  Open-system approaches are gaining traction in the simulation of charge
transport in nanoscale and molecular electronic devices. In particular,
"extended reservoir" simulations, where explicit reservoir degrees of freedom
are present, allow for the computation of both real-time and steady-state
properties but require relaxation of the extended reservoirs. The strength of
this relaxation, $\gamma$, influences the conductance, giving rise to a
"turnover" behavior analogous to Kramers' turnover in chemical reaction rates.
We derive explicit, general expressions for the weak and strong relaxation
limits. For weak relaxation, the conductance increases linearly with $\gamma$
and every electronic state of the total explicit system contributes to the
electronic current according to its "reduced" weight in the two extended
reservoir regions. Essentially, this represents two conductors in series -- one
at each interface with the implicit reservoirs that provide the relaxation. For
strong relaxation, a "dual" expression -- one with the same functional form --
results, except now proportional to $1/\gamma$ and dependent on the system of
interest's electronic states, reflecting that the strong relaxation is
localizing electrons in the extended reservoirs. Higher order behavior (e.g.,
$\gamma^2$ or $1/\gamma^2$) can occur when there is a gap in the frequency
spectrum. Moreover, inhomogeneity in the frequency spacing can give rise to a
pseudo-plateau regime. These findings yield a physically motivated approach to
diagnosing numerical simulations and understanding the influence of relaxation,
and we examine their occurrence in both simple models and a realistic,
fluctuating graphene nanoribbon.
