Quantum corrections to the momentum distribution of moderately dense Boltzmann gases. I. Smooth potentials

The momentum distribution of a quantum Boltzmann gas at equilibrium is shown to be the usual Maxwell–Boltzmann term plus a correction term of higher order in density and fully quantum -mechanical in origin. For smooth potentials, the correction may be described by a semiclassical expansion, and the leading quantum part depends on momentum in the simple manner postulated by Imam-Rahajoe and Curtiss, although higher-order terms have a more complicated momentum dependence. At lowest order in density, a Sonine polynomial expansion and expressions for the moments of the quantum distribution are derived. The pressure tensor and energy density of a quantum gas, as determined from kinetic theory, are shown to be consistent with thermodynamics in the equilibrium limit, but only when the quantum part of the momentum distribution is properly taken into account.