Death and resurrection of a current by disorder, interaction or activity

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.