  Simulations are increasingly employing explicit reservoirs - internal, finite
regions - to drive electronic or particle transport. This naturally occurs in
simulations of transport via ultracold atomic gases. Whether the simulation is
numerical or physical, these approaches rely on the rapid development of the
steady state. We demonstrate that steady state formation is a manifestation of
the Gibbs phenomenon well-known in signal processing and in truncated discrete
Fourier expansions. Each particle separately develops into an individual steady
state due to the spreading of its wave packet in energy. The rise to the steady
state for an individual particle depends on the particle energy - and thus can
be slow - and ringing oscillations appear due to filtering of the response
through the electronic bandwidth. However, the rise to the total steady state -
the one from all particles - is rapid, with timescale $\pi/W$, where $W$ is the
bandwidth. Ringing oscillations are now also filtered through the bias window,
and they decay with a higher power. The Gibbs constant - the overshoot of the
first ring - can appear in the simulation error. These results shed light on
the formation of the steady state and support the practical use of explicit
reservoirs to simulate transport at the nanoscale or using ultracold atomic
lattices.
