  Landauer's formula is the standard theoretical tool to examine ballistic
transport in nano- and meso-scale junctions, but it necessitates that any
variation of the junction with time must be slow compared to characteristic
times of the system, e.g., the relaxation time of local excitations. Transport
through structurally dynamic junctions is, however, increasingly of interest
for sensing, harnessing fluctuations, and real-time control. Here, we calculate
the steady-state current when relaxation of electrons in the reservoirs is
present and demonstrate that it gives rise to three regimes of behavior: weak
relaxation gives a contact-limited current; strong relaxation localizes
electrons, distorting their natural dynamics and reducing the current; and in
an intermediate regime the Landauer view of the system only is recovered. We
also demonstrate that a simple equation of motion emerges, which is suitable
for efficiently simulating time-dependent transport.
