Time irreversibility in active matter, from micro to macro

Active matter encompasses systems whose individual constituents dissipate energy to exert propelling forces on their environment. These systems exhibit dynamical phenomena with no counterpart in passive systems. In this Review, we disentangle the respective roles of the arrow of time and the non-Boltzmann nature of steady-state fluctuations in this rich phenomenology. We show that effective, time-reversible descriptions of active systems may be found at all scales and discuss how interactions, either between constituents or with external operators, may reveal the nonequilibrium nature of the microscopic source of energy. At a time when engineering active materials appears to be becoming possible, we argue that methods stemming from equilibrium statistical mechanics may guide the design of new active materials. Much of the rich phenomenology of active matter can be traced back to violations of time-reversal symmetry and departure from Boltzmann statistics. This Review disentangles these nonequilibrium signatures for interacting and dilute systems, in the presence of obstacles and external potentials.