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 two baths at different temperatures. We propose to take the weighted average temperature as the effective temperature of the system. The weight factors are related to the couplings 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. It is found that the fundamental thermodynamic relations in nonequilibrium steady states possess similar concise forms as those in equilibrium thermodynamics, provided that we replace the temperature in equilibrium with the weighted average temperature in steady states. We also illustrate the procedure to explicitly calculate the effective temperatures via three examples.