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We combine the swap Monte Carlo algorithm to long multi-CPU molecular dynamics simulations to analyse the equilibrium relaxation dynamics of model supercooled liquids over a time window covering ten orders of magnitude for temperatures down to the experimental glass transition temperature $T_g$. The analysis of \rev{several} time correlation functions coupled to spatio-temporal resolution of particle motion allow us to elucidate the nature of the equilibrium dynamics in deeply supercooled liquids. We find that structural relaxation starts at early times in rare localised regions characterised by a waiting time distribution that develops a power law near $T_g$. At longer times, relaxation events accumulate with increasing probability in these regions as $T_g$ is approached. This accumulation leads to a power-law growth of the linear extension of relaxed domains with time with a large, temperature-dependent dynamic exponent. Past the average relaxation time, unrelaxed domains slowly shrink with time due to relaxation events happening at their boundaries. Our results provide a complete microscopic description of the particle motion responsible for key experimental signatures of glassy dynamics, from the shape and temperature evolution of relaxation spectra to the core features of dynamic heterogeneity. They also provide a microscopic basis to understand the emergence of dynamic facilitation in deeply supercooled liquids and allow us to critically reassess theoretical descriptions of the glass transition.

Glass is everywhere. We use and are surrounded by glass objects which make tangible the reality of glass as a distinct state of matter. Yet, glass as we know it is usually obtained by cooling a liquid sufficiently rapidly below its melting point to avoid crystallisation. The viscosity of this supercooled liquid increases by many orders of magnitude upon cooling, until the liquid becomes essentially arrested on experimental timescales below the ``glass transition'' temperature. From a structural viewpoint, the obtained glass still very much resembles the disordered liquid, but from a mechanical viewpoint, it is as rigid as an ordered crystal. Does glass qualify as a separate state of matter? We provide a pedagogical perspective on this question using basic statistical mechanical concepts. We recall the definitions of states of matter and of phase transitions between them. We review recent theoretical results suggesting why and how an ``ideal glass'' can indeed be defined as a separate equilibrium state of matter. We discuss recent success of computer simulations trying to analyse this glass state. We close with some experimental perspectives.

We numerically study the equilibrium relaxation dynamics of a two-dimensional Mari-Kurchan glass model. The tree-like structure of particle interactions forbids both non-trivial structural motifs and the emergence of a complex free-energy landscape leading to a thermodynamic glass transition, while the finite-dimensional nature of the model prevents the existence of a mode-coupling singularity. Nevertheless, the equilibrium relaxation dynamics is shown to be in excellent agreement with simulations performed in conventional glass-formers. Averaged time-correlation functions display a phenomenology typical of supercooled liquids, including the emergence of an excess signal in relaxation spectra at intermediate frequencies. We show that this evolution is accompanied by strong signatures of collective and heterogeneous dynamics which cannot be interpreted in terms of single particle hopping and emerge from dynamic facilitation. Our study demonstrates that an off-lattice interacting particle model with extremely simple structural correlations displays quantitatively realistic glassy dynamics.

We explore the emergence of nonequilibrium collective motion in disordered non-thermal active matter when persistent motion and crowding effects compete, using simulations of a two-dimensional model of size polydisperse self-propelled particles. In stark contrast with monodisperse systems, we find that polydispersity stabilizes a homogeneous active liquid at arbitrary large persistence times, characterized by remarkable velocity correlations and irregular turbulent flows. For all persistence values, the active fluid undergoes a nonequilibrium glass transition at large density. This is accompanied by collective motion, whose nature evolves from near-equilibrium spatially heterogeneous dynamics at small persistence, to a qualitatively different intermittent dynamics when persistence is large. This latter regime involves a complex time evolution of the correlated displacement field

We use atomistic computer simulations to provide a microscopic description of the brittle failure of amorphous materials, and we assess the role of rare events and quenched disorder. We argue that brittle yielding originates at rare soft regions, similarly to Griffiths effects in disordered systems. We numerically demonstrate how localized plastic events in such soft regions trigger macroscopic failure via the propagation of a shear band. This physical picture, which no longer holds in poorly annealed ductile materials, allows us to discuss the role of finite size effects in brittle yielding and reinforces the similarities between yielding and other disorder-controlled nonequilibrium phase transitions.