  We consider a random diffusion dynamics for an infinite system of hard
spheres of two different sizes evolving in $\mathbb{R}^d$, its reversible
probability measure, and its projection on the subset of the large spheres. The
main feature is the occurrence of an attractive short-range dynamical
interaction -- known in the physics literature as a depletion interaction --
between the large spheres, which is induced by the hidden presence of the small
ones. By considering the asymptotic limit for such a system when the density of
the particles is high, we also obtain a constructive dynamical approach to the
famous discrete geometry problem of maximisation of the contact number of $n$
identical spheres in $\mathbb{R}^d$. As support material, we propose numerical
simulations in the form of movies.
