  Reversible to irreversible (R-IR) transitions arise in numerous periodically
driven collectively interacting systems that, after a certain number of driving
cycles, organize into a reversible state where the particle trajectories
repeat, or remain irreversible with chaotic motion. R-IR transitions were first
systematically studied for periodically sheared dilute colloids, and appear in
a wide variety of both soft and hard matter systems, including amorphous
solids, crystals, vortices in type-II superconductors, and magnetic textures.
In some cases, the reversible transition is an absorbing phase transition with
a critical divergence in the organization time scale. R-IR systems can store
multiple memories and exhibit return point memory. We give an overview of R-IR
transitions including recent advances in the field, and discuss how the general
framework of R-IR transitions could be applied to a much broader class of
periodically driven nonequilibrium systems, including soft and hard condensed
matter systems, astrophysics, biological systems, and social systems. Some
likely candidate systems are commensurate-incommensurate states, systems
exhibiting hysteresis or avalanches, and nonequilibrium pattern forming states.
Periodic driving could be applied to hard condensed matter systems to see if
R-IR transitions occur in metal-insulator transitions, semiconductors, electron
glasses, electron nematics, cold atom systems, or Bose-Einstein condensates.
R-IR transitions could also be examined in dynamical systems where
synchronization or phase locking occurs. We discuss the use of complex periodic
driving such as changing drive directions or multiple frequencies as a method
to retain complex multiple memories. Finally, we describe features of classical
and quantum time crystals that could suggest the occurrence of R-IR transitions
in these systems.
