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 (log N)/m, where m 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 (log N)/m. 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/refolding experiments on DNA hairpins of varying loop size and dissipation, displaying both near-Gaussian and non-Gaussian work distributions.