Imagine we have n point particles in $\mathbb R^3$ of mass 1, with gravitational constant $G > 0$, moving according to Newtonian gravitation. Suppose we assign, to each initial position coordinate and each initial velocity component of each particle, a (possibly separate for each one) compactly supported continuous probability density. Then we imbue those particles with initial positions and initial velocities drawn from those distributions. What is the probability that there's eventually a collision?
I assume the answer is zero; it seems "hard" to get a collision with infinitely small particles. But I wasn't sure how to prove it.
It's very easy to show that the probability of a collision at initial time t = 0 (i.e., choosing two particles to initially start at the same place) is zero. Furthermore, it seems intuitive that running the n-body problem for a small time (after randomly selecting initial conditions) should be the same as initializing all the particles from different probability distributions which still satisfy the above assumptions. I'm not sure if this can be turned into a real argument, though.