Nonequilibrium Steady States in Active Systems: A Helmholtz–Hodge Perspective

We revisit the question of the existence of a potential function, the Cole-Hopf transform of the stationary measure, for nonequilibrium steady states, in particular those found in active matter systems. This has been the subject of ongoing research for more than fifty years, but continues to be relevant. In particular, we want to make a connection to some recent work on the theory of Helmholtz-Hodge decompositions and address the recently suggested notion of typical trajectories in such systems.