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Distribution Functions in Physics and Mathematics Why Fermi–Dirac and Bose–Einstein Balance Equations are not Distributions from an Axiomatic Perspective

Pascal Moser

Published Aug 3, 2026• 0 citations
Publisher

Abstract

The term ‘Verteilungsfunktion’ (distribution function) is used in physics as a semantic container for at least four ontologically distinct mathematical objects: proper probability density functions (PDFs), probability mass functions (PMFs), expectation values of occu-pation numbers, and phase-space measure densities. This conflation arose historically because the statistical mechanics of Boltzmann, Maxwell, Fermi, Dirac and Bose predates Kolmogorov’s axiomatic probability theory (1933) by decades; no systematic back-mapping was ever performed. The central finding concerns the Fermi–Dirac and Bose–Einstein functions specifically: they are not merely mislabeled — they are relative measurements, bounded observable expectation values ⟨nk⟩-normalised against their capacity. Each of the five objects routinely called a ‘distribution function’ is audited against the Kolmogorov taxonomy: the Maxwell–Boltzmann speed distribution is a proper PDF that is merely mislabeled; the Boltzmann canonical distribution is a genuine probability measure; the Boltzmann-transport phase-space density is normalised toNrather than to unity; and the Fermi–Dirac and Bose–Einstein functions are occupation-number expectation values that acquire physical meaning only as convolution kernels against the density of states. A short supporting section shows that this is not an accident of two particular formulas. One elementary theorem establishes that a bounded occupation of capacityMunder Boltz-mann weighting with an additive constraint yields⟨n⟩M= −d ln ZM/dx; atM=1 this is exactly the logistic function, and at M→∞it diverges. Fermi–Dirac and Bose–Einstein are therefore the two branches of a single construction, which is why the first is bounded in [0, 1]and the second admits no bounded normalisation at all. The same unit-capacity form appears outside physics wherever a bounded quantity accumulates additive evidence, which suggests that such objects are best read as relative measurements against a capacity rather than as distributions. The paper reconstructs the historical sequence that produced the terminological divergence and proposes notational conventions to restore conceptual clarity in pedagogy and literature. Keywords: probability density function, distribution function, Fermi–Dirac, Bose–Einstein, Maxwell–Boltzmann, Kolmogorov, Jaynes, ontology, statistical mechanics, probability theory, quantum statistics, foundations of physics,philosophy of science, physics education, information theory, thermodynamics.

DOI
10.5281/zenodo.21772119

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The term ‘Verteilungsfunktion’ (distribution function) is used in physics as a semantic container for at least four ontologically distinct mathematical objects: proper probability density functions (PDFs), probability mass functions (PMFs), expectation values of occu-pation numbers, and phase-space measure densities. This conflation arose historically because the statistical mechanics of Boltzmann, Maxwell, Fermi, Dirac and Bose predates Kolmogorov’s axiomatic probability theory (1933) by decades; no systematic back-mapping was ever performed. The central finding concerns the Fermi–Dirac and Bose–Einstein functions specifically: they are not merely mislabeled — they are relative measurements, bounded observable expectation values ⟨nk⟩-normalised against their capacity. Each of the five objects routinely called a ‘distribution function’ is audited against the Kolmogorov taxonomy: the Maxwell–Boltzmann speed distribution is a proper PDF that is merely mislabeled; the Boltzmann canonical distribution is a genuine probability measure; the Boltzmann-transport phase-space density is normalised toNrather than to unity; and the Fermi–Dirac and Bose–Einstein functions are occupation-number expectation values that acquire physical meaning only as convolution kernels against the density of states. A short supporting section shows that this is not an accident of two particular formulas. One elementary theorem establishes that a bounded occupation of capacityMunder Boltz-mann weighting with an additive constraint yields⟨n⟩M= −d ln ZM/dx; atM=1 this is exactly the logistic function, and at M→∞it diverges. Fermi–Dirac and Bose–Einstein are therefore the two branches of a single construction, which is why the first is bounded in [0, 1]and the second admits no bounded normalisation at all. The same unit-capacity form appears outside physics wherever a bounded quantity accumulates additive evidence, which suggests that such objects are best read as relative measurements against a capacity rather than as distributions. The paper reconstructs the historical sequence that produced the terminological divergence and proposes notational conventions to restore conceptual clarity in pedagogy and literature. Keywords: probability density function, distribution function, Fermi–Dirac, Bose–Einstein, Maxwell–Boltzmann, Kolmogorov, Jaynes, ontology, statistical mechanics, probability theory, quantum statistics, foundations of physics,philosophy of science, physics education, information theory, thermodynamics.
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Addendum July 19, 2021 The Fermi-Dirac and Bose-Einstein statistical cases, like the Maxwell-Boltzmann case, may be formulated in terms of a reaction balance between a regular two body elastic collision and its time reversed reaction: e1+e2 = e3 + e4 and f(e1)[1-/+f(e3)]f(e2)[1-/+f(e4)] = f(e3)[1-/+ f(e1)] f(e4)[1-/+ f(e2)]. Both the Fermi-Dirac and Bose-Einstein cases reduce to the Maxwell-Boltzmann case under certain conditions. An unusual feature of bound state quantum mechanics is that there is a spatial density (or probability to be at x = P(x)), just as in classical statistical mechanics, but also a wavefunction W(x) such that P(x)=W(x)W(x). In other words, W(x) for the bound state is like a square root of probability. Furthermore, there is a P(p)=a(p)a(p) where W(x)=Sum over p a(p)exp(ipx). In this note, we try to argue that P(p), which is based on the measurement of momenta of particles knocked out of an ensemble of identical single particle bound states, does not have physical meaning within the bound state, unlike P(p,x) which has physical meaning for a Maxwell-Boltzmann gas both inside the gas and for particles released from a small hole in the gas wall. This forces one to create another function to “link” the two physically observed probabilities P(x) and P(p).
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
We analyze the so-called classical limit of the quantum-mechanical canonical partition function. In order to do that, we define accurately the density matrix for symmetrized and antisymmetrized wave functions only (Bose-Einstein and Fermi-Dirac), and find an exact relation between them and the density matrix for non symmetrized functions (Maxwell-Boltzmann). Our results differ from the generally assumed in a numerical factor N!, for which we suggest a physical interpretation. We derive as well the reverse (and also exact) relation, to find the canonical partition function for non-symmetrized wave functions in terms of the corresponding function for fermions and bosons.
If one tosses a die or coin, different outcomes occur with a probability p(i) where i is the outcome. This probability may be measured by performing trial runs. For N trial runs, where N is very large, one might expect Np(outcome i) results of an outcome i. In other words, p(outcome i) is both the probability and describes the trial run. Even in the Maxwell-Boltzmann case, one has a probability p(ei) where ei is kinetic energy (no V(x)) and reaction balance p(e1)p(e2) = p(e3)p(e4) represents trial runs. Thus the trial run is again characterized by p(ei) as ei’s contribution. It is, however, possible to have coupling of probabilities in certain physical situations even though p(outcome i) still exists. In such a case, the number of outcome i results for N large trials is not N p(outcome i). In other words, p(i) does not fully describe the trial run. There may be other physical rules in place. If elastic scattering of particles is the way trial runs occur, e1+e2 = e3 + e4 represents the salient information in the problem and even if there is nonindependence of p(outcome i) in the scattering, one should at least obtain a function Q(p(e1)) such that Q(p(e1)) = -e1/T. Q(p(ei)), however, need not be p(ei). We consider the physical rule given by Pauli stating that only one particle may fit in an energy level (no spin). Thus there is either 0 or 1 particle in the level. This should inhibit scattering, so the trial run is no longer simply described by p(ei). Rather the probability to scatter into level ej is (1-n(ej)) and so one has: n(e1)[1-n(e3)] n(e2)[1-n(e4)]= n(e3)[1-n(e1)]n(e4)[1-n(e2)] where n(ej) is the average occupancy, a number between 0 and 1. The opposite of only 0 or 1 particle in a state ei, is to have any possible number of particles in this state and an enhancement, not a hindrance given by 1+p(ej). This is only a guess, but experiment shows that bosons follow this rule. Given a definition of trial runs in terms of the inhibition and enhancement factors, one may do two things. First, since the trial rule seems to involve different numbers of particles in a state associated with a p(ei), one may consider sums of p(ei) (power number of particles) when describing the system statistically (i.e. grand canonical scheme). This approach gives rise to linear terms (a+bp(ei)) which may in turn be formulated in terms of a factorial/arrangement equation which may be maximized subject to the constraints Sum over i ei n(ei) and Sum over i n(ei). We note that this maximization is not simply in terms of all degrees of freedom in the system, but is specifically linked with the rules associated with the trial runs and the constraints. We note, however, that issues seem to arise with the factorial scheme in the boson case. Thus, overall we argue that a single particle probability and its trial run scheme are key to establishing the statistical behaviour of a system. (We also note that single particle independent probability, which is an approximation, also affects the final results.)
Bose gases and Fermi gases represent two fundamental particle systems in statistical mechanics, governed by Bose-Einstein and Fermi-Dirac distributions, respectively. This paper provides a comprehensive analysis of the thermodynamic quantities, phase transitions, chemical potentials, and energy level relationships of these gases. The study utilizes Laurent series expansions and group theory analysis to explore the statistical properties and behavior of Bose and Fermi gases under various conditions. Additionally, the paper derives a new energy-mass equation based on metric and curvature RRR.