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 $\sigma^2(t)$ of atom displacements depends on the initial condition. For the initial condition with nonzero momentum, $\sigma^2(t)$ goes as $t^\alpha$ with $\alpha=1$ and 0 for incommensurate Frenkel-Kontorova (FK) model at $V$ below and above $V_c$ respectively; and $\alpha=1$ for uniform, quasiperiodic and random chains. It is also found that $\alpha=1-\beta$ with $\beta$ the exponent of distribution function of frequency at zero frequency, i.e., $\rho(\omega)\sim \omega^{\beta}$ (as $\omega\to 0$). For the initial condition with zero momentum, $\alpha=0$ for all systems studied. The underlying physical meaning of this diffusive behavior is discussed.