Diffusion dynamics for an infinite system of two-type spheres and the associated depletion effect

Abstract

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.

Figures (6)

• Figure 1: Identical hard spheres in a bath of identical small particles. An ideal mathematical representation (left) and a culinary realisation (right, jelly doughnuts in particles of frying oil). Left: the orange depletion shells around the brown spheres overlap.
• Figure 2: Examples of hard discs with depletion shell; the shaded areas represent the overlap between the depletion shells. (a) For $\rho$ smaller but close to the critical case $\rho_2$, where only pair interactions occur; (b) For $\rho_2<\rho\leq\rho_3$, there cannot be four-body interactions; (c) For $\rho >\rho_3$, four-body interactions occur.
• Figure 3: Intersection of two depletion discs.
• Figure 4: Behaviour of $u \mapsto \mathcal{V}_{\rm{ovlap}}(u)$ (left) and $u \mapsto \mathcal{V}_{\rm{ovlap}}'(u)$ (right) for discs in the plane.
• Figure 5: Behaviour of $u \mapsto \mathcal{V}_{\rm{ovlap}}(u)$ (left) and $u \mapsto \mathcal{V}_{\rm{ovlap}}'(u)$ (right) for balls in ${\mathbb R}^3$.

Theorems & Definitions (29)

• Theorem 2.1
• Proposition 2.2
• Remark 2.3
• proof : Proof of Proposition \ref{['prop:existencesolrevSnmR']}
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Lemma 2.6
• Lemma 2.7