  The nature of particle and entropy flow between two superfluids is often
understood in terms of reversible flow carried by an entropy-free, macroscopic
wavefunction. While this wavefunction is responsible for many intriguing
properties of superfluids and superconductors, its interplay with excitations
in non-equilibrium situations is less understood. Here, we observe large
concurrent flows of both particles and entropy through a ballistic channel
connecting two strongly interacting fermionic superfluids. Both currents
respond nonlinearly to chemical potential and temperature biases. We find that
the entropy transported per particle is much larger than the prediction of
superfluid hydrodynamics in the linear regime and largely independent of
changes in the channel's geometry. In contrast, the timescales of advective and
diffusive entropy transport vary significantly with the channel geometry. In
our setting, superfluidity counterintuitively increases the speed of entropy
transport. Moreover, we develop a phenomenological model describing the
nonlinear dynamics within the framework of generalised gradient dynamics. Our
approach for measuring entropy currents may help elucidate mechanisms of heat
transfer in superfluids and superconducting devices.
