Optimization of an active heat engine

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