  The understanding of heat transport in nonequilibrium thermodynamics is an
important research frontier, which is crucial for implementing novel
thermodynamic devices, such as heat engines and refrigerators. The convection,
conduction, and radiation are the well-known basic ways to transfer thermal
energy. Here we demonstrate a new mechanism of phonon heat transport between
two spatially separated nanomechanical resonators coupled by the
cavity-enhanced long-range interactions. The single trajectory for
thermalization and non-equilibrium dynamics is monitored in real-time. We find
that, in the strong coupling regime, instant heat flux counterintuitively
oscillates back and forth in nonequilibrium steady states, which occurs
spontaneously above the critical point during the thermal energy transfer
process. The universal bound on the precision of nonequilibrium steady-state
heat flux, i.e. the thermodynamic uncertainty relation, is verified in such a
temperature gradient driven far-off equilibrium system. Our results reveal a
previously unobserved phenomenon for heat transfer with mechanical oscillators,
and provide a playground for testing fundamental theories in non-equilibrium
thermodynamics.
