We give a new and elementary proof that the number of elastic collisions of a finite number of balls in the Euclidean space is finite. We show that if there are $n$ balls of equal masses and radii 1, and at the time of a collision between any two balls the distance between any other pair of balls is greater than $n^{-n}$, then the total number of collisions is bounded by $n^{(5/2+\epsilon )n}$, for any fixed $\epsilon >0$ and large $n$. We also show that if there is a number of collisions larger than $n^{cn}$ for an appropriate $c>0$, then a large number of these collisions occur within a subfamily of balls that form a very tight configuration.

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关于硬球碰撞的次数

我们给出了一个新的基本证明，即在欧几里得空间中有限数量的球的弹性碰撞次数是有限的。我们证明如果有$ñ$ 质量等于半径1的球，并且在任何两个球发生碰撞时，其他任何一对球之间的距离都大于 ${ñ}^{-ñ}$，则总碰撞次数以 ${ñ}^{（5/2+\epsilon ）ñ}$，用于任何固定 $\epsilon >0$ 和大 $ñ$。我们还表明，如果碰撞次数大于${ñ}^{Cñ}$ 适当的 $C>0$，则大量此类碰撞会在形成非常紧密配置的球子族中发生。