A particle is bouncing randomly in a two-dimensional box.

probability/particle.in.box

Problem

A particle is bouncing randomly in a two-dimensional box.  How far does it
travel between bounces, on average?

Suppose the particle is initially at some random position in the box and is
traveling in a straight line in a random direction and rebounds normally
at the edges.

Show solution

Let theta be the angle of the point's initial vector.  After traveling a
distance r, the point has moved r*cos(theta) horizontally and r*sin(theta)
vertically, and thus has struck r*(sin(theta)+cos(theta))+O(1) walls.  Hence
the average distance between walls will be 1/(sin(theta)+cos(theta)).  We now
average this over all angles theta:
        2/pi * intg from theta=0 to pi/2 (1/(sin(theta)+cos(theta))) dtheta
which (in a computation which is left as an exercise) reduces to
        2*sqrt(2)*ln(1+sqrt(2))/pi = 0.793515.