Supercooled liquids exhibit a pronounced slowdown of their dynamics on cooling 1 without showing any obvious structural or thermodynamic changes 2 . Several theories relate this slowdown to increasing spatial correlations 3-6 . However, no sign of this is seen in standard static correlation functions, despite indirect evidence from considering specific heat 7 and linear dielectric susceptibility 8 . Whereas the dynamic correlation function progressively becomes more non-exponential as the temperature is reduced, so far no similar signature has been found in static correlations that can distinguish qualitatively between a high-temperature and a deeply supercooled glass-forming liquid in equilibrium. Here, we show evidence of a qualitative thermodynamic signature that differentiates between the two. We show by numerical simulations with fixed boundary conditions that the influence of the boundary propagates into the bulk over increasing length scales on cooling. With the increase of this static correlation length, the influence of the boundary decays non-exponentially. Such long-range susceptibility to boundary conditions is expected within the random first-order theory 4,9,10 (RFOT) of the glass transition. However, a quantitative account of our numerical results requires a generalization of RFOT, taking into account surface tension fluctuations between states.Inspired by critical phenomena, it is natural to expect that the slowing down of the dynamics is related to the vicinity of a thermodynamic phase transition, where some kind of long-range order would set in 11 . This is the spirit of different recent theories 4,9,[12][13][14] , but seems at odds with others 5,15 , at least at first sight. In particular, the crucial physical mechanism at the root of random first-order theory 4 (RFOT) is the emergence of long-range amorphous order, the precise definition and quantitative characterization of which is however far from obvious. Dynamic heterogeneities 16 do show a growing dynamic correlation length accompanying the glass transition, both experimentally 17 and numerically 18 . This is certainly a first important step, but not sufficient to prune down-even at a qualitative level-different theories of the glass transition. In particular, it is not clear whether this phenomenon is due to an underlying static or purely dynamic phase transition.The approach followed here is based on the very definition of a thermodynamic phase transition, where the effect of boundary conditions becomes long-ranged. The problem is that for glasses there are no natural boundary conditions, because these should be as 'random' as the bulk amorphous states that they favour. A possible solution is to use equilibrium liquid configurations to define the boundary 19 . In the context of RFOT, this was suggested in ref. 9 (and further discussed in ref. 11), but the scope and some conclusions of this Gedankenexperiment are more general [19][20][21] . Starting from a given equilibrium configuration, we freeze the motion of all particles outside a ...

Glasses are amorphous solids, in the sense that they display elastic behaviour. In crystalline solids, elasticity is associated with phonons, which are quantized vibrational excitations. Phonon-like excitations also exist in glasses at very high (terahertz; 10(12) Hz) frequencies; surprisingly, these persist in the supercooled liquids. A universal feature of such amorphous systems is the boson peak: the vibrational density of states has an excess compared to the Debye squared-frequency law. Here we investigate the origin of this feature by studying the spectra of inherent structures (local minima of the potential energy) in a realistic glass model. We claim that the peak is the signature of a phase transition in the space of the stationary points of the energy, from a minima-dominated phase (with phonons) at low energy to a saddle-point-dominated phase (without phonons). The boson peak moves to lower frequencies on approaching the phonon-saddle transition, and its height diverges at the critical point. Our numerical results agree with the predictions of euclidean random matrix theory on the existence of a sharp phase transition between an amorphous elastic phase and a phonon-free one.

We compute the thermodynamic properties of the glass phase in a binary mixture of soft spheres. Our approach is a generalization to mixtures of the replica strategy, recently proposed by Mezard and Parisi, providing a first principle statistical mechanics computation of the thermodynamics of glasses. The method starts from the inter-atomic potentials, and translates the problem into the study of a molecular liquid. We compare our analytical predictions to numerical simulations, focusing onto the values of the thermodynamic transition temperature and the configurational entropy.

According to the mosaic scenario, relaxation in supercooled liquids is ruled by two competing mechanisms: surface tension, opposing the creation of local excitations, and entropy, providing the drive to the configurational rearrangement of a given region. We test this scenario through numerical simulations well below the Mode Coupling temperature. For an equilibrated configuration, we freeze all the particles outside a sphere and study the thermodynamics of this sphere. The frozen environment acts as a pinning field. Measuring the overlap between the unpinned and pinned equilibrium configurations of the sphere, we can see whether it has switched to a different state. We do not find any clear evidence of the mosaic scenario. Rather, our results seem compatible with the existence of a single (liquid) state. However, we find evidence of a growing static correlation length, apparently unrelated to the mosaic one.It is common opinion that in finite dimension a divergence of a relaxation time τ at nonzero temperature is associated to a diverging characteristic length ξ. The idea is that when this length increases, relaxation proceeds through the rearrangement of ever larger regions, taking a longer and longer time. The relation between τ and ξ depends on the physical mechanism of relaxation. Two main mechanisms are activated relaxation of a ψ-dimensional droplet of size ξ, giving τ ∼ exp(Aξ ψ /T ), and critical slowing down, whereGlass-forming liquids are tricky: relaxation times grow spectacularly (more than ten decades) upon lowering the temperature, without clear evidence of a growing static cooperative lenght. In particular, density fluctuations are thought to remain correlated over short distances close to the glass transition (although there are some indication that energy fluctuations might develop larger correlations [2,3]). Thus the concept of dynamic heterogeneities is central to several theories of the glass transition [4,5,6,7], where the role of order parameter is played by dynamic quantities such as local time correlators, which become correlated over the growing dynamic lenght scale ξ dyn . No thermodynamic singularity is present in these theories. Dynamic singularities are also typically absent at finite temperatures, with the notable exception of mode coupling theory (MCT) [8], recently recast in terms of dynamic heterogeneities [9]. Note, however, that the experimental values of ξ dyn [10,11,12,13] are barely in the nm range, the same as density correlations [14].The mosaic scenario (MS) [15,16,17], working within the conceptual framework of nucleation theory, identifies on the other hand a static correlation length. Deeply rooted in the physics of mean-field spin glasses, the MS crucially assumes the existence of exponentially many inequivalent states exp(N Σ), below the mode coupling temperature T MC (Σ is called complexity or configurational entropy, and N is the size of the system). Suppose the system is in a state α and ask: what is the free energy cost for a region of linear size R to rearrange...