  We introduce a quantum thermodynamic system that can operate as a thermal
clock in analogy with the thermal clock first introduced by Ernst Mach in which
the temperature difference between cooling bodies in contact can be used as a
kind of relational time. Our model is based on an optomechanical system in
which photons are transferred irreversibly between two cavities due to the
modulation of the cavity cooling rate by a mechanical system coupled to a heat
bath. We contrast the ensemble average view with a single system view using the
theory of continuous measurement and we show that, by making a weak continuous
measurement of the energy difference between the systems, a natural Mach
thermal clock arises in an appropriate semiclassical limit. We investigate how
quantum fluctuations modify this result.
