Fast Brownian cluster dynamics

We present an efficient method to perform overdamped Brownian dynamics simulations in external force fields and for particle interactions that include a hardcore part. The method applies to particle motion in one dimension, where it is possible to update particle positions by repositioning particle clusters as a whole. These clusters consist of several particles in contact. They form because particle collisions are treated as completely inelastic rather than elastic ones. Updating of cluster positions in time steps is carried out by cluster fragmentation and merging procedures. The presented method is particularly powerful at high collision rates in densely crowded systems, where collective movements of particle assemblies is governing the dynamics. As an application, we simulate the single-file diffusion of sticky hard spheres in a periodic potential.

💡 Research Summary

The paper introduces a novel O(N) algorithm for simulating overdamped Brownian dynamics of one‑dimensional systems that contain hard‑core (excluded‑volume) interactions. Traditional hard‑sphere Brownian dynamics either treat collisions elastically or avoid overlaps via rejection, both of which become computationally prohibitive when the collision rate is high, scaling as O(N²) or worse. The authors instead treat particle collisions as completely inelastic, instantly binding contacting particles into “clusters”. Within each cluster, the total force acting on each particle (external force, pairwise interaction, and stochastic noise) is computed, and the mean forces of prospective sub‑clusters are compared. Two sets of conditions—non‑splitting (Eq. 4a) and splitting (Eq. 4b)—determine whether a contact remains stable or becomes unstable, thereby dictating how the cluster fragments.

Fragmentation is performed efficiently through an iterative “pair‑splitting” procedure. Although an n‑particle cluster has 2ⁿ⁻¹ possible fragmentations, the algorithm only examines O(n²) pair‑splittings by repeatedly selecting the division point where the difference between the mean forces of the left and right sub‑clusters is maximal and positive. This greedy selection, proved by induction, yields a sequence of pair‑splittings that respects the required ordering of sub‑cluster velocities (v₁ < v₂ < …), which in turn guarantees that all splitting conditions are satisfied without exhaustive combinatorial checks.

After fragmentation, each sub‑cluster moves with a velocity proportional to its mean total force (v = F̄/μ). When two moving clusters come into contact, they merge in a completely inelastic fashion: the new cluster’s velocity is the mass‑weighted average of the colliding clusters (Eq. 6). The authors adopt a fixed time step Δt and treat forces as constant over each interval, allowing a straightforward Euler‑Maruyama integration. A “pre‑merging” step determines which clusters will collide during the current Δt without sorting events chronologically, preserving the O(N) scaling even in densely packed regimes where collisions are frequent.

To validate the method, the authors simulate single‑file diffusion of sticky hard spheres (Baxter’s model) in a periodic external potential. The system is a classic testbed: particles cannot pass each other, and sticky interactions promote cluster formation. The BCD results reproduce known analytical and Monte‑Carlo benchmarks for mean‑square displacement, velocity autocorrelation, and cluster‑size distributions. Moreover, at high densities and strong stickiness, the simulations reveal collective phenomena such as solitary cluster waves—coherent propagating clusters—that are difficult to capture with traditional algorithms due to prohibitive computational cost.

In summary, the paper delivers a mathematically rigorous, computationally efficient framework for one‑dimensional Brownian dynamics with hard‑core and additional interactions. By converting frequent elastic collisions into cluster‑based inelastic events and using a pair‑splitting fragmentation scheme, the method reduces the computational complexity from quadratic to linear in particle number. This enables the study of highly crowded, strongly interacting systems such as colloidal suspensions, nanoparticle transport, and diffusion in biological channels. The approach is readily extensible to more complex force fields and, with appropriate modifications, could inspire analogous cluster‑based algorithms in higher dimensions.