The discontinuous ideal glass transition, a soft crossover at finite waiting times

The \emph{non-equilibrium} self-consistent generalized Langevin equation (NE-SCGLE) theory of irreversible processes in liquids is shown to provide a coherent and conceptually simple picture of the crossover from ergodic equilibration to non-equilibrium aging in structural glass-forming liquids. According to this picture, the glass transition is in essence a discontinuous, ``mode-coupling''--like transition, characterized by the abrupt passage from ergodic to dynamically arrested states and by the divergence of the \emph{equilibrium} $\alpha$-relaxation time at the transition. The same picture, however, also predicts that such discontinuous and singular scenario will be blurred in real life by the unavoidable finiteness of the time window of any experimental observation. The comparison of these predictions with pertinent simulation experiments involving hard-sphere glass-forming liquids reconciles the lack of observable diverging relaxation times in (real or simulated) experiments, with the predicted critical condition at which the \emph{equilibrium} $\alpha$-relaxation time is expected to diverge.