Improving Free-Energy Estimates from Unidirectional Work Measurements: Theory and Experiment

We derive analytical expressions for the bias of the Jarzynski free-energy estimator from N nonequilibrium work measurements, for a generic work distribution. To achieve this, we map the estimator onto the random energy model in a suitable scaling limit parametrized by (logN)/mu, where mu measures the width of the lower tail of the work distribution, and then compute the finite-N corrections to this limit with different approaches for different regimes of (logN)/mu. We show that these expressions describe accurately the bias for a wide class of work distributions and exploit them to build an improved free-energy estimator from unidirectional work measurements. We apply the method to optical tweezers unfolding and refolding experiments on DNA hairpins of varying loop size and dissipation, displaying both near-Gaussian and non-Gaussian work distributions.