Energy relaxation in disordered lattice ϕ ⁴ system: The combined effects of disorder and nonlinearity*

Abstract

We address the issue of how disorder together with nonlinearity affect energy relaxation in the lattice ϕ 4 system. The absence of nonlinearity leads such a model to only supporting fully localized Anderson modes whose energies will not relax. However, through exploring the time decay behavior of each Anderson mode’s energy–energy correlation, we find that adding nonlinearity, three distinct relaxation details can occur. (i) A small amount of nonlinearity causes a rapid exponential decay of the correlation for all modes. (ii) In the intermediate value of nonlinearity, this exponential decay will turn to power-law with a large scaling exponent close to –1. (iii) Finally, all Anderson modes’ energies decay in a power-law manner but with a quite small exponent, indicating a slow long-time tail decay. Obviously, the last two relaxation details support a new localization mechanism. As an application, we show that these are relevant to the nonmonotonous nonlinearity dependence of thermal conductivity. Our results thus provide new information for understanding the combined effects of disorder and nonlinearity on energy relaxation.

Thermal conductivity in one-dimensional nonlinear disordered lattices: Two kinds of scattering effects of hard-type and soft-type anharmonicities

“…C E n (t) with n being some number in the middle of mode range (e.g., n = 512 and 1024) decays faster than those with n at boundaries (e.g., n = 100 and 3996 in our simulations). This indicates that the 512th and 1024th Anderson modes have shorter lifetimes than those of the 100th and 3996th modes, which is consistent with the previous results observed only in the HT models [41].…”

“…First, the harmonic disordered model (ξ = 1) has a minimal κ = 0 because in this case all the phonon modes become Anderson localized as expected. Second, in the HT type model a usual nonmonotonous ξdependence of κ can be identified [41], i.e., as ξ increases from ξ = 1 (the on-site potential becomes hard), κ first increases, then reaches its maximum at certain ξ value, and finally decreases. This nonmonotonous behavior has been attributed to the delocalization and re-localization of the Anderson modes induced by the combined effects of disorder and nonlinearity [41].…”

“…Second, in the HT type model a usual nonmonotonous ξdependence of κ can be identified [41], i.e., as ξ increases from ξ = 1 (the on-site potential becomes hard), κ first increases, then reaches its maximum at certain ξ value, and finally decreases. This nonmonotonous behavior has been attributed to the delocalization and re-localization of the Anderson modes induced by the combined effects of disorder and nonlinearity [41]. Bearing this in mind, let us now turn to the results of the ST type model.…”

Thermal conductivity in one-dimensional nonlinear disordered lattices: two kinds of scattering effects of hard-type and soft-type anharmonicities

“…Second, in the harmonic disordered model (ξ = 1), we observe a minimal value of κ = 0, which is expected to be due to the Anderson localization of all phonon modes [51]. Third, for the HT-type model, a non-monotonic dependence of κ on ξ can be identified [48]. Specifically, as ξ increases from 1 (resulting in a harder on-site potential), κ initially increases, reaches its maximum at around ξ ≃ 1.5, and then decreases.…”

“…Specifically, as ξ increases from 1 (resulting in a harder on-site potential), κ initially increases, reaches its maximum at around ξ ≃ 1.5, and then decreases. This non-monotonic behavior has been attributed to the combined effects of disorder and nonlinearity inducing delocalization and re-localization of Anderson modes [48]. Keeping this in mind, let us now focus on the results obtained from the ST-type model.…”