Very Low-Temperature Fermi Gas

The momentum distribution in a low-temperature Fermi gas is investigated using the methods of quantum statistics developed by Lee and Yang together with the linked-pair expansions of the previous paper. It is shown that in order to determine the momentum distribution at very low temperatures two coupled integral equations must be considered, one in momentum variables and due to Lee and Yang, and the other in temperature variables. It is also shown that the dominant low-temperature behavior of the momentum distribution can be extracted in terms of a certain function ${\nu }^{\prime }\left(\mathrm{k}\right)$. For a low-density Fermi gas with strong, short-range, two-body interactions, it is shown to third order in the scattering parameters of the interaction that at $T=0\mathrm{}$ the function ${\nu }^{\prime }\left(\mathrm{k}\right)$ is equal to the free particle momentum distribution. Also, the energy and other thermodynamic quantities are expressed in terms of ${\nu }^{\prime }\left(\mathrm{k}\right)$, so that the theory permits a generalization of perturbation theoretic results to nonzero temperatures. The ground-state energy, momentum distribution, and thermodynamic potential are calculated to third order in the scattering parameters.