Marginally stable solids have peculiar physical properties that were first analyzed in the context of the jamming transition. We theoretically investigate the existence of marginal stability in a prototypical model for structural glass formers, combining analytical calculations in infinite dimensions to computer simulations in three dimensions. While mean-field theory predicts the existence of a Gardner phase transition towards a marginally stable glass phase at low temperatures, simulations show no hint of diverging time scales or length scales, but reveal instead the presence of sparse localized defects. Our results suggest that the Gardner transition is deeply affected by finite dimensional fluctuations, and raise issues about the relevance of marginal stability in structural glasses far away from jamming.

A vertically shaken granular medium hosts a blade rotating around a fixed vertical axis, which acts as a mesorheological probe. At high densities, independently from the shaking intensity, the blade's dynamics show strong caging effects, marked by transient sub-diffusion and a maximum in the velocity power density spectrum (vpds), at a resonant frequency ∼ 10 Hz. Interpreting the data through a diffusing harmonic cage model allows us to retrieve the elastic constant of the granular medium and its collective diffusion coefficient. For high frequencies f , a tail ∼ 1/f in the vpds reveals non-trivial correlations in the intra-cage micro-dynamics. At very long times (larger than 10 s), a super-diffusive behavior emerges, ballistic in the most extreme cases. Consistently, the distribution of slow velocity inversion times τ displays a power-law decay, likely due to persistent collective fluctuations of the host medium.

The configurational entropy is one of the most important thermodynamic quantities characterizing supercooled liquids approaching the glass transition. Despite decades of experimental, theoretical, and computational investigation, a widely accepted definition of the configurational entropy is missing, its quantitative characterization remains fraud with difficulties, misconceptions and paradoxes, and its physical relevance is vividly debated. Motivated by recent computational progress, we offer a pedagogical perspective on the configurational entropy in glass-forming liquids. We first explain why the configurational entropy has become a key quantity to describe glassy materials, from early empirical observations to modern theoretical treatments. We explain why practical measurements necessarily require approximations that make its physical interpretation delicate. We then demonstrate that computer simulations have become an invaluable tool to obtain precise, non-ambiguous, and experimentally-relevant measurements of the configurational entropy. We describe a panel of available computational tools, offering for each method a critical discussion. This perspective should be useful to both experimentalists and theoreticians interested in glassy materials and complex systems. I. CONFIGURATIONAL ENTROPY AND GLASS FORMATION A. The glass transitionWhen a liquid is cooled, it can either form a crystal or avoid crystallization and become a supercooled liquid. In the latter case, the liquid remains structurally disordered, but its relaxation time increases so fast that there exists a temperature, called the glass temperature T g , below which structural relaxation takes such a long time that it becomes impossible to observe. The liquid is then trapped virtually forever in one of many possible structurally disordered states: this is the basic phenomenology of the glass transition. 1-4 Clearly, T g depends on the measurement timescale and shifts to lower temperatures for longer observation times. The experimental glass transition is not a genuine phase transition, as it is not defined independently of the observer.The rich phenomenology characterizing the approach to the glass transition has given rise to a thick literature. It is not our goal to review it, and we refer instead to previous articles. 1-9 There are convincing indications that the dynamic slowing down of supercooled liquids is accompanied by an increasingly collective relaxation dynamics. This is seen directly by the measurement of growing lengthscales for these dynamic heterogeneities, 10-12 or more indirectly by the growth of the apparent activation energy for structural relaxation, as seen in its non-Arrhenius temperature dependence. These observations suggest an interpretation of the experimental glass transition in terms of a generic, collective mechanism possibly controlled by a sharp phase transition. 13 'Solving the glass problem' thus amounts to identifying and obtaining direct experimental signatures about the fundamental nature and the mathematical des...