Weighted average temperature as the effective temperature of a system in contact with two thermal baths

We investigate the effective temperature of a harmonic chain whose two ends are coupled to baths at different temperatures. We propose taking a weighted average temperature as the effective temperature of the system. The weighting factors are determined by the coupling strengths between the system and two baths, as well as the asymmetry of interactions between oscillators. We revisit the thermodynamics of nonequilibrium steady states based on the weighted average temperature and find that the fundamental thermodynamic relations in such states take on similarly concise forms as those in equilibrium thermodynamics, provided that we replace the equilibrium temperature with the weighted average temperature. Additionally, we demonstrate how to explicitly calculate effective temperatures through three illustrative examples.