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Non-thermal vibrations in biased molecular junctions

We study vibrational statistics in current-carrying model molecular junctions using master equation approach. Especially, we concentrate on the validity of using an effective temperature $T_{\rm eff}$ to characterize the nonequilibrium steady state of a vibrational mode. We identify cases where a single $T_{\rm eff}$ can not fully describe one vibrational state. In such cases, the probability distribution among different vibrational states does not follow the Boltzmann type. Consequently, the actual entropy (free energy) of the vibrational mode is lower (higher) than the corresponding thermal value given by $T_{\rm eff}$, indicating extra work can be extracted from these states. Our results will be useful for the study of non-thermal vibrational state in thermodynamics of nanoscale systems, and its usage in nanoscale heat engines.