The second law of thermodynamics prohibits the extraction of work from a system that is, on average, greater than the decrease in its free energy. However, for small systems there can be stochastic fluctuations which allow individual trajectories to exceed this bound while still satisfying the second law at the ensemble level. This motivates the following question: how can we maximise the likelihood of violating the second law in a probabilistic sense? It is known that the Jarzynski equality places an upper bound on how large this likelihood can be given a fixed free energy change, and protocols saturating this bound have been previously proposed. However, to reach this limit it is necessary to perform a concatenation of quasi-static processes where the system remains in instantaneous equilibrium. In real experiments we must account for finite-time deviations from equilibrium, and so this upper bound is not generally achievable. Here I demonstrate how to maximise the probability of extracting work above the decrease in free energy in finite time using a series of imperfect isothermal steps and adiabats. This technique is applicable to open systems in both quantum and classical regimes. It is shown that the optimal protocol requires a work distribution given by a convex combination of two Gaussian peaks with a minimal amount of dissipation created along the way. Therefore, paths that minimise the entropy production along the isotherms provide the best chance of exceeding the free energy change stochastically. These findings can be used to explain the results of a recent experiment that attempted this probabilistic work extraction using a single electron transistor.

This is joint work with Martí Perarnau-Llobet. This work was supported by Royal Commission for the Exhibition of 1851.