Sommaire:

If a liquid is cooled fast enough, it is possible to avoid the phase transition towards a crystalline solid, and the system enters a metastable state, called supercooled liquid. If the cooling continues, the viscosity and the relaxation times grow so steeply that the system becomes effectively solid, even though the system did not undergo any thermodynamic phase transition. Such a solid is called a glass. Glassy materials appear in both in everyday life and in high-tech devices. Further, many systems in biology, computer science and mathematics also exhibit a glassy behavior. In this talk, we try to understand the mechanisms underlying the rigidity of glasses. In phase space, the dynamics of such systems can be regarded as a random walk in a high-dimensional energy landscape. This landscape is rugged, so in order to sample new configurations the system must 'hop' over an energy barrier from one local minimum (basin) to another. This type of process is called activation. We adopt a bottom-up strategy to understand this phenomenon. We start from the simplest models, attempt to fully characterize the behavior of their random walk in phase space, and only then add features representing more realistic glasses. We analyze three paradigmatic mean-field models: the Trap Model, the Random Energy Model (REM) and the p-spin model. In particular, in the REM, we see that despite the motion in configuration space is complex on intermediate time-scales, there is a very large separation of time scales that washes out correlations induced by the exploration of the landscape. As a result, by coarse-graining the dynamical evolution in terms of basins of the energy landscape, the dynamics can be effectively described through a hopping between disconnected traps. This type of description is not applicable anymore in systems with a higher degree of correlation in the energy landscape, such as the p-spin model, where the energy landscape exhibits large basins which are connected one to the other through non-trivial boundaries.

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