Phonon excitation and energy redistribution in phonon space for energy transport in one-dimensional lattice with nonlinear dispersion

Abstract

We first propose fundamental solutions of wave propagation in one-dimensional dispersive chain subject to a localized initial perturbation in the displacement. Analytical solutions are obtained for both second order nonlinear dispersive chain and homogenous harmonic chain. Solution was also compared with numerical results from molecular dynamics (MD) simulations. Locally dominant phonon modes (k-space) are introduced based on these solutions. These locally defined phonon modes k(x,t) spatially and temporally varying with x and t are critical to the establishment of the local thermodynamic equilibrium (LTE). The establishment of LTE should require a uniform distribution of energy in these modes at any given time t and location x. Wave propagation accompanying with the nonequilibrium dynamics leads to the excitation of these locally defined phonon modes. It was found that the system energy is redistributed among these excited phonons modes (k-space). This redistribution process is only possible with nonlinear dispersion and requires a finite amount of time to achieve a steady state distribution depending on the spatial distribution (or frequency content) of initial perturbation and the dispersion relation, where sharper and concentrated perturbation leads to a faster redistribution. Ballistic type of heat transport along the harmonic chain reveals that at any given position the lowest mode (k=0) is excited first and gradually expanding to the highest mode, which can only asymptotically approach the maximum mode of the first Brillouin zone. Energy is shown to be uniformly distributed in all available phonon modes lower than the highest mode. The energy flux along the chain is shown to be a constant with time and proportional to the sound speed (ballistic transport). Comparison with the Fourier's law leads to a time-dependent thermal conductivity that diverges with time.