Free energy inference from partial work measurements in small systems

Abstract: Fluctuation relations (FRs) are among the few existing general results in non-equilibrium systems. Their verification requires the measurement of the total work (or entropy production) performed on a system. Nevertheless in many cases only a partial measurement of the work is possible. Here we consider FRs in dual-trap optical tweezers where two different forces (one per trap) are measured. With this setup we perform pulling experiments on single molecules by moving one trap relative to the other. We demonstrate that work should be measured using the force exerted by the trap that is moved. The force that is measured in the trap at rest fails to provide the full dissipation in the system leading to a (incorrect) work definition that does not satisfy the FR. The implications to single-molecule experiments and free energy measurements are discussed. In the case of symmetric setups a new work definition, based on differential force measurements, is introduced. This definition is best suited to measure free energies as it shows faster convergence of estimators. We discuss measurements using the (incorrect) work definition as an example of partial work measurement. We show how to infer the full work distribution from the partial one via the FR. The inference process does also yield quantitative information, e.g. the hydrodynamic drag on the dumbbell. Results are also obtained for asymmetric dual-trap setups. We suggest that this kind of inference could represent a new and general application of FRs to extract information about irreversible processes in small systems.