  A two-lane exclusion process is studied where particles move in the two lanes
in opposite directions and are able to change lanes. The focus is on the steady
state behavior in situations where a positive current is constrained to an
extended subsystem (either by appropriate boundary conditions or by the
embedding environment) where, in the absence of the constraint, the current
would be negative. We have found two qualitatively different types of steady
states and formulated the conditions of them in terms of the transition rates.
In the first type of steady state, a localized cluster of particles forms with
an anti-shock located in the subsystem and the current vanishes exponentially
with the extension of the subsystem. This behavior is analogous to that of the
one-lane partially asymmetric simple exclusion process, and can be realized
e.g. when the local drive is induced by making the jump rates in two lanes
unequal. In the second type of steady state, which is realized e.g. if the
local drive is induced purely by the bias in the lane change rates, and which
has thus no counterpart in the one-lane model, a delocalized cluster of
particles forms which performs a diffusive motion as a whole and, as a
consequence, the current vanishes inversely proportionally to the extension of
the subsystem. The model is also studied in the presence of quenched
disordered, where, in case of delocalization, phenomenological considerations
predict anomalously slow, logarithmic decay of the current with the system size
in contrast with the usual power-law.
