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The past decade has witnessed the development of systematic effectivetheories for dissipative thermal systems. Here, we describe an analogouseffective theory framework that applies to the classical stochastic dynamics ofnonequilibrium systems. We illustrate this approach using a range of examples,including nonreciprocal (predator-prey) dynamics, dissipative and drivenrigid-body motion, and active chiral fluids and solids. Many of these systemsexhibit a generalized time-reversal symmetry, which plays a crucial role withinour formalism, and in many cases can be implemented within theMartin-Siggia-Rose path integral. This effective theory formalism yieldsgeneralizations of the fluctuation-dissipation theorem and second law ofthermodynamics valid out of equilibrium. By stipulating a stationarydistribution and a set of symmetries -- rather than postulating the stochasticequations of motion directly -- this formalism provides an alternative route tobuilding phenomenological models of driven and active matter. We hope that thisapproach facilitates a systematic investigation of the universality classes ofactive matter, and provides a common language for nonequilibrium many-bodyphysics from high energy to condensed matter.

Generalized time-reversal symmetry and effective theories for nonequilibrium matter