Inelastic collapse in one-dimensional driven systems under gravity

We study the inelastic collapse in the one-dimensional $N$-particle systems in the situation where the system is driven from below under the gravity. We investigate the hard-sphere limit of the 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 the systems with large enough $N \agt 20$, we find three regimes in $e$ depending on the behavior of $n_{coll}$ in the hard-sphere limit: (i) uncollapsing regime for $1 \ge e > e_{c1}$, where $n_{coll}$ converges to a finite value, (ii) logarithmically collapsing regime for $e_{c1} > e > e_{c2}$, where $n_{coll}$ diverges as $n_{coll} \sim \log k$, and (iii) power-law collapsing regime for $e_{c2} > e > 0$, where $n_{coll}$ diverges as $n_{coll} \sim k^\alpha$ with an exponent $\alpha$ 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 $e_{c1} \simeq 1-2.6/N$ and $e_{c2} \simeq 1-3.0/N$. We demonstrate that this difference between large and small systems exists already in the inelastic collapse without the external drive and the gravity.