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.

基于自洽声子理论，通过正则变换和傅立叶变换计算光谱能量密度。通过洛伦兹分布拟合谱能量密度，在一维非线性晶格中获得声子频率以及声子弛豫时间，这在 Fermi-Pasta-Ulam- β (FPU - β ) 和ϕ ^{4 中得到验证}不同温度下的晶格。然后根据声子弛豫时间和声子群速度评估声子平均自由程。结果表明，在 FPU- β在晶格中，声子平均自由程以及声子弛豫时间显示出不同的幂律行为。发散指数与派生自 Peierls-Boltzmann 理论的弱非谐非线性指数相吻合。发散指数的值期望随系统大小出现幂律发散热导率，这违反了傅立叶定律。对于φ ⁴晶格，声子弛豫时间和平均自由程都是有限的，这保证了正常的热传导。