Markov Chains with Kinetic Constraints and Applications (20w5154)


(CNRS and University Paris Dauphine)

Shirshendu Ganguly (UC Berkeley)

Fabio Martinelli (Universita' Roma Tre)


The Banff International Research Station will host the "Markov Chains with Kinetic Constraints and Applications" workshop in Banff from July 12 to July 17, 2020.

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 scientists. At the heart of this puzzle lies the intriguing fact that glasses display properties of both solids and liquids. Even though glasses are rigid, their disordered arrangement of atoms and molecules is essentially indistinguishable from that of a liquid. Thus, when physicists or chemists look at a drinking glass filled with water and ice cubes, it's not clear to them whether the glass is more like the water or the ice! Even though scientists cannot yet explain how glass is formed, we have we been manufacturing glasses for the last 2000 years. The secret is to cool a liquid mixture of silica, sodium carbonate and calcium oxyde very fast in order to prevent the formation of the ordered crystal structure. Then molecules move slower and slower forming a thick syrup and eventually they get stuck in the structureless glass state. A very rich and fascinating phenomenology occurs before the formation of the glass including aging, hysteresis, rejuvenation, anomalous transport phenomena, cooperative and heterogeneous motion. Furthermore, a dynamical arrest displaying similar features occurs in a variety of different materials: grains in powders, emulsions, foams, colloidal suspensions, plastics, However, despite a great deal of experimental and theoretical investigation, understanding these phenomena is far out of reach and the subject is still hotly debated. To use a joke of David Weitz, a physics professor at Harvard, "There are more theories of the glass transition than there are theorists who propose them."

So, how do mathematicians come into play? in the 1980's, a class of models called "Kinetically Constrained Models" (KCM), have been proposed by physicists to shed some light on the glassy behavior. KCM live on a discrete grid, each site of the grid being either empty or occupied by one particle, and dynamics evolves via a stochastic process in which moves at a certain location occur only if there are enough empty sites around. Despite being an oversimplified model, KCM display a rich phenomenology including the key dynamical features of real glassy systems. However, understanding the behavior of KCM turned out to be a hard task. On the one hand the interesting regimes are often beyond reach of numerical simulations, since KCM display an extremely slow dynamics (as do the real glassy systems). On the other hand most of the tools that have been developed by physicists and mathematicians to study particle systems with stochastic dynamics cannot be applied to KCM. This is a direct consequence of the presence of the constraints requiring enough empty neighbouring sites, which are in turn essential to model glassy dynamics. In the last decade, an increasing number of mathematical works have been devoted to the study of KCM. These rigorous results have already had an important impact in the glassy community, confirming or disproving some conjectures that had been put forward on the basis of numerical simulations. However, several key issues remain widely open. The aim of this workshop is to gather highly qualified experts covering complementary areas that are related to the different facets of the KCM dynamics, to foster interactions among different communities and to develop novel mathematical tools to solve the open problems.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).