  Consider two unit balls in a $d$-dimensional flat torus with edge length $r$,
for $d\geq 2$. The balls do not move by themselves but they are pushed by a
Brownian motion. The balls never intersect---they reflect if they touch. It is
proved that the joint distribution of the processes representing the centers of
the balls converges to the distribution of two independent Brownian motions
when $r\to \infty$, assuming that we use a proper clock and proper scaling. The
diffusion coefficient of the limit process depends on the dimension. The
positions of the balls are asymptotically independent also in the following
sense. The rescaled stationary distributions of the centers of the balls
converge to the product of the stationary (hence uniform) distributions for
each ball separately, as $r\to\infty$.
