Disturbance spreading in incommensurate and quasiperiodic systems

The propagation of an initially localized excitation in one-dimensional incommensurate, quasiperiodic and random systems is investigated numerically. It is discovered that the time evolution of variances ${\sigma }^2\mathrm{}\left(t\right)\mathrm{}$ of atom displacements depends on the initial condition. For the initial condition with nonzero momentum, ${\sigma }^2\mathrm{}\left(t\right)\mathrm{}$ goes as $t^{\alpha }$ with $\alpha =1\mathrm{}$ and $0$ for incommensurate Frenkel-Kontorova model at $V$ below and above $V_c$ respectively, and $\alpha =1\mathrm{}$ for uniform, quasiperiodic and random chains. It is also found that $\alpha =1\mathrm{}-\beta$ with $\beta$ the exponent of distribution function of frequency at zero frequency, i.e., $\rho \left(\omega \right)\sim {\omega }^{\beta }$ (as $\overrightarrow{\omega }0).$ For the initial condition with zero momentum, $\alpha =0\mathrm{}$ for all systems studied. The underlying physical meaning of this diffusive behavior is discussed.