Abstract

Active matter is made out of elementary agents that are able to transform energy from their environment to self-propel. This self-propulsion drives an active system far from equilibrium at the microscopic scale, which, in turn, can make its study quite challenging. However, coarse-graining the dynamics of large assemblies of active particles can sometimes restore an effective, large-scale, time-reversal symmetry. Active systems demonstrating such property can then be analyzed (and controlled) with the toolbox of equilibrium statistical mechanics. To characterize such systems, one can introduce a generalization of the curl operator to functional spaces, called the functional exterior derivative. When the application of this operator to the stochastic hydrodynamics of the system vanishes, this means that this system is (macroscopically) time-reversible. When it is not the case, this operator reveals the structure of the probability currents (which live in an abstract functional space) associated to the system, as well as the manifestation of these currents in the real, physical space. The functional exterior derivative thus constitutes an important tool to study irreversibility of stochastic field theories, and its range of applicability goes actually far beyond the realm of active matter.