  The role of quantum coherence and correlations in heat flow and equilibration
is investigated by exploring the Rayleigh's dynamical problem to equilibration
in the quantum regime and following Onsager's approach to thermoelectricity.
Specifically, we consider a qubit bombarded by two-qubit projectiles from a
side. For arbitrary collision times and initial states, we develop the master
equation for sequential and collective collisions. By deriving the
Fokker-Planck equation out of the master equation, we identify the quantum
version of the Rayleigh's heat conduction equation. We find that quantum
discord and entanglement shared between the projectiles can contribute to
genuine heat flow only when they are associated with so-called heat-exchange
coherences. Analogous to Onsager's use of Rayleigh's principle of least
dissipation of energy, we use the entropy production rate to identify the
coherence current. Both coherence and heat flows can be written in the form of
quantum Onsager relations, from which we predict coherent Peltier and coherent
Seebeck effects. The effects can be optimized by the collision times and
collectivity. Finally, we discuss some of the possible experimental
realizations and technological applications of the thermocoherent phenomena in
different platforms.
