Periodic Reporting for period 4 - MALIG (A mathematical approach to the liquid-glass transition: kinetically constrained models, cellular automata and mixed order phase transitions)
Reporting period: 2021-03-01 to 2022-07-31
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 it forms still remains a challenge for condensed matter physicists. At the heart of this puzzle lies the intriguing fact that the glass displays properties of both solids and liquid: despite its rigidity, the microscopic structure of a glass has the same disordered arrangement of molecules and atoms as a liquid. Experimentally, the glass is formed when a liquid is cooled
sufficiently fast: the nucleation of the crystal is prevented and the liquid enters a metastable supercooled phase characterised by dramatically long relaxation times and by a very cooperative motion.
The dynamics slows down as temperature is further lowered, the liquid becomes a thick sirup and when it can't flow any more the glass is formed.
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 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. 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 external parameter. This "jamming transition" takes place in several materials with industrial applications: grains in powders, emulsions, foams, colloidal suspensions, polymers, plastics, ceramics, etc.
The main objective of this project has been the mathematical study of different types of models that have been proposed to study the liquid-glass and more general jamming transitions.
We have developed in particular new mathematical tools to study kinetically constrained models (KCM), which have been first introduced in the '80s and have been subject of several numerical and theoretical works in physics literature in the last thirty years. From a mathematical point of view, KCM belong to the class of interacting particle systems with stochastic dynamics on discrete lattices. Each lattice site is empty or occupied by one particle. The evolution is given by a continuous time Markov process in which the elementary moves are eith birth/death of particles or jumps of particles. The key feature in both cases is that a move can occur only if the configuration verifies a "local constraint".
Modelization of the liquid glass and jamming transitions via KCM relies on the idea that these are purely dynamical phenomena with static interactions playing a minor role.
In spite of their simplicity, KCM display many key dynamical features of real materials that undergo glass or jamming transitions. These include in particular: anomalous ergodicity breaking transition, percolation of blocked structures, dynamical arrest, super-Arrehenius divergence of the relaxation time, non-trivial spatio-temporal
fluctuations corresponding to dynamical heterogeneities, transport decoupling and aging. Mathematically, KCM pose
very challenging and interesting problems. Indeed, the presence of the constraints induce the existence of clusters of blocked particles, the occurrence of several invariant measures, non-attractiveness, the failure of classic coercive inequalities to analyze relaxation to equilibrium, ...We stress that it is not "technical problems" which prevent using the standard techniques: the behavior of KCM is qualitatively different from the models without constraints, thus they truly open a new chapter for interacting particle systems and require the development of new mathematical tools.
Besides the study of KCM, we also made progress on a deeply related subject: the study of bootstrap percolation (BP) models, a rich class of deterministic cellular automata which can be regarded as a deterministic version of KCM. Though several works have been devoted in the last thirty years in the probabilistic and combinatorics community to bootstrap percolation, several issues remained open. and our works have contributed to the study of BP models that feature a percolation transition of blocked clusters.
sufficiently fast: the nucleation of the crystal is prevented and the liquid enters a metastable supercooled phase characterised by dramatically long relaxation times and by a very cooperative motion.
The dynamics slows down as temperature is further lowered, the liquid becomes a thick sirup and when it can't flow any more the glass is formed.
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 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. 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 external parameter. This "jamming transition" takes place in several materials with industrial applications: grains in powders, emulsions, foams, colloidal suspensions, polymers, plastics, ceramics, etc.
The main objective of this project has been the mathematical study of different types of models that have been proposed to study the liquid-glass and more general jamming transitions.
We have developed in particular new mathematical tools to study kinetically constrained models (KCM), which have been first introduced in the '80s and have been subject of several numerical and theoretical works in physics literature in the last thirty years. From a mathematical point of view, KCM belong to the class of interacting particle systems with stochastic dynamics on discrete lattices. Each lattice site is empty or occupied by one particle. The evolution is given by a continuous time Markov process in which the elementary moves are eith birth/death of particles or jumps of particles. The key feature in both cases is that a move can occur only if the configuration verifies a "local constraint".
Modelization of the liquid glass and jamming transitions via KCM relies on the idea that these are purely dynamical phenomena with static interactions playing a minor role.
In spite of their simplicity, KCM display many key dynamical features of real materials that undergo glass or jamming transitions. These include in particular: anomalous ergodicity breaking transition, percolation of blocked structures, dynamical arrest, super-Arrehenius divergence of the relaxation time, non-trivial spatio-temporal
fluctuations corresponding to dynamical heterogeneities, transport decoupling and aging. Mathematically, KCM pose
very challenging and interesting problems. Indeed, the presence of the constraints induce the existence of clusters of blocked particles, the occurrence of several invariant measures, non-attractiveness, the failure of classic coercive inequalities to analyze relaxation to equilibrium, ...We stress that it is not "technical problems" which prevent using the standard techniques: the behavior of KCM is qualitatively different from the models without constraints, thus they truly open a new chapter for interacting particle systems and require the development of new mathematical tools.
Besides the study of KCM, we also made progress on a deeply related subject: the study of bootstrap percolation (BP) models, a rich class of deterministic cellular automata which can be regarded as a deterministic version of KCM. Though several works have been devoted in the last thirty years in the probabilistic and combinatorics community to bootstrap percolation, several issues remained open. and our works have contributed to the study of BP models that feature a percolation transition of blocked clusters.
Our major progresses since the beginning of the project concern the study KCM. In particular we have:
1. developed a toolbox to determine upper bounds for the relaxation time (inverse of the spectral gap)
2. identified the universality classes for KCM in two dimensions
3. devised an algorithmic construction of the bottleneck of the KCM dynamics
4. developed some variational tools that allow the study of KCM with random constraints
5. devised a toolbox that allows to establish for KCM with conservative dynamics a diffusive behavior for the relaxation time and a positive self diffusion coefficient for the tagged particle
6. proven a law of large numbers and a central limit theorem for the front of the one dimensional FA-1f model at high temperature
7. determined exponential convergence to equilibrium for the East model in any dimension and for the so called supercritical models.
Concerning bootstrap percolation we have:
1) studied the sharp asymptotics for the critical probability of the so-called Duarte model
3) studied the logarithmic corrections to the sharp threshold for 2-neighbour bootstrap percolation
4) studied two dimensional models that display a percolation transition (subcritical models)
5) studied the critical exponent for two-dimensional models and algorithms to evaluate this exponent
6) studied the metastability phenomena for bootstrap percolation on Galton Watson trees
Besides KCM and bootstrap percolation models, we have also studied other models with "glassy dynamics" proposed by physicists. In particular we have
1) initiated the mathematical study of a class of ferromagnetic spin models with local frustration, the so called plaquette models
2) studied a one dimensional interface model, and showed that the presence of disorder in the underlying medium causes (as in glasses) the presence of many metastable states that induce anomalous fluctuations.
Finally, I wish to mention a bulk of activity on other combinatorial problems performed by different members of the team, in particular Rob Morris and Erik Slivken (post doc), which are complementary to the study of bootstrap percolation.
All results have been published in high level international journals and disseminated at several workshops.
1. developed a toolbox to determine upper bounds for the relaxation time (inverse of the spectral gap)
2. identified the universality classes for KCM in two dimensions
3. devised an algorithmic construction of the bottleneck of the KCM dynamics
4. developed some variational tools that allow the study of KCM with random constraints
5. devised a toolbox that allows to establish for KCM with conservative dynamics a diffusive behavior for the relaxation time and a positive self diffusion coefficient for the tagged particle
6. proven a law of large numbers and a central limit theorem for the front of the one dimensional FA-1f model at high temperature
7. determined exponential convergence to equilibrium for the East model in any dimension and for the so called supercritical models.
Concerning bootstrap percolation we have:
1) studied the sharp asymptotics for the critical probability of the so-called Duarte model
3) studied the logarithmic corrections to the sharp threshold for 2-neighbour bootstrap percolation
4) studied two dimensional models that display a percolation transition (subcritical models)
5) studied the critical exponent for two-dimensional models and algorithms to evaluate this exponent
6) studied the metastability phenomena for bootstrap percolation on Galton Watson trees
Besides KCM and bootstrap percolation models, we have also studied other models with "glassy dynamics" proposed by physicists. In particular we have
1) initiated the mathematical study of a class of ferromagnetic spin models with local frustration, the so called plaquette models
2) studied a one dimensional interface model, and showed that the presence of disorder in the underlying medium causes (as in glasses) the presence of many metastable states that induce anomalous fluctuations.
Finally, I wish to mention a bulk of activity on other combinatorial problems performed by different members of the team, in particular Rob Morris and Erik Slivken (post doc), which are complementary to the study of bootstrap percolation.
All results have been published in high level international journals and disseminated at several workshops.
We have made several progresses beyond the state of the art, detailed in the above sections.