) the energy, momenta, and other dynamical properties of an isolated system are quantized, and 3) for the purpose of determining multiplicity, identical particles are, indistinguishable from one another. Thus, (3) gives the multiplicity Ω of an ideal gas composed of N distinguishable particles that occupy volume V and share an energy E.

If Ω ( E , V , N ) is the multiplicity of an ideal gas composed of N distinguishable particles with total energy E and placed in a volume V, it follows that Ω ( E / N , V ,1 ) should be the multiplicity of just a single particle of ideal gas, characterized by E/N and, and occupying volume V. This particle composes a larger system of N distinguishable particles. A single particle of an ideal gas occupies Ω ( E / N , V ,1 ) equally probable microstates. Accordingly,

Ω ( E , V , N ) = [ Ω ( E / N , V ,1 ) ] N . (4)

A crucial insight is here gotten: the number of cells that can be occupied by a single particle of ideal gas must be independent of whether that particle is itself distinguishable or indistinguishable from the other particles that compose the gas system and if indistinguishable whether fermion or boson. A system of N distinguishable particles of ideal gas may occupy [ Ω ( E / N , V ,1 ) ] N equally probable microstates [4].

Focus attention now upon an ideal Fermi gas. Let us call n the number of cells that can be occupied by a single fermion E/N. We have n / N ≥ 1 . According to (3) we have

n = V ( E / N ) 3 / 2 ( 4 e π m / 3 h 2 ) 3 / 2 , (5)

with

V = 4 3 π r 3 . (6)

We face the problem of filling n cells with N ≤ n identical fermions. Thus the multiplicity becomes [4]

Ω = n ! N ! ( n − N ) ! . (7)

This multiplicity derived above is the result of the so-called average energy approximation. According to this approximation, each gas particle possesses the same energy E/N for all fermions in the system. That the average energy approximation produces the exact multiplicity when identical particles are considered distinguishable. The average energy approximation has been extensively used and produces quite reasonable results [4].

We approximate the Gamma function using the Stirling recipe

Γ ( z ) = 2 π z z − 1 2 e − z , (8)

and find ( Θ is Heaviside’s step function)

Ω = ee ( n + 1 ) ln ( n + 1 ) − Θ ( n − N + 1 ) ( n − N + 1 ) ln ( n − N + 1 ) 2 π ( N + 1 ) ( N + 1 ) , (9)

and introduce the notationally simplifying definition (change of variables from r to s)

s = ( 4 π 3 N ) 1 3 ( E N ) 1 2 ( 4 π e m 3 h 2 ) 1 2 r , (10)

so that

Ω = ee ( 1 + N s 3 ) l n ( 1 + N s 3 ) e − Θ [ N s 3 − ( N − 1 ) ] [ N s 3 − ( N − 1 ) ] l n [ N s 3 − ( N − 1 ) ] 2 π ( N + 1 ) ( N + 1 ) . (11)

Of course, it must be

Ω ≥ 1 ;       l n ( Ω ) ≥ 0. (12)

From (11) one gets

ln ( Ω ) = ( 1 + N s 3 ) ln ( 1 + N s 3 ) − Θ [ N s 3 − ( N − 1 ) ] [ N s 3 − ( N − 1 ) ]     × ln [ N s 3 − ( N − 1 ) ] + 1 − 1 2 ln ( 2 π ) − ( N + 1 ) ln ( N + 1 ) , (13)

and we ask for a possible vanishing of l n ( Ω ) . This is the critical novel issue that we address in this paper.

Since N ≫ 1 we are led to answer with the relation

l n ( Ω ) N ≃ s 3 l n ( s 3 ) − Θ ( s 3 − 1 ) ( s 3 − 1 ) l n ( s 3 − 1 ) = 0, (14)

that has two possible solutions, namely,

s = 0 ,       s = 1. (15)

Figure 1 shows that the multiplicity Ω would be negative for 0 ≤ s ≤ 1 , which is absurd. We gather that the system is subjected to a kind of “phase transition” at s = 0 and forced to “jump” to s = 1 . One could putatively associate this jump to a sort of inflation-phenomenon motivated by the Pauli principle that would forbid N fermions to be accommodated in a too small volume V. A critical “accommodating” volume is reached at s = 1 . Emphasize that this happens for free fermions micro-canonically described. Equilibrium prevails.

The paper could well finish here. However, it is too tempting to extrapolate a bit further. We do this below, after discussing Bosons.

The multiplicity is, for n microstates and N bosons [4],

Ω = ( N + n − 1 ) ! N ! ( n − 1 ) ! . (16)

The counting performed above is structurally identical to counting the number of distinct manners of ordering a set of N identical balls and n − 1 identical white dividers. If placed in a row, these n − 1 dividers separate our N balls into n ordered (distinct) groups [4].

Thus, things are quite different for bosons, not subjected to the exclusion principle. The above Ω , for N bosons of mass m contained in V with total energy E will yield [4]

Ω = e ( N + N s 3 ) ln ( N + N s 3 ) e − ( N s 3 ) ln ( N s 3 ) 2 π N N , (17)

with s given by (2).

ln ( Ω ) = ( N + N s 3 ) ln ( N + N s 3 ) − ( N s 3 ) ln ( N s 3 ) − 1 2 ln ( 2 π ) − N ln ( N ) . (18)

From (7) it follows that, in the spirit of last Subsection,

l n ( Ω ) N ≃ ( 1 + s 3 ) l n ( 1 + s 3 ) − Θ ( s ) s 3 l n ( s 3 ) = 0, (19)

whose solution is

s = 0 , (20)

as illustrated in Figure 2.

What might have happened when fermions began to emerge at the Big-bang?

A tiny fraction of a second after the singularity, some fermions began to emerge, out of a quark-gluon plasma [5], N of them, at a tiny region of size V, whose total fermion-energy was E (this variables determine a microcanonical ensemble). This region had to “explode” in order to accommodate them (transition from s = 0 to s = 1 above).

If we accept Siegel’s estimate for the Universe’s radius at the end of the inflation period of r = r 0 = 17   cm [5], then, from Equation (11), one gathers that one has for N, when s = 1 ,

N = 4 π 3 ( m c ) 3 2 ( 4 π e m 3 h 2 ) 3 2 r 0 . (21)

If we set E N = m c 2 (lower bound for E), with m = 1.6 × 10 − 27 kg the baryon

mass, then the fermion number becomes N = 1.15 × 10 48 at the critical volume referred to above, a very small number compared to today’s estimate of 10⁷⁹. Of course, radiation, neutrinos and plasma predominate at this stage [5].

The entropy at the end of the inflationary period ( s = 1 ) reads

S | s = 1 = k B ln Ω | s = 1 , (22)

with k B the Boltzmann constant. For the temperature T we have

1 T | s = 1 = ∂ S ∂ E | ( V , N , s = 1 ) , (23)

or

1 T | s = 1 = ∂ S ∂ s 3 | ( V , N , s = 1 ) ∂ s 3 ∂ E | ( V , N , s = 1 ) . (24)

More explicitly, one has

1 T | s = 1 = 2 π r 0 3 k B m 2 c h 3 ( 4 π e 3 ) 3 2 ln ( 1 + N ) . (25)

Using the appropriate values for k B , h, etc., we obtain T = 3.9 × 10 18 Kelvin degrees, which agrees with the value estimated in reference [6]. Note that in our case we are advancing a statistical prediction.