Extending Kroemer's two-sided bound on the Bose–Fermi partition-function difference beyond the spinless, noninteracting, continuum classical regime

Kroemer (American Journal of Physics 48, 962 (1980)) quantified how far the classical Maxwell–Boltzmann partition function $Z_1^N/N!$ of the ideal gas can be from the exact bosonic or fermionic partition functions: in the classical regime, the multiple-occupancy miscounting satisfies the two-sided bound $f\,B_N>B_N-F_N>f\,F_N$ with $f=2^{-3/2}n^2V/n_Q$, a factor of order $10^{13}$ for one cubic centimeter of helium at room temperature, while observables remain correct because they depend on $\log Z$. The bound is proved only for spinless, monatomic, noninteracting particles with a continuum box spectrum, in systems large enough that removing two particles does not change the chemical potential. The open problem is to extend the bound and its prefactor to the four regimes the note's own restrictions exclude: (a) spin and orbital degeneracy; (b) small systems and stronger quantum degeneracy where the chemical-potential step fails; (c) interparticle interactions; and (d) discrete level structures or confining geometries. No published extension of the bound is known.

Background

In the canonical ensemble of statistical mechanics, the partition function $Z_N=\sum_s e^{-E_s/kT}$ — the sum of Boltzmann factors $e^{-E_s/kT}$ over the exact quantum states $s$ of an $N$-particle system at temperature $T$ — generates all equilibrium thermodynamics through its logarithm: the Helmholtz free energy is $-kT\log Z_N$, and the entropy, pressure, and energy follow by differentiation. For identical particles the exact sum depends on quantum statistics. Bosons (particles whose wavefunction is symmetric under exchange) allow any number of particles to occupy a given one-particle state, called an orbital; fermions (antisymmetric wavefunction) obey the Pauli exclusion principle, which permits at most one particle per orbital. The textbook classical (Maxwell–Boltzmann) recipe for the ideal gas instead writes $Z_N=Z_1^N/N!$, where $Z_1$ is the single-particle partition function and the divisor $N!$ corrects for indistinguishability. This recipe is justified only when multiple occupancy of orbitals is negligible, that is, in the classical regime where the concentration $n=N/V$ is much smaller than the quantum concentration $n_Q=(MkT/2\pi\hbar^2)^{3/2}$, the inverse cube of the thermal de Broglie wavelength of a particle of mass $M$ — a condition equivalently expressed through the chemical potential $\mu$ as $\exp(\mu/kT)=n/n_Q$.

Herbert Kroemer's 1980 note in the American Journal of Physics asked how badly the classical recipe miscounts when this condition holds but multiple occupancy is not strictly absent. Let $B_N$ and $F_N$ denote the exact canonical partition functions of $N$ bosons and $N$ fermions in a box of volume $V$ at temperature $T$. The difference $B_N-F_N$ isolates the net contribution of multiply occupied orbitals (present for bosons, forbidden for fermions) and therefore brackets the absolute miscounting of any counting scheme lying between the two statistical extremes, including the classical $Z_1^N/N!$. Kroemer derived the elementary two-sided bound

$f\,B_N\;>\;B_N-F_N\;>\;f\,F_N,$

with the dimensionless prefactor

$f=\frac{1}{2^{3/2}}\frac{n^2V}{n_Q}=\frac{1}{2^{3/2}}\frac{N^2}{n_QV},$

where the continuum box spectrum gives $Z_1=n_QV$ (so that $Z_1$ evaluated at half the temperature is $n_QV/2^{3/2}$). For $1\ \mathrm{cm}^3$ of helium at 300 K and atmospheric pressure, $f\approx 2.7\times 10^{13}$: the classical value can differ from the exact quantum value by a multiplicative factor of order $10^{13}$ even well inside the classical regime. The apparent paradox dissolves because all observables depend on $\log Z_N$, whose fractional error is negligible; the classical treatment thus yields correct observables despite a wildly wrong partition function.

The derivation — which orders contributions by the energy of the lowest multiply occupied orbital, extracts a factor $\exp(-2\epsilon_n/kT)$, and relates the residual $(N-2)$-particle counting to $B_{N-2}$ and $F_{N-2}$ — rests on four scope restrictions: spin is neglected; the gas is noninteracting and monatomic; the single-particle spectrum is the continuum box spectrum; and $N$ is large enough that removing two particles leaves the chemical potential unchanged, $B_{N-2}\approx B_N\exp(2\mu/kT)$, used with $\exp(\mu/kT)=n/n_Q$. The surrounding pedagogical literature on the $1/N!$ prescription — Grandy's maximum-entropy treatment of indistinguishability, Baierlein's analysis of the probability that all particles occupy different orbitals, and Singh and Kaur's exact numerical enumeration of $B$, $F$, and the classical value for small discrete model systems, which verifies the ordering $B>Z_1^N/N!>F$ at all temperatures — treats adjacent questions but leaves the bound itself unextended.

Problem Statement

Kroemer's two-sided bound $f\,B_N>B_N-F_N>f\,F_N$ with $f=2^{-3/2}n^2V/n_Q=2^{-3/2}N^2/(n_QV)$ is proved only for a spinless, monatomic, noninteracting gas in a continuum box spectrum, in the classical regime $\exp(\mu/kT)=n/n_Q$, and for systems large enough that $B_{N-2}\approx B_N\exp(2\mu/kT)$. The open problem is to extend this quantitative error analysis to each regime the note's own restrictions excluded. For each of the following relaxations, determine whether — and, if so, in what modified form — the corresponding two-sided bound on the Bose–Fermi partition-function difference holds; where it holds, prove it, give an explicit modified prefactor expressed in terms of the system's parameters together with the bound's domain of validity, and show that it reduces to Kroemer's bound in the spinless, noninteracting, continuum classical limit:

(a) Spin and degeneracy: particles of spin $s$, with the associated degeneracy of the orbitals.

(b) Stronger degeneracy and small systems: many-body systems with $N\ge 3$, in regimes in which removing two particles appreciably changes the chemical potential, so that the step $B_{N-2}\approx B_N\exp(2\mu/kT)$ fails; the cases $N\le 2$ lie outside this regime because the step $B_{N-2}\approx B_N\exp(2\mu/kT)$ presupposes $N-2\ge 1$.

(c) Interparticle interactions: interacting systems of identical particles of mass $M$ in volume $V$ with a specified two-body interaction, whose dimensionless interaction strength at the given $N$, $V$, and $T$ is of order unity or larger, so that occupation correlations and interaction-induced energy shifts alter the counting and the single-particle spectrum and the orbital decomposition underlying the original derivation must be reformulated. For these systems, the object to bound is the difference between the exact canonical partition functions of the interacting $N$-boson and $N$-fermion systems at the given $N$, $V$, and $T$.

(d) Discrete level structure and confining geometries: systems in which $Z_1$ and $n_Q$ are not given by the continuum ideal-gas expressions.

The single scientific objective is a controlled, explicit, two-sided quantification of quantum-statistical miscounting outside the strict classical regime, recovering the known bound inside it. The intended form of the result is a two-sided bound whose modified prefactor is an explicit closed-form function of the system's parameters, independent of the exact partition functions $B_N$ and $F_N$ or of ratios of them, with a domain of validity covering the regime in question — or, where no single such prefactor exists across that regime, the demonstration of that impossibility.

References

1. Herbert Kroemer, "How incorrect is the classical partition function for the ideal gas?", American Journal of Physics 48(11), 962–963 (1980), DOI 10.1119/1.12196, https://doi.org/10.1119/1.12196
2. Sushil K. Singh and Savinder Kaur, "On the validity of classical partition function", arXiv:2104.01315 [physics.ed-ph] (2021), https://arxiv.org/abs/2104.01315; published as "Justifying the classical partition function for bosonic and fermionic systems using computer based experiments", European Journal of Physics 43(1), 015101 (2022), DOI 10.1088/1361-6404/ac3064, https://doi.org/10.1088/1361-6404/ac3064
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