In granular systems temperature does not play a major role. The dynamics of such systems are dominated by the geometric packing fraction of their constituents and may be described by various kinetically-constrained models [1]. We focus on the Kob-Andersen model [2], which is defined as a lattice gas with only on-site exclusion, for which particles can move to neighboring sites only if they have less than a certain threshold of occupied neighbors. Most of the previous research considered infinite systems, while actual experimental systems are finite. We use the Kob-Andersen model to describe a semi-infinite two-dimensional system, such that it is infinite in the horizontal direction, but finite in the vertical direction [3].

By increasing the density of particles in the system, it becomes jammed, i.e. almost all the particles can never move according to the kinetic constraints. It has been proven [4] that a system which is infinite in both directions, gets jammed only at a density of 1, namely when all lattice sites are occupied by particles. We find, analytically and numerically, that finite-width systems become jammed at some finite width-dependent density which nears 1 as the width of the tunnel is increased. We obtain exact analytical results for the fraction of jammed particles for thin systems (width 1 or 2), and good approximations for arbitrary width.

We also investigate the structures and the heterogeneity of the jammed particles along the vertical direction. We find that the particles can be divided into two groups: those in the top and bottom rows, and all the rest. The particles in the top and bottom rows have a higher probability of being jammed than those in the middle, since the system's boundary acts as a constraint that blocks them, while the others do not feel the system being finite.

[1] F. Ritort and P. Sollich, Adv. Phys. 52, 219 (2003).

[2] W. Kob and H. C. Andersen, Phys. Rev. E 48, 4364 (1993).

[3] E. Teomy and Y. Shokef, Accepted to Phys. Rev E. Preprint arXiv:1211.0860

[4] C. Toninelli, G. Biroli and D. S. Fisher, Phys. Rev. Lett. 92, 185504 (2004).