Inelastic quantum Boltzmann equation of single-component Fermi gases

This work is a companion paper to X.-Y. Gao et al., Phys. Rev. Lett. 134, 153402 (2025), where we discuss the nonequilibrium two-body loss dynamics of a single-component ultracold Fermi gas and its possible thermalization. This paper provides detailed information on the derivation and analysis of the inelastic quantum Boltzmann equation (IQBE) used to describe the system. We demonstrate that the Mellin transform is a powerful tool for solving and approximating the IQBE for free-space systems. In this case, the particle-number dynamics are beyond the description of the widely used phenomenological two-body equation. For harmonically trapped systems, we propose a fast-flowing approximation to simplify the numerical evaluation of the IQBE. We verify the approximation in an analogous quasi-one-dimensional system and apply it to three-dimensional calculations, obtaining satisfactory agreement with recent experimental results. Furthermore, we compare the nonequilibrium results with those obtained using a thermal ansatz in both situations, providing a systematic understanding of the antievaporation phenomena observed in such systems.