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Infinite-dimensional diffusions and depletion interaction for a model of colloids

Myriam Fradon, Alexander Zass

TL;DR

The paper addresses the dynamic and statistical mechanics of two-type colloidal suspensions modeled by the Asakura--Oosawa framework in infinite volume. It develops a rigorous construction of an infinite-dimensional diffusion with collision local times and reflection, proving a unique strong solution and reversibility under Gibbs measures $\mathcal{G}_{\mathring{z},\dot{z}}$, and it uncovers an emergent depletion attraction via projection to the hard-sphere subsystem, linking to a gradient dynamics with effective energy $\mathcal{E}$. The work shows a clear separation of regimes: in the low-activity regime there is no percolation, while in the high-activity regime the system concentrates near the closest packing, highlighting the interplay between depletion forces, percolation, and packing in colloidal suspensions. The results connect stochastic dynamics, equilibrium Gibbs measures, and geometric packing phenomena, providing a rigorous path from microscopic diffusion to macroscopic phase-like behavior in multi-type hard-core systems. All mathematical expressions are stated with explicit $\,$delimiters, reinforcing the precise, utilitarian nature of the contributions.

Abstract

We consider infinite-dimensional random diffusion dynamics for the Asakura--Oosawa model of interacting hard spheres of two different sizes. We construct a solution to the corresponding SDE with collision local times, analyse its reversible measures, and observe the emergence of an attractive short-range depletion interaction between the large spheres. We study the Gibbs measures associated to this new interaction, exploring connections to percolation and optimal packing.

Infinite-dimensional diffusions and depletion interaction for a model of colloids

TL;DR

The paper addresses the dynamic and statistical mechanics of two-type colloidal suspensions modeled by the Asakura--Oosawa framework in infinite volume. It develops a rigorous construction of an infinite-dimensional diffusion with collision local times and reflection, proving a unique strong solution and reversibility under Gibbs measures , and it uncovers an emergent depletion attraction via projection to the hard-sphere subsystem, linking to a gradient dynamics with effective energy . The work shows a clear separation of regimes: in the low-activity regime there is no percolation, while in the high-activity regime the system concentrates near the closest packing, highlighting the interplay between depletion forces, percolation, and packing in colloidal suspensions. The results connect stochastic dynamics, equilibrium Gibbs measures, and geometric packing phenomena, providing a rigorous path from microscopic diffusion to macroscopic phase-like behavior in multi-type hard-core systems. All mathematical expressions are stated with explicit delimiters, reinforcing the precise, utilitarian nature of the contributions.

Abstract

We consider infinite-dimensional random diffusion dynamics for the Asakura--Oosawa model of interacting hard spheres of two different sizes. We construct a solution to the corresponding SDE with collision local times, analyse its reversible measures, and observe the emergence of an attractive short-range depletion interaction between the large spheres. We study the Gibbs measures associated to this new interaction, exploring connections to percolation and optimal packing.

Paper Structure

This paper contains 23 sections, 14 theorems, 126 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Allowed configurations of spheres and particles
  3. Dynamics for the AO model
  4. Main results
  5. Strategy of proof
  6. Construction of the approximating process
  7. Penalisation functions
  8. Penalised SDE
  9. Mixture of penalised processes
  10. Convergence of the penalised processes
  11. Proof of Proposition \ref{['prop:Convergence_y']}: convergence on a subset
  12. Proof of Proposition \ref{['prop:FullMeasure']}: almost sure convergence
  13. Reversibility of the limit under Gibbs measures
  14. Occurrence of a depletion interaction between hard spheres
  15. The projection of the two-type reversible measure
  16. ...and 8 more sections

Key Result

Theorem 1.3

For any values of the activities ${\mathring{z}},{\dot{z}}>0$, for any Gibbs measure $\mu\in\mathcal{G}_{{\mathring{z}},{\dot{z}}}$, the two-type infinite-dimensional SDE eq:SDE2infty admits a unique $\mathcal{D}$-valued strong solution, for $\mu$-almost every deterministic initial condition. Moreov

Figures (1)

  • Figure 1: Highlighted, an example of an ${\varepsilon}$-chain of length $\kappa = 9$.

Theorems & Definitions (33)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Proposition 2.1
  • Remark 2.2
  • ...and 23 more