Inelastic collapse in one-dimensional driven systems under gravity

We study inelastic collapse in a one-dimensional N-particle system when the system is driven from below under gravity. We investigate the hard-sphere limit of inelastic soft-sphere systems by numerical simulations to find how the collision rate per particle n_coll increases as a function of the elastic constant of the sphere k when the restitution coefficient e is kept constant. For systems with large enough Na≳³20, we find three regimes in e depending on the behavior of n_coll in the hard-sphere limit: (i) an uncollapsing regime for 1≥e>e_c1, where n _coll converges to a finite value, (ii) a logarithmically collapsing regime for e_c1>e>e_c2, where n_coll diverges as n_coll∼logk, and (iii) a power-law collapsing regime for e_c2>e>0, where n_coll diverges as n _coll∼kα with an exponent α that depends on N. The power-law collapsing regime shrinks as N decreases and seems not to exist for the system with N=3, while, for large N, the size of the uncollapsing and the logarithmically collapsing regime decreases as e_c1≃1-2.6/N and e_c2≃1-3.0/N. We demonstrate that this difference between large and small systems exists already in the inelastic collapse without external drive and gravity.