Many physical systems e.g. atomic or molecular systems, granular matter and colloidal suspensions, when rapidly cooled or compressed undergo a glass or jamming transition from a dynamical fluid-like state to a frozen solid-like state. Such transitions are characterized by spatial disorder accompanied by a divergent relaxation time such that structures are almost static over observation time scales. A simple way to theoretically model such non-equilibrium transitions is to use discrete lattice models with a set of kinetic rules for their dynamics. Jamming percolation models become non-ergodic even in the thermodynamic limit, contrary to other kinetically-constrained models which exhibit an apparent transition only due to finite-size effects.
Inspired by the two-dimensional spiral model, we introduced a kinetically-constrained model in three dimensions, which we have proved to jam at the critical density of three-dimensional directed percolation. We investigated its static and dynamic properties numerically and provided a theoretical understanding of the observed critical behavior based on the theory of directed percolation. Our numerical results show that the model exhibits a mixed transition at a non-trivial critical density, where on one hand the fraction of frozen particle jumps discontinuously from zero to a finite value, and on the other hand the cluster size of frozen particles follows a power law scaling near the critical point. This transition is indeed of the same nature as observed in jamming transitions in other physical systems and qualitatively similar to that found in two dimensions. However, the dynamics of our model are qualitatively different, exhibiting diffusive behavior even beyond the critical density unlike its two-dimensional counterpart. This occurs due to the way percolated structures form - in both cases, the existence of one dimensional percolating paths of mutually blocked particles leads to jamming. However, in three dimensions, the mobile particles can still bypass through the third dimension.
Jamming percolation in three dimensions
A. Ghosh, E. Teomy, and Y. Shokef
Europhysics Letters 106, 16003 (2014)