Brownian dynamics simulations with hard-body interactions: Spherical particles

A novel approach to account for hard-body interactions in (overdamped) Brownian dynamics simulations is proposed for systems with non-vanishing force fields. The scheme exploits the analytically known transition probability for a Brownian particle on a one-dimensional half-line. The motion of a Brownian particle is decomposed into a component that is affected by hard-body interactions and into components that are unaffected. The hard-body interactions are incorporated by replacing the affected component of motion by the evolution on a half-line. It is discussed under which circumstances this approach is justified. In particular, the algorithm is developed and formulated for systems with space-fixed obstacles and for systems comprising spherical particles. The validity and justification of the algorithm is investigated numerically by looking at exemplary model systems of soft matter, namely at colloids in flow fields and at protein interactions. Furthermore, a thorough discussion of properties of other heuristic algorithms is carried out.

💡 Research Summary

The paper introduces a rigorously grounded algorithm for handling hard‑body (excluded‑volume) interactions in overdamped Brownian dynamics simulations, especially when non‑zero external force fields are present. The core idea exploits the exact transition probability for a one‑dimensional Brownian particle confined to a half‑line (x ≥ 0). By decomposing each particle’s displacement into a component that may be affected by a hard‑body encounter and a component that is not, the method replaces the affected component with a stochastic step drawn from the half‑line propagator, while the unaffected component is updated with the standard Euler‑Maruyama scheme. This construction guarantees that the reflecting boundary condition at a hard surface is satisfied at the level of the probability distribution, eliminating unphysical overlaps without ad‑hoc corrections.

The authors first formulate the approach for a particle moving near a fixed obstacle and then extend it to pairwise interactions between spherical particles. They discuss the necessary assumptions: the integration time step Δt must be sufficiently small that the probability of crossing the hard surface more than once within a single step is negligible, and the force field should be smooth on the scale of Δt. Under these conditions the algorithm is both accurate and computationally efficient.

Numerical validation is performed on two representative soft‑matter systems. In the first case, a colloidal sphere is driven through a steady flow field past a stationary wall; the new scheme reproduces the exact analytical distribution of reflected positions and the mean first‑passage time, outperforming conventional “bounce‑back” and “skip‑overlap” heuristics. In the second case, two protein‑sized spheres interact via a short‑range attractive potential while being subjected to external forces; again the half‑line based method matches results from finely resolved Monte‑Carlo simulations, whereas heuristic methods exhibit systematic bias and energy non‑conservation.

A thorough comparison with existing heuristic algorithms highlights several shortcomings of the latter: dependence on the chosen time step, violation of detailed balance, and occasional generation of spurious forces during collision handling. The proposed half‑line approach, by contrast, preserves detailed balance by construction and remains stable across a wide range of Δt values. The authors also discuss limitations, such as the increased complexity when multiple simultaneous collisions occur in dense suspensions and the need for extensions to non‑spherical particles or deformable boundaries.

In summary, the paper provides a mathematically exact yet practically simple framework for incorporating hard‑body interactions into Brownian dynamics. By leveraging the analytically known half‑line propagator, it achieves physically correct reflection at obstacles while keeping computational overhead low. This makes it a valuable tool for simulations of colloids, polymers, proteins, and other soft‑matter systems where excluded‑volume effects and external forces coexist.