  Open system simulations of quantum transport provide a platform for the study
of true steady states, Floquet states, and the role of temperature,
time-dynamics, and fluctuations, among other physical processes. They are
rapidly gaining traction, especially techniques that revolve around "extended
reservoirs" - a collection of a finite number of degrees of freedom with
relaxation that maintain a bias or temperature gradient - and have appeared
under various guises (e.g., the extended or mesoscopic reservoir, auxiliary
master equation, and driven Liouville-von Neumann approaches). Yet, there are
still a number of open questions regarding the behavior and convergence of
these techniques. Here, we derive general analytical solutions, and associated
asymptotic analyses, for the steady-state current driven by finite reservoirs
with proportional coupling to the system/junction. In doing so, we present a
simplified and unified derivation of the non-interacting and many-body
steady-state currents through arbitrary junctions, including outside of
proportional coupling. We conjecture that the analytic solution for
proportional coupling is the most general of its form for isomodal relaxation
(i.e., relaxing proportional coupling will remove the ability to find compact,
general analytical expressions for finite reservoirs). These results should be
of broad utility in diagnosing the behavior and implementation of extended
reservoir and related approaches, including the convergence to the Landauer
limit (for non-interacting systems) and the Meir-Wingreen formula (for
many-body systems).
