  Charge and energy are expected to diffuse in interacting systems of fermions
at finite temperatures, even in the absence of disorder, with the interactions
inducing a crossover from the coherent and ballistic streaming of
quasi-particles at early times, to incoherent diffusive behavior at late times.
The relevant crossover timescales and the transport coefficients are both
controlled by the strength of interactions. In this work we develop a numerical
method to simulate such systems at high temperatures, applicable in a wide
range of interaction strengths, by adapting Dissipation-assisted Operator
Evolution (DAOE) to fermions. Our fermion DAOE, which approximates the exact
dynamics by systematically discarding information from high $n$-point
functions, is tailored to capture non-interacting dynamics exactly, thus
providing a good starting point for the weakly interacting problem. Applying
our method to a microscopic model of weakly interacting fermions, we
numerically demonstrate that the crossover from ballistic to diffusive
transport happens at a time $t_D\sim1/\Delta^{2}$ and that the diffusion
constant similarly scales as $D \sim 1/\Delta^2$, where $\Delta$ is the
interaction strength. We substantiate this scaling with a Fermi's golden rule
calculation in the operator spreading picture, interpreting $t_D$ as the
fermion-fermion scattering time and lifetime of the single-particle Green's
function.
