Collision location for hard spheres in stationary regime

Consider two balls with radius r ∈ ( 0 , 1 ) r∈ (0,1) whose centers are at a distance 2, positioned symmetrically with respect to the origin in R d { R}^d . Suppose that the initial velocities are independent standard normal vectors. We prove that when r → 0 r→ 0 , the collision probability goes to 0 as r d − 1 rᵈ⁻¹ , and the asymptotic collision location distribution is a (defective) t t -distribution. This distribution is rotationally symmetric about the origin for no apparent reason.