We consider a random diffusion dynamics for an infinite system of hard spheres of two different sizes evolving in Rd, 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 Rd. As support material, we propose numerical simulations in the form of movies.