  Low temperature analysis of nonequilibrium systems requires finding the
states with the longest lifetime and that are most accessible from other
states. We determine these dominant states for a one-dimensional diffusive
lattice gas subject to exclusion and with nearest neighbor interaction. They do
not correspond to lowest energy configurations even though the particle current
tends to zero as the temperature reaches zero. That is because the dynamical
activity that sets the effective time scale, also goes to zero with
temperature. The result is a non-trivial asymptotic phase diagram, which
crucially depends on the interaction coupling and the relative chemical
potentials of the reservoirs.
