Efficient protocols for Stirling heat engines at the micro-scale

We investigate the thermodynamic efficiency of sub-micro-scale Stirling heat engines operating under the conditions described by overdamped stochastic thermodynamics. We show how to construct optimal protocols such that at maximum power the efficiency attains for constant isotropic mobility the universal law $\eta =2{\eta }_C/(4-{\eta }_C)$, where ${\eta }_C$ is the efficiency of an ideal Carnot cycle. We show that these protocols are specified by the solution of an optimal mass transport problem. Such solution can be determined explicitly using well-known Monge-Ampère-Kantorovich reconstruction algorithms. Furthermore, we show that the same law describes the efficiency of heat engines operating at maximum work over short time periods. Finally, we illustrate the straightforward extension of these results to cases when the mobility is anisotropic and temperature dependent. (letter)