  On approaching the glass transition, the microscopic kinetic unit spends
increasing time rattling in the cage of the first neighbours whereas its
average escape time, the structural relaxation time $\tau_\alpha$, increases
from a few picoseconds up to thousands of seconds. A thorough study of the
correlation between $\tau_\alpha$ and the rattling amplitude, expressed by the
Debye-Waller factor (DW), was carried out. Molecular-dynamics (MD) simulations
of both a model polymer system and a binary mixture were performed by varying
the temperature, the density $\rho$, the potential and the polymer length to
consider the structural relaxation as well as both the rotational and the
translation diffusion. The simulations evidence the scaling between the
$\tau_\alpha$ and the Debye-Waller factor. An analytic model of the master
curve is developed in terms of two characteristic length scales pertaining to
the distance to be covered by the kinetic unit to reach a transition state. The
model does not imply $\tau_\alpha$ divergences. The comparison with the
experiments supports the numerical evidence over a range of relaxation times as
wide as about eighteen orders of magnitude. A comparison with other scaling and
correlation procedures is presented. The study suggests that the equilibrium
and the moderately supercooled states of the glassformers possess key
information on the huge slowing-down of their relaxation close to the glass
transition. The latter, according to the present simulations, exhibits features
consistent with the Lindemann melting criterion and the free-volume model.
