Geometric view of stochastic thermodynamics for non-equilibrium steady states

We explore the idea that non-equilibrium steady states breaking detailed balance are obtained by deforming trajectories (lines in space-time) that have been sampled in a reference system with stochastic dynamics obeying detailed balance, and we ask for the work required to perform this task. These geometric deformations are not arbitrary but arise through interactions with the environment, either the manipulation of conserved quantities by an external agent, or by their exchange with a work reservoir. This view allows to consistently model the breaking of detailed balance and the accompanying entropy production without non-conservative forces, and to systematically extend the notion of thermodynamic ensembles to non-equilibrium steady states. We illustrate the usefulness of this approach by applying it to suspensions of active colloidal particles and deriving their thermodynamically consistent equations of motion.