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In addition, both the direction and magnitude of the vapor temperature gradient, as well as the temperature jump at the liquid-vapor interface, are functions of the applied heat flux. The interfacial entropy generation rate calculated from the vibrational density of state of the interfacial liquid and vapor molecules shows a positive production during evaporation, and the results qualitatively agree with the predictions from non-equilibrium thermodynamics.We present a new approach to incorporate the geometric phase in the time-dependent