Although extensive experimental and theoretical works have been conducted to understand the ballistic and diffusive phonon transport in nanomaterials recently, direct observation of temperature and thermal nonequilibrium of different phonon modes has not been realized. Herein, we have developed a method within the framework of molecular dynamics to calculate the temperatures of phonon in both real and phase spaces. Taking silicon thin film and graphene as examples, we directly obtained the spectral phonon temperature (SPT) and observed the local thermal nonequilibrium between the ballistic and diffusive phonons. Such nonequilibrium also generally exists across interfaces and is surprisingly large, and it provides an additional thermal interfacial resistance mechanism. Our SPT results directly show that the vertical thermal transport across the dimensionally mismatched graphene/substrate interface is through the coupling between flexural acoustic phonons of graphene and the longitudinal phonons in the substrate with mode conversion. In the dimensionally matched interfaces, e.g. graphene/graphene junction and graphene/boron nitride planar interfaces, strong coupling occurs between the acoustic phonon modes on both sides, and the coupling decreases with interfacial mixing. The SPT method together with the spectral heat flux can eliminate the size effect of the thermal conductivity prediction induced from ballistic transport. Our work shows that in thin films and across interfaces, phonons are in local thermal nonequilibrium. * λQλ ,whereQ λ (t) is the time derivative of normal mode amplitude, which is given by the Fourier transform of atomic arXiv:1703.10957v1 [cond-mat.mes-hall]

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