Efficient protocols for Stirling heat engines at the micro-scale

We investigate the thermodynamic efficiency of sub-micro-scale heat engines operating under the conditions described by overdamped stochastic thermodynamics. We prove that at maximum power the efficiency obeys the universal law η=2ηC/(4−ηC) for ηC the efficiency of an ideal Carnot cycle. The corresponding power optimizing protocol is specified by the solution of an optimal mass transport problem. It can be determined explicitly using well known Monge--Amp\`ere--Kantorovich reconstruction algorithms. Furthermore, we show that the same law describes the efficiency of heat engines operating at maximum work over short time periods.