  Mechanisms of resistivity can be divided into two basic classes: one is
dissipative (like scattering on phonons) and another is quasi-elastic (like
scattering on static impurities). They are often treated by the empirical
Matthiessen rule, which says that total resistivity is just the sum of these
two contributions, which are computed separately. This is quite misleading for
two reasons. First, the two mechanisms are generally correlated. Second,
computing the elastic resistivity alone masks the fundamental fact that the
linear-response approximation has a vanishing validity interval at vanishing
dissipation. Limits of zero electric field and zero dissipation do not commute
for the simple reason that one needs to absorb the Joule heat quadratic in the
applied field. Here, we present a simple model that illustrates these two
points. The model also illuminates the role of variational principles for
non-equilibrium steady states.
