Classical and quantum harmonic mean-field models coupled intensively and extensively with external baths

We study the nonequilibrium steady-state of a fully-coupled network $N$ quantum harmonic oscillators, interacting with two thermal reservoirs. Given long-range nature couplings, we consider setups: one in which number particles coupled to baths is fixed (intensive coupling) and it proportional size (extensive coupling). In both cases, compute analytically heat fluxes kinetic temperature distributions using Green's function approach, classical regimes. large limit, derive asymptotic expressions quantities as difference between baths. discuss peculiar feature model, namely that bulk vanishes thermodynamic due decoupling dynamics inner part system from At variance usual this implies steady state depends on initial bulk. also show effects are relevant only below characteristic $1/N$. low-temperature regime energy flux universal conductance.