Impurity in a zero-temperature three-dimensional Fermi gas

We consider an impurity in a sea of zero-temperature fermions uniformly distributed throughout the space. The impurity scatters on fermions. On average, the momentum of impurity decreases with time as $t^{-1/(d+1)}$ in $d$ dimensions, and the momentum distribution acquires a scaling form in the long time limit. We solve the Lorentz-Boltzmann equation for the scaled momentum distribution of the impurity in three dimensions. The solution is a combination of confluent hypergeometric functions. In two spatial dimensions, the Lorentz-Boltzmann equation is analytically intractable, so we merely extract a few exact predictions about asymptotic behaviors when the scaled momentum of the impurity is small or large.