Optimal work in a harmonic trap with bounded stiffness

Abstract

We apply Pontryagin’s principle to drive rapidly a trapped overdamped Brownian particle in contact with a thermal bath between two equilibrium states corresponding to different trap stiffness κ. We work out the optimal time dependence κ(t) by minimizing the work performed on the particle under the nonholonomic constraint 0 κ κmax, an experimentally relevant situation. Several important differences arise, as compared with the case of unbounded stiffness that has been analyzed in the literature. First, two arbitrary equilibrium states may not always be connected. Second, depending on the operating time tf and the desired compression ratio κf/κi, different types of solutions emerge. Finally, the differences in the minimum value of the work brought about by the bounds may become quite large, which may have a relevant impact on the optimization of heat engines