Using the general formulation for obtaining chemical potentialμ of an ideal Fermi gas of particles at temperature T, with particle rest mass m₀ and average density 〈N〉/V, the dependence of the mean square number fluctuation 〈ΔN²〉/V on the particle mass m₀ has been calculated explicitly. The numerical calculations are exact in all cases whether rest mass energym₀c² is very large (non-relativistic case), very small (ultra-relativistic case) or of the same order as the thermal energy k_BT. Application of our results to the detection of the universal very low energy cosmic neutrino background (CNB), from any of the three species of neutrinos, shows that it is possible to estimate the neutrino mass of these species if from approximate experimental measurements of their momentum distribution one can extract, someday, not only the density 〈N_v〉/V but also the mean square fluctuation 〈Δ_v²〉/V. If at the present epoch, the universe is expanding much faster than thermalization rate for CNB, it is shown that our analysis leads to a scaled neutrino massm_v instead of the actual massm_0v.

We use different determinantal Hartree–Fock (HF) wave functions to calculate true variational upper bounds for the ground state energy of 𝑁 spin-half fermions in volume $V_{0}$, with mass 𝑚, electric charge zero, and magnetic moment 𝜇, interacting through magnetic dipole–dipole interaction. We find that at high densities when the average interparticle distance $r_{0}$ becomes small compared to the magnetic length $r_{m} \equiv 2m\mu^{2}/\hbar^{2}$, a ferromagnetic state with spheroidal occupation function $n_{\uparrow} (\vec{k})$, involving quadrupolar deformation, gives a lower upper bound compared to the variational energy for the uniform paramagnetic state or for the state with dipolar deformation. This system is unstable towards infinite density collapse, but we show explicitly that a suitable short-range repulsive (hard core) interaction of strength $U_{0}$ and range a can stop this collapse. The existence of a stable equilibrium high density ferromagnetic state with spheroidal occupation function is possible as long as the ratio of coupling constants $\Gamma_{cm} \equiv (U_{0}a^{3}/\mu^{2})$ is not very smallcompared to 1.