The problem of optimizing system work output has been a fundamental part of thermodynamics ever since the introduction of the Carnot engine. The solutions to such optimization problems often include sharp changes in system parameters, such as found in the transitions between isothermal and adiabatic regimes in the Carnot cycle. Of course, an engine employing reversible processes takes an infinite time to complete and is irrelevant to real engines. A commonly used alternative is to examine engines at maximal power output.  Here we study the cycles that maximize the output of stochastic pumps. Stochastic pumps are small systems that are driven by periodic variation of externally controlled variables. Such models were used to study the dynamics of artificial molecular machines.

We use a combination of numerical and analytical approaches to find the cycles that maximize the power output of a simple model of a stochastic pump. Surprisingly, the optimal cycle is a singular point in the state space, with external parameters being switched at an infinite rate.