Crossover from entropic to thermal dynamics in glassy models

We study the emergence of a crossover from entropically driven to thermally activated dynamics in different versions of the `entropic' phase space model introduced by Barrat and M\'ezard, and previously considered in the zero temperature limit. We first focus on the low temperature (T < Tg) aging phase of the original model and show that a short time singularity appears in the correlation function for Tg/2 < T < Tg, leading to dynamical ultrametricity at T=Tg. We then consider the finite size version of this model, showing that the long time dynamics is always thermally activated beyond a size dependent crossover time scale. We also generalize the model, introducing a threshold energy so as to mimic a phase space composed of saddles above the threshold, and minima below. In this case, the crossover time scale becomes much smaller than the equilibration time, and both kinds of aging dynamics are successively found, inducing a non trivial aging scaling which does not reduce to the usual t/tw one.