Quantum diffusion: A simple, exactly solvable model

•Exactly solvable quantum diffusion model.•Power-law decay.•Entropy production and diffusion current.•Quantum diffusion and overshoot effects.

We propose a simple quantum mechanical model describing the time dependent diffusion current between two fermion reservoirs that were initially disconnected and characterized by different densities or chemical potentials. The exact, analytical solution of the model yields the transient behavior of the coupled fermion systems evolving to a final steady state, whereas the long-time behavior is determined by a power law rather than by exponential decay. Similar results are obtained for the entropy production which is proportional to the diffusion current.