  Hysteresis in the current-voltage characteristic in a superconducting
nanowire reflects an underlying bistability. As the current is ramped up
repeatedly, the state switches from a superconductive to a resistive one, doing
so at random current values below the equilibrium critical current. Can a
single phase-slip event somewhere along the wire--during which the
order-parameter fluctuates to zero--induce such switching, via the local
heating it causes? We address this and related issues by constructing a
stochastic model for the time-evolution of the temperature in a nanowire whose
ends are maintained at a fixed temperature. The model indicates that although,
in general, several phase-slip events are necessary to induce switching, there
is indeed a temperature- and current-range for which a single event is
sufficient. It also indicates that the statistical distribution of switching
currents initially broadens, as the temperature is reduced. Only at lower
temperatures does this distribution show the narrowing with cooling naively
expected for resistive fluctuations consisting of phase slips that are
thermally activated.
