On approaching the glass transition, the microscopic kinetic unit spends increasing time rattling in the cage of the first neighbors, whereas its average escape time, the structural relaxation time τα, increases from a few picoseconds up to thousands of seconds. A thorough study of the correlation between τα and the rattling amplitude, expressed by the Debye-Waller factor, was carried out. Molecular-dynamics simulations of both a model polymer system and a binary mixture were performed by...