Optimization of an active heat engine

Optimisation of heat engines at the micro-scale has applications in biological and artificial nano-technology, stimulates theoretical research non-equilibrium statistical physics. Here we consider non-interacting overdamped particles confined by an external harmonic potential, contact either with a thermal reservoir or stochastic self-propulsion force (active Ornstein-Uhlenbeck model). A cyclical machine is produced periodic variation parameters potential noise. An exact mapping between passive active model allows us to define effective temperature $T_{eff}(t)$ which meaningful for thermodynamic performance engine. We show that different from all other known temperatures, typically used static situations. The optimise engine, whatever are values persistence time velocity. In particular - through linear irreversible thermodynamics (small amplitude cycle) give explicit formula optimal cycle period phase delay (between two modulated parameters, stiffness temperature) achieving maximum power Curzon-Ahlborn efficiency. quasi-static limit, simplifies coincides recently proposed thermodynamics, bearing compact expression point overlooked recent literature made about difficulty defining efficiency without consistent definition temperature.