Inelastic collisions of three particles on a line as a two-dimensional billiard

Constantin, Peter; Grossman, E. L.; Mungan, Muhittin

doi:10.1016/0167-2789(95)00042-3

Cited by 72 publications

(17 citation statements)

References 5 publications

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“…Our result shows that the partially condensed state appears below e c1 , which means N (1 − e) 2.6 from Eq. (25). These results are in agreement with each other and show that the point where the system starts to condense is not sensitive to the driving mode.…”

Section: Summary and Discussionsupporting

confidence: 93%

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Inelastic collapse in one-dimensional driven systems under gravity

et al. 2013

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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 N>/~20, 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)~k(α) 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.

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Section: Summary and Discussionsupporting

confidence: 93%

“…The sequence of m (n) and m (n) , which are completely determined by the collision laws and the initial condition m (0) (or m (n) ), has been studied by Constantin et al [25]. We briefly summarize their results that are relevant for our purpose in this Appendix.…”

Section: Appendix Amentioning

confidence: 97%

Inelastic collapse in one-dimensional driven systems under gravity

et al. 2013

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“…One-dimensional particle Systems performing inelastic collisions have been recently studied as a model for the time évolution of granular media [1][2][3][4]. The main features of these Systems are the possibility of the occurrence of inelastic collapses (namely infinitely many collisions in a finite time) and the tendency of the system to clusterize, that is to create states of concentration of the density, as sand grains over a shaken sheet of paper.…”

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confidence: 99%

A kinetic equation for granular media

Benedetto¹,

Caglioti²,

Pulvirenti³

1999

ESAIM: M2AN

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“…Using energy and momentum conservation, we can determine the state of the system after each collision and thereby find the number of collisions before the three particles mutually recede. However, this approach is complicated and provides minimal physical insight (see Appendix and also [14,15,18,19]). We now present a much simpler solution by mapping the original three-particle system onto a billiard in an appropriately-defined domain.…”

Section: Three Particles On An Infinite Linementioning

confidence: 99%

“…A particularly intriguing feature of inelastic systems is the phenomenon of "inelastic collapse", where clumps of particles with negligible relative motion form. This collapse occurs when the number of particles is sufficiently large or when collisions are sufficiently inelastic [13,14,15]. Some of the methods described here may be useful to understand these systems.…”

Section: Introductionmentioning

confidence: 99%

A billiard-theoretic approach to elementary one-dimensional elastic collisions

Redner¹

2004

American Journal of Physics

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A simple relation is developed between elastic collisions of freely-moving point particles in one dimension and a corresponding billiard system. For two particles with masses m1 and m2 on the half-line x > 0 that approach an elastic barrier at x = 0, the corresponding billiard system is an infinite wedge. The collision history of the two particles can be easily inferred from the corresponding billiard trajectory. This connection nicely explains the classic demonstrations of the "dime on the superball" and the "baseball on the basketball" that are a staple in elementary physics courses. It is also shown that three elastic particles on an infinite line and three particles on a finite ring correspond, respectively, to the motion of a billiard ball in an infinite wedge and on on a triangular billiard table. It is shown how to determine the angles of these two sets in terms of the particle masses.

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Inelastic collisions of three particles on a line as a two-dimensional billiard

Cited by 72 publications

References 5 publications

Inelastic collapse in one-dimensional driven systems under gravity

Inelastic collapse in one-dimensional driven systems under gravity

A kinetic equation for granular media

A billiard-theoretic approach to elementary one-dimensional elastic collisions

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