  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.
