We consider the adiabatic evolution of glassy states under external perturbations. Although the formalism we use is very general, we focus here on infinite-dimensional hard spheres where an exact analysis is possible. We consider perturbations of the boundary, i.e. compression or (volume preserving) shear-strain, and we compute the response of glassy states to such perturbations: pressure and shear-stress. We find that both quantities overshoot before the glass state becomes unstable at a spinodal point where it melts into a liquid (or yields). We also estimate the yield stress of the glass. Finally, we study the stability of the glass basins towards breaking into sub-basins, corresponding to a Gardner transition. We find that close to the dynamical transition, glasses undergo a Gardner transition after an infinitesimal perturbation.Introduction -Glasses are long lived metastable states of matter, in which particles are confined around an amorphous structure [1,2]. For a given sample of a material, the glass state is not unique: depending on the preparation protocol, the material can be trapped in different glasses, each displaying different thermodynamic properties. For example, the specific volume of a glass prepared by cooling a liquid depends strongly on the cooling rate [1,2]. Other procedures, such as vapor deposition, produce very stable glasses, with higher density than those obtained by simple cooling [3,4]. When heated up, glasses show hysteresis: their energy (specific volume) remains below the liquid one, until a "spinodal" point is reached, at which they melt into the liquid (see e.g. [2, Fig.1] and [4, Fig.2]).The behavior of glasses under shear-strain also shows similarly complex phenomena. Suppose to prepare a glass by cooling a liquid at a given rate until some low temperature T is reached. After cooling, a strain γ is applied and the stress σ is recorded. At small γ, an elastic (linear) regime where σ ∼ µγ is found. At larger γ, the stress reaches a maximum and then decreases until an instability is reached, where the glass yields and starts to flow (see e.g. [5, Fig.3c] and [6, Fig.2]). The amplitude of the shear modulus µ and of the stress overshoot increase when the cooling rate is decreased, and more stable glasses are reached.Computing these observables theoretically is a difficult challenge, because glassy states are always prepared through non-equilibrium dynamical protocols. Firstprinciple dynamical theories such as Mode-Coupling Theory (MCT) [7] are successful in describing properties of supercooled liquids close to the glass state (including the stress overshoot [8]), but they fail to describe glasses at low temperatures and high pressures [9]. The dynamical facilitation picture can successfully describe calorimetric properties of glasses [10], but for the moment it does not allow one to perform first-principles calculations starting from the microscopic interaction potential. To bypass the difficulty of describing all the dynamical details of glass formation, one can exploit a standa...

Jönsson, Yoshino, and Nordblad Reply: In their Comment to our work [1], Berthier and Bouchaud [2] report results of temperature shifts T in a numerical simulation of a four-dimensional Edwards-Anderson Ising spin-glass model using the same thermal protocol as in [1]. Their scaling analysis yields an apparently similar result to ours.In our experiments the data are limited to a narrow temperature region jTj < 0:02 T g . The expected saturation of L eff to the overlap length L T [3] is not observed for these small values of jTj. In this Reply, we instead point out that our experimental results can be understood in terms of temperature chaos as partial rejuvenation in a weakly perturbed regime.Within the droplet theory [3], it is assumed that there are dangerously irrelevant droplets whose free-energy gaps F L are arbitrarily small. The gap can be smaller than dF with probability 0dF=L=L 0 , wherẽ 0 > 0. Consequently, a perturbation which amounts to a small free-energy gain of order UL=L 0 can induce a droplet excitation with a probability pL=L U 0L=L U . Here L U L 0 U= ÿ1= is the overlap length with the chaos exponent ÿ . In the case of T shifts, U k B jTj and d s =2, where d s is the surface fractal dimension of the droplets. Importantly, the probability pL=L U is nonzero in the weakly perturbed regime L < L U and increases with L if > 0 smoothly connecting to the strongly perturbedSupposing that L T > L T i t w , all length scales smaller than the domain size L T i t w belong to the weakly perturbed regime. A finite region of size L within a domain can then become out of equilibrium with the probability pL=L T . This reflects the breaking up of domains into smaller ones, yielding a broad distribution of domain sizes up to the maximum L max L T i t w . Then a simple scaling argument gives that the effective domain size surrounding an arbitrary point of the system on average reaches L eff L T FL max =L T with Fx x ÿ cx 1 . Here the second term in Fx, with c being a positive constant, yields a correction term, which is the first order of 0U= to the no-chaos limit of L eff . Higher order terms can be neglected for small enough jTj. It can be seen in Fig. 1 that the ansatz fits our data for jTj & 0:4 K.In the simulations of Berthier-Bouchaud [2], length scales L & 5 are probed while L T * 20. In [5] the spatial correlation function is found to decay to 0 only after L T i t w , being consistent with L max L T i t w . Then the rejuvenation found in [2] can be understood as partial rejuvenation due to the temperature chaos effect in a weakly perturbed regime. We note that emergence of the temperature chaos effect has been confirmed in the same model as in [2] with modest system sizes [6]. Indeed, we have found that the ansatz proposed above also fits the result of Berthier-Bouchaud [2] very well with c 0:15 and 1. However, the largest T shifts used in [2] involve temperatures in the range 0:8-0:9T g , within which numerical (but not experimental) length/ time scales are dominated by critical fluctuations as shown by Fig...

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Using Monte Carlo simulations, we have studied isothermal aging of three-dimensional Ising spin-glass model focusing on quasi-equilibrium behavior of the spin auto-correlation function. Weak violation of the time translational invariance in the quasi-equilibrium regime is analyzed in terms of effective stiffness for droplet excitations in the presence of domain walls. Within the range of computational time window, we have confirmed that the effective stiffness follows the expected scaling behavior with respect to the characteristic length scales associated with droplet excitations and domain walls, whose growth law has been extracted from our simulated data. Implication of the results are discussed in relation to experimental works on ac susceptibilities.