Onsager coefficients in a coupled-transport model displaying a condensation transition

We study nonequilibrium steady states of a one-dimensional stochastic model, originally introduced as an approximation the Discrete Nonlinear Schr\"odinger equation. This model is characterized by two conserved quantities, namely mass and energy; it displays ``normal", homogeneous phase, separated condensed (negative-temperature) where macroscopic fraction energy localized on single lattice site. When steadily maintained out equilibrium external reservoirs, system exhibits coupled transport herein studied within framework linear response theory. find that Onsager coefficients satisfy exact scaling relationship, which allows reducing their dependence thermodynamic variables to density for unitary density. also determine structure in proximity critical line, proving existence paths partially enter region. phenomenon consequence Joule effect: temperature increase induced current so strong drive negative temperatures. Finally, since attains diverging at finite energy, such limit energy-mass conversion efficiency reaches ideal Carnot value.