  Interacting particle systems with many degrees of freedom may undergo phase
transitions to sustain atypical fluctuations of dynamical observables such as
the current or the activity. This leads in some cases to symmetry-broken
space-time trajectories which enhance the probability of such events due to the
emergence of ordered structures. Despite their conceptual and practical
importance, these dynamical phase transitions (DPTs) at the trajectory level
are difficult to characterize due to the low probability of their occurrence.
However, during the last decade advanced computational techniques have been
developed to measure rare events in simulations of many-particle systems that
allow for the first time the direct observation and characterization of these
DPTs. Here we review the application of a particular rare-event simulation
technique, based on cloning Monte Carlo methods, to characterize DPTs in
paradigmatic stochastic lattice gases. In particular, we describe in detail
some tricks and tips of the trade, paying special attention to the measurement
of order parameters capturing the physics of the different DPTs, as well as to
the finite-size effects (both in the system size and number of clones) that
affect the measurements. Overall, we provide a consistent picture of the
phenomenology associated with DPTs and their measurement.
