Low-dimensional Fermi and Bose gases

In the ideal Fermi gas a physical interpretation can be given to a curious “hump” that develops in the chemical potential μ(T) as a function of absolute temperature T for any spatial dimensionality d<2, integer or not. This contrasts with the more familiar monotonic decrease for d≥2. The hump height increases without limit as d decreases toward zero where μ=+∞. This positive divergence at d→0 is argued to be a clear manifestation of the Pauli Exclusion Principle in configuration space, whereby two spinless fermions cannot sit on top of each other. The observed hump is thus an obvious precursor of this manifestation, otherwise well understood, say, in the 1s level of the H atom. The ideal Bose gas for d<2 is also reexamined and found impossible to be confined at all in d→0 as it exhibits the opposite divergence μ=−∞ there. Both divergences are seen to follow from the Heisenberg Uncertainty Principle.