  We consider fluctuations of the time-averaged current in the one-dimensional
weakly-asymmetric exclusion process on a ring. The optimal density profile
which sustains a given fluctuation exhibits an instability for low enough
currents, where it becomes time-dependent. This instability corresponds to a
dynamical phase transition in the system fluctuation behavior: while typical
current fluctuations result from the sum of weakly-correlated local events and
are still associated with the flat, steady-state density profile, for currents
below a critical threshold the system self-organizes into a macroscopic jammed
state in the form of a coherent traveling wave, that hinders transport of
particles and thus facilitates a time-averaged current fluctuation well below
the average current. We analyze in detail this phenomenon using advanced Monte
Carlo simulations, and work out macroscopic fluctuation theory predictions,
finding very good agreement in all cases. In particular, we study not only the
current large deviation function, but also the critical current threshold, the
associated optimal density profiles and the traveling wave velocity, analyzing
in depth finite-size effects and hence providing a detailed characterization of
the dynamical transition.
