Sommaire:

The main advantage of effective theories in statistical mechanics is their simple (polynomial) structure that allows for an analytical treatment and suggests that large length-scales physics is insensitive to microscopic details. In general, however, their predictions are quantitatively wrong, except close to a critical point, where universality ensures that all systems behave in the same way. On the contrary, when there is no genuine critical point there is no guarantee that they are appropriate to describe a given system. This problem is becoming particularly pressing since recent advances have shown that a dynamical effective theory of supercooled liquids display a rich phenomenology, namely: i) dynamical crossover from power-law to exponential relaxation, ii) dynamical heterogeneities iii) decreasing dynamical correlation lengths. A partial answer to these problems comes from the existence of a number of academic models that are exactly described by the dynamical effective theory. Thus it can be viewed, at worst, as a toy model of supercooled liquids. For a given realistic system, instead, the problem cannot be decided a priori but one can devise a certain number of tests to determine whether the system under consideration should be described by the effective theory in the glass crossover region.

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