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Fluctuation relations (FRs) are among the few existing general results innon-equilibrium systems. Their verification requires the measurement of thetotal work (or entropy production) performed on a system. Nevertheless in manycases only a partial measurement of the work is possible. Here we consider FRsin dual-trap optical tweezers where two different forces (one per trap) aremeasured. With this setup we perform pulling experiments on single molecules bymoving one trap relative to the other. We demonstrate that work should bemeasured using the force exerted by the trap that is moved. The force that ismeasured in the trap at rest fails to provide the full dissipation in thesystem leading to a (incorrect) work definition that does not satisfy the FR.The implications to single-molecule experiments and free energy measurementsare discussed. In the case of symmetric setups a new work definition, based ondifferential force measurements, is introduced. This definition is best suitedto measure free energies as it shows faster convergence of estimators. Wediscuss measurements using the (incorrect) work definition as an example ofpartial work measurement. We show how to infer the full work distribution fromthe partial one via the FR. The inference process does also yield quantitativeinformation, e.g. the hydrodynamic drag on the dumbbell. Results are alsoobtained for asymmetric dual-trap setups. We suggest that this kind ofinference could represent a new and general application of FRs to extractinformation about irreversible processes in small systems.

Free energy inference from partial work measurements in small systems