The ballistic to diffusive crossover, that occurs when quasiparticles transport heat or charge, is important in small systems. Propagation of energy from an initial localized pulse provides a useful picture of the process. This paper looks at the simplest example, vibrational pulses on a one-dimensional harmonic chain of atoms. Pure ballistic propagation occurs in ordered chains, and a crossover toward diffusive propagation can be seen in disordered chains. A full analysis is inhibited by non-perturbative effects, especially Anderson localization. A partial analysis using Boltzmann theory is given.
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