Evolution of a vibrational wave packet on a disordered chain

A linear chain of point masses coupled by harmonic springs is a standard model used to introduce concepts of solid state physics. The well-ordered chain has sinusoidal standing wave normal modes (if the ends are fixed) or traveling wave normal modes (if the ends are connected in a ring). Ballistically propagating wave packets can be built from these normal modes, and illustrate the mechanism of heat propagation in insulating crystals. When the chain is disordered, new effects arise. Ballistic propagation is replaced by diffusive propagation on length scales larger than the mean free path for ballistic motion. However, a new length scale, the localization length, also enters. On length scales longer than the localization length, neither ballistic nor diffusive propagation occurs, and energy is trapped unless there are anharmonic forces. These ideas are illustrated by a computer experiment.