  e discuss the relevance of the Stillinger and Weber approach to the glass
transition investigating the non-equilibrium behavior of models with
non-trivial dynamics, but with simple equilibrium properties. We consider a
family of 1D constrained kinetic models, which interpolates between the
asymmetric chain introduced by Eisinger and J\"ackle [Z. Phys. {\bf B84}, 115
(1991)] and the symmetric chain introduced by Fredrickson and Andersen [Phys.
Rev. Lett {\bf 53}, 1244 (1984)], and the 1D version of the Backgammon model
[Phys. Rev. Lett. {\bf 75}, 1190 (1995)]. We show that the configurational
entropy obtained from the inherent structures is the same for all models
irrespective of their different microscopic dynamics. We present a detailed
study of the coarsening behavior of these models, including the relation
between fluctuations and response. Our results suggest that any approach to the
glass transition inspired by mean-field ideas and resting on the definition of
a configurational entropy must rely on the absence of any growing
characteristic coarsening pattern.
