  Causality is an important assumption underlying nonequilibrium
generalizations of the second law of thermodynamics known as fluctuation
relations. We here experimentally study the nonequilibrium statistical
properties of the work and of the entropy production for an optically trapped,
underdamped nanoparticle continuously subjected to a time-delayed feedback
control. Whereas the non-Markovian feedback depends on the past position of the
particle for a forward trajectory, it depends on its future position for a
time-reversed path, and is therefore acausal. In the steady-state regime, we
show that the corresponding fluctuation relations in the long-time limit
exhibit a clear signature of this acausality, even though the time-reversed
dynamics is not physically realizable.
