  When a quantum system is coupled to several heat baths at different
temperatures, it eventually reaches a non-equilibrium steady state featuring
stationary internal heat currents. These currents imply that entropy is
continually being produced in the system at a constant rate. In this paper we
apply phase-space techniques to the calculation of the Wigner entropy
production on general linear networks of harmonic nodes. Working in the
ubiquitous limit of weak internal coupling and weak dissipation, we obtain
simple closed-form expressions for the entropic contribution of each individual
quasi-probability current. Our analysis highlights the essential role played by
the internal unitary interactions (node-node couplings) in maintaining a
non-equilibrium steady-state and hence a finite entropy production rate. We
also apply this formalism to the paradigmatic problem of energy transfer
through a chain of oscillators subject to self-consistent internal baths that
can be used to tune the transport from ballistic to diffusive. We find that the
entropy production scales with different power law behaviors in the ballistic
and diffusive regimes, hence allowing us to quantify what is the "entropic cost
of diffusivity."
