# Classical unpredictability

Consider, a box with $$N$$ particles each of mass $$m$$, arranged at time $$t=0$$ to be moving parallel to the $$x$$-direction, with mean velocity v and mean distance l. The particles are spheres of mean radius r. Assume, the collisions are elastic and head-on, collisions occur when two particles move along the same direction, the collision with walls are also elastic. It seems, that the system would preserve its initial condition for a long time under this setting, but it is not the case.

How do I show that a small force, perhaps due to gravitational force of some distant particle-$$f$$ can destroy the predictability of the system? and how do I find the approximate number of collisions of particles that would take for the system to lose its predictability?

• You say "mean velocity," "mean radius," and "mean distance". Are these quantities drawn from some random distribution? If so what distribution? Specifying this might make an it possible to find an interesting answer to your question. (Also, distance from what?) Aug 1, 2020 at 5:41