Relation between structure and dynamics in kinetically constrained models

In a liquid all the particles are mobile, while in a glass only some of them are mobile at any given time. Although overall the structure is amorphous in both cases, the difference is that in glasses there are local structures that inhibit the movement of particles inside them. We investigate these structures by considering the minimum number of particles that need to move before a specific particle can move. By mapping the dynamics of the particles to diffusion of mobile vacancies, we find a general algebraic relation between the mean size of the structures and the mean persistence time, which is the time until a particle moves for the first time. The exponent relating these two quantities depends on the system's properties. 

We investigated this relation analytically and numerically in several kinetically-constrained models: the Fredrickson-Andersen, Kob-Andersen and Spiral models. These models are either lattice gas models or Ising-like models, in which a particle can move or a spin can flip only if the local environment satisfies some model-dependent rule. Due the discrete nature of these models and relative simplicity, we were able to analytically find the relation between the structure and the dynamics and found an excellent agreement between our analytical results, our numerical simulations, and the heuristic arguments presented above. In these simple models, the minimum number of particles that need to move before a specific particle can move is easily found by using a culling algorithm, also called bootstrap or threshold percolation.