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On the number of hard ball collisions
Author(s) -
Burdzy Krzysztof,
Duarte Mauricio
Publication year - 2020
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12274
Subject(s) - ball (mathematics) , bounded function , euclidean space , finite set , collision , combinatorics , mathematics , physics , euclidean geometry , elastic collision , geometry , mathematical analysis , quantum mechanics , computer science , computer security , electron
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 + ε ) n , for any fixed ε > 0 and large n . We also show that if there is a number of collisions larger than n c nfor an appropriate c > 0 , then a large number of these collisions occur within a subfamily of balls that form a very tight configuration.