Nondeterministic Billiards

We consider a new type of billiard trajectories of point-particles moving freely in $$d$$ -dimensional space until collision. At collisions of two or more particles we have scattering at a subspace with a co-dimension of at least $$d$$ preserving only the total momentum of the colliding particles, but the internal direction and kinetic energy can change arbitrary and even an exchange of mass is possible. Hence the future of the trajectory is nondeterministic. Motivated by questions concerning non-collision singularities in the $$n$$ -body problem, for which these systems might serve as approximations, we are mainly interested in the asymptotic growth rate for trajectories that have infinitely many collisions and are expanding. For this case we provide as our main results exponential lower bounds for the diameter and the kinetic energy of the system in the number of so-called chain-closing collisions.