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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - MALIG (A mathematical approach to the liquid-glass transition: kinetically constrained models, cellular automata and mixed order phase transitions)

Teaser

Summary

\"Glass is widely present in our daily life: it is a very versatile material, easily produced and manipulated on an industrial scale. And yet a microscopic understanding of this state of matter and of how the glass forms still remains a challenge for condensed matter physicists. Nobel prize Philip W. Anderson, wrote in 1995: \"\"The deepest and most interesting unsolved problem in solid state theory is probably the theory of the nature of glass and the glass transition.\"\" He added, \"\"This could be the next breakthrough in the coming decade.\"\" And yet, more than twenty years later, physicists still disagree about the nature of glass and on how it forms.
At the heart of this puzzle lies the intriguing fact that the glass displays properties of both solids and liquids.
Despite its rigidity, the microscopic structure of a glass has the same disordered arrangement of molecules and atoms as a liquid.
Experimentally, the amorphous solid structure of the glass is formed when a liquid mixture of silica, sodium carbonate and calcium oxyde
sufficiently fast: the nucleation of the crystal is prevented and the liquid enters a metastable supercooled phase.
Very roughly speaking, the liquid-crystal transition is avoided because molecules do not have enough time to organise themselves to form the ordered crystal structure.
The molecules move slower and slower forming a thick syrup and eventually they get trapped in the structureless glass state.
Even if this state is not thermodynamically stable, relaxation times are out of reach of any reasonable experiment and the system gets stuck.
The slowing down of dynamics is extremely sharp: relaxation times can increase of 14 orders of magnitude upon a small decrease in temperature. This dramatic growth of time scales is related to the fact that when the temperature is lowered the density is augmented: molecules tend to obstruct each other, blocked structures may arise, and the motion becomes very cooperative.
A key experimental observation of the cooperative motion is the fact that when a glass cools, the molecules do not slow down uniformly. There is indeed a clear coexistence of fast and slow regions, a phenomenon that is called dynamical heterogeneities: some regions of the liquid jam first, while in other regions molecules continue to move around.
Thus, even if a change of structure does not occur when the glass is formed, an underlying dynamical phase transition separating slow and fast trajectories seems to occur.
Besides dynamical heterogeneities and the anomalous divergence of relaxation times, a very rich phenomenology occurs in the vicinity of the glass transition: aging, hysteresis, rejuvenation and anomalous transport phenomena,.. Despite a great deal of experimental and theoretical investigation, a complete understanding of the glass transition is still far out of reach. None of the numerous theories covers all the phenomenology and a common consensus around \"\"the\"\" theory of the glass transition is certainly still lacking in the physics community. A central theoretical difficulty is the fact that from the point of view of critical phenomena the situation is very peculiar: the liquid/glass transition displays a \"\"mixed character\"\". Indeed diverging time and length scales (typical of second order phase transitions) are accompanied by a discontinuous order parameter (typical of first order transitions). The jump of the order parameter corresponds to the discontinuous emergence of an amorphous density profile.
Furthermore, both from the experimental and the theoretical point of view, the huge degeneracy of ground states complicates the problem. Therefore physicists agree that this is certainly a non standard type of ergodicity breaking transition.
The fervent research activity around the glass transition is also enhanced by the fact that a dynamical arrest towards an amorphous state displaying similar properties occurs in a large variety of physical systems upon tuning a proper exte\"

Work performed

Our major progresses since the beginning of the project concern the study of the equilibrium dynamics for KCM (objective 1 in the DoA). We have in particular:

1. developed a toolbox to determine upper bounds for the relaxation time (inverse of the spectral gap) in a paper published in Annals of Probability (authors: PI+team member Fabio Martinelli);
2. identified the universality classes for KCM in two dimensions in a paper published in Communications in Mathematical Physics (authors: team members Fabio Martinelli+Rob Morris+PI);
3. devised an algorithmic construction of the bottleneck of the dynamics that allows to prove lower bounds for the relaxation times when the energy barriers dominate and therefore the divergence of time scales is much faster than for the corresponding bootstrap models.
This is the subject of two papers, the first one in Annals of Probability (authors: team members Fabio Martinelli and Laure Mareche, PhD of PI) + a preprint (authors: team members Laure Mareche [PhD student of PI] , Ivailo Hartarsky [PhD student of PI]+ PI)
4. we have developed some variational tools that allow the study of KCM with random constraints (preprint by Assaf Shapira [PhD of PI])
5. we have devised a toolbox that allows to establish for KCM with conservative dynamics a diffusive behavior for the relaxation time (preprint by team members Fabio Martinelli, Assaf Shapira [PhD student of PI] + PI) and a positive self diffusion coefficient for the tagged particle (in a paper published in Annals of IHP, by O.Blondel + PI).

Besides the study of the equilibrium regime, we have initiated the study of the out of equilibrium dynamics for KCM, a particularly challenging task due to the non monotonicity of the dynamics.
1. We have studied the non equilibrium dynamics of the so called Friedrickson Andersen model in one dimension started from a configuration entirely occupied on the left half-line and focused on the evolution of the front, namely the position of the leftmost zero. We prove, in a proper density regime, a law of large numbers and a central limit theorem for the front, as well as the convergence to an invariant measure of the law of the process seen from the front. This work, appeared in a paper in Electronic Journal of Probability by Oriane Blondel, Aurelia Deshayes (team member, postdoc) and PI, is a first step for the study of shape theorems for KCM (objective 4 of DoA).
2. We have determined exponential convergence to equilibrium when the dynamics starts far from equilibrium for the East model in dimension greater than one and for the so called supercritical models. In both cases, model specific techniques had to be developed to prove convergence to equilibrium due to the non monotonicity of the dynamics and to the the existence of blocked structures that prevent the use of standard coercive inequalities. This is the subject of two preprints by team member Laure Mareche (PhD of PI). These works are the first steps for the development of a general toolbox to study convergence to equilibrium of KCM (objective 2 of DoA).

We have also studied the dynamical phase transition that is responsible for the coexistence of slow and mobile regions for glassy dynamics (objective 5 in the DoA). This dynamical transition leads to a singularity in the large deviations of the activity, the quantity which encodes the fluctuations of the number of configuration changes during a long time interval. In a series of works by team member Vivien Lecomte and co-authors we study the finite size scaling around this transition, we propose a possible connection between this transition and the experimentally observed dynamical heterogeneities in glass forming liquids, and we study and optimise numerical algorithms (the so called cloning algorithms) adapted to evaluate large deviation functions.

Concerning the study of boostrap percolation (see in particular objectives 6 and 8 in DoA) , our main results include:
1) the study of the critical pr

Final results

\"In the first half of the project we have made several progresses beyond the state of the art. We have in particular established the universality classes for two dimensional KCM and developed a robust toolbox that allows to establish critical time scales for the equilibrium dynamics of KCM (see the above section \"\"performed work\"\" for more details and additional results).

We have also made some first steps towards the study of the out of equilibrium dynamics. However all the non equilibrium results apply so far only to some specific models. In the second half of the project we intend to build on this first results to develop robust mathematical tools to study the out of equilibrium dynamics.
A particularly interesting regime is the evolution occurring after a density quench, namely when the system is initialized at a density which is different from the equilibrium value. The basic issues are: (i) whether the system converges at large times to equilibrium; (ii) in the affirmative case, the form of convergence and otherwise the study of the asymptotic non-equilibrium measure.
The natural conjecture, supported by numerical simulations, is that convergence is ubiquitous if both the initial and the equilibrium density are in the ergodic regime (i.e. the regime in which blocked clusters do not percolate).
Proving this conjecture is mathematically very challenging: due to the existence of blocked structures the behavior is not uniform on the choice of the initial configuration, classic coercive inequalities to analyze relaxation to equilibrium fail and standard coupling techniques cannot be applied due to the non-attractiveness of dynamics. We expect that the tools that we will develop in this context can be useful for other models different from KCM with non attractive dynamics.
An even more challenging situation occurs when the equilibrium density is instead in the non ergodic regime. In this case the evolution should be dominated by the coarsening of increasingly large clusters which can be unblocked only from their boundary. We will analyze this coarsening dynamics with the aim of unveiling the properties of the measure towards which the system evolves. The analysis of this regime is also particularly relevant from the physicists\'s point of view. On the one hand interesting aging and coarsening phenomena are expected to occur, on the other hand numerical simulations do not give clearcut results due to the extremely slow dynamics.

Numerical simulations show that non-random limit shapes emerge for KCM in the large time limit for the set of sites which have been already updated when the initial condition are such that \"\"infection\"\" can initially spread only from a certain set of points.
Since KCM are neither monotone nor additive, the usual tools to prove shape theorems cannot be applied and this conjecture is still open. We expect this phenomenon to be ubiquitous and strongly related to the fact that glassy dynamics occurs at the phase coexistence among active and non-active regions, corresponding to a first order transition among slow and mobile trajectories. Building on the results on the first half of the project on this dynamical phase transition, we will work towards the proof of this conjecture.

Concerning bootstrap percolation, our main objective in the second half of the project will be to study the behavior around criticality for the so called \"\"subcritical bootstrap percolation models\"\" in two dimension, namely those models that display a percolation transition, a study which we already initiated in the first half of the project. We intend also to study bootstrap models on different types of graphs, with a particular focus on models that display a mixed order discontinuous/critical transition which is particularly relevant for the study of glassy dynamics.

Finally, we wish to continue the study of KCM with random constraints initiated in the first half of the project, and in particular to study the behavior o\"