Bose and Fermi statistics

For a noninteracting single-particle mode of energy $\epsilon$, define $r=e^{-\beta(\epsilon-\mu)}$. A spinless fermionic mode can be empty or singly occupied, so its grand partition factor is $1+r$. A bosonic mode admits every nonnegative occupation, giving $1+r+r^2+\cdots=1/(1-r)$ when $r<1$. Differentiating the logarithm with respect to the occupation weight produces the mean occupation. The sign difference in the final distributions comes directly from these allowed state counts.

A mode includes all quantum numbers needed to specify one single-particle state. Two fermions with opposite spins may share a spatial orbital because they occupy distinct spin modes; exclusion does not forbid that. The factorization into independent mode sums assumes the Hamiltonian is a sum of mode energies times occupations. Interactions generally correlate occupations and invalidate this simple product, even though the underlying bosonic or fermionic exchange symmetry remains exact.

A fermionic occupation is Bernoulli, with variance $\bar n_F(1-\bar n_F)$. It vanishes as the mode becomes certainly empty or certainly full. A bosonic geometric occupation has variance $\bar n_B(1+\bar n_B)$, which exceeds its mean. The extra term expresses enhanced occupation fluctuations in this ideal thermal mode. It is not a claim that every bosonic state has bunching: a fixed-number Fock state has zero number variance and is a different preparation.

When $r\ll1$, both means approach $r$, and their variances approach their means. Exchange effects then become small, recovering Maxwell–Boltzmann counting. For a gas, the appropriate dilute condition involves the density of occupied states, often summarized by $n\lambda_T^3\ll1$ with spin degeneracy handled consistently. Comparing formulas at fixed $\mu$ does not keep density fixed. To compare two gases at the same total density, determine the chemical potential separately from the number constraint in each statistics.

Mean occupations are not yet total particle number or energy. Sum them over the actual single-particle spectrum, or integrate against a density of states only when the level spacing is sufficiently fine. At low temperature, ideal fermions fill states up to a Fermi energy at fixed density, so appreciable kinetic energy survives even without thermal excitation. For ideal bosons in suitable dimensions and confinement, excited states can saturate, allowing a macroscopically occupied ground mode in the thermodynamic limit.

Dimensionality, dispersion, and trapping geometry determine whether that saturation argument works. A uniform ideal gas in one or two dimensions does not have ordinary finite-temperature Bose condensation in its standard thermodynamic limit. A finite box always retains discrete levels and smooth finite partition functions. Interactions can support other collective phenomena and alter fluctuations. A divergent grand canonical ground-mode mean as $\mu$ approaches its energy signals the need to enforce the physical number constraint; it is not a prediction of infinitely many particles in a finite fixed-number experiment.