Maximum number of collisions of elastic particles

Abstract

Consider n point-particles, all of the same mass, moving freely in space and interacting only by elastic collisions. There are two degrees of freedom in the outcome of any collision. Let C(n) be the maximum number of collisions possible under Newtonian laws and C*(n) the maximum number possible in relativistic mechanics. This paper is directed towards finding those two functions but is far from achieving that. The following results are established: C(n) ≥1/2n ( n — 1), (C*(n) ≥ 1/2n( n — 1); C(3) = C*(3) = 3; C(4) = 6 (but only for a certain collision-topology) and C*(4)≥ 7. The problem of finding C(n) and C*(n) is of interest in kinetic theory and elsewhere.