Approach to Phonon Relaxation Time and Mean Free Path in Nonlinear Lattices

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-beta (FPU-beta) and phi (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-beta 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 phi (4) lattice, both the phonon relaxation time and mean free path are finite, which ensures normal heat conduction.

Automated summary

Based on the self-consistent phonon theory, this study calculated the spectral energy density, fitted the spectral energy density, and obtained phonon frequency and relaxation time in one-dimensional nonlinear lattices, validating the phonon mean free path and behavior at different temperatures.