Through the integration of the power spectral density, we obtain temperature profiles of both multisegment harmonic and anharmonic systems, showing the presence of an anomalous negative temperature gradient inside the interfacial segment. Via investigating patterns of the power spectral density, we found that the counterintuitive phenomenon comes from the presence of interfacial localized phonon modes. Two out-band localized modes of the harmonic model, which make no contributions to local temperature due to the absence of phonon interactions, result in the concave temperature profile and overcooling effect. For the anharmonic model, thanks to the phonon-phonon interactions, the localized modes are excited and make considerable contributions to interfacial temperature, which is clearly shown by examining the temperature accumulation function. When anharmonicity is considerably large, the negative temperature gradient is absent since the localized phonon modes are fully mixed. The presence of localized modes are evidently demonstrated by the inverse participation ratio and normal mode analysis for the isolated harmonic model. The localized modes make contribution to interfacial temperature profiles of the harmonic system when they are excited in initial conditions of simulations.

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Based on the self-consistent phonon theory, the spectral energy density is calculated by the canonical transformation and the Fourier transformation. Through fitting the spectral energy density by the Lorentzian profile, the phonon frequency as well as the phonon relaxation time is obtained in one-dimensional nonlinear lattices, which is validated in the Fermi–Pasta–Ulam-β (FPU-β) and ϕ 4 lattices at different temperatures. The phonon mean free path is then evaluated in terms of the phonon relaxation time and phonon group velocity. The results show that, in the FPU-β lattice, the phonon mean free path as well as the phonon relaxation time displays divergent power-law behavior. The divergent exponent coincides well with that derived from the Peierls–Boltzmann theory at weak anharmonic nonlinearity. The value of the divergent exponent expects a power-law divergent heat conductivity with system size, which violates Fourier’s law. For the ϕ 4 lattice, both the phonon relaxation time and mean free path are finite, which ensures normal heat conduction.

Harmonics are quite common in pulsating stars. They are always considered to mimic the behaviors of their independent parent pulsation modes, and are not taken for key information in asteroseismology. Here, we report an SX Phoenicis star XX Cygni, whose periodogram is dominated by the fundamental frequency f 0 = 7.41481 ± 0.00004 c day−1 and its 19 harmonics. According to the analysis of the archival data from the Transiting Exoplanet Survey Satellite (TESS), we find that both the amplitudes and frequencies of the fundamental mode and the harmonics vary within TESS Sectors 14–17 and 54–57, which might be caused by the contamination of neighboring stars. What is more interesting is that the harmonics show significantly uncorrelated amplitude and frequency variations over time. Some possible origins and interesting issues are proposed to scheme further research of this hidden corner in current asteroseismology.