  Because of disorder the current-field characteristic may show a first order
phase transition as function of the field, at which the current jumps to zero
when the driving exceeds a threshold. The discontinuity is caused by adding a
finite correlation length in the disorder. At the same time the current may
resurrect when the field is modulated in time, also discontinuously: a little
shaking enables the current to jump up. Finally, in trapping models exclusion
between particles postpones or even avoids the current from dying, while
attraction may enhance it. We present simple models that illustrate those
dynamical phase transitions in detail, and that allow full mathematical
control.
