--- title: "A crowd can grow without anyone pulling" description: "Swimmers accumulate where they slow down. Follow the feedback that can turn a density fluctuation into motility-induced clustering." canonical_url: "https://sandboxphysics.com/stories/active-matter/crowding-without-attraction/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/active-matter/crowding-without-attraction/" last_modified: "2026-10-28" --- [Active matter · A course with experiments](https://sandboxphysics.com/stories/active-matter/) L2 · Chapter 3 # A crowd can grow without anyone pulling A crowded patch grows when swimmers enter faster than they leave. If crowding slows their swimming, a chance pile-up makes escape even slower, and the patch can collect more particles. The mechanism needs a strong enough feedback between density and speed. Sandbox Physics · 2026-10-28 In this chapter [All eight chapters](https://sandboxphysics.com/stories/active-matter/) Before you start The first chapter’s persistence length is enough. Read the free-swimmer calculation if you want a quantitative reference. ## Slow places collect traffic Imagine a corridor with a thick carpet halfway along it. People keep trying to walk, but they move more slowly over the carpet. Even with no desire to gather there, the slow half becomes busier: people spend more time in it. A swimmer that slows in a region behaves similarly. In a simple steady state, the number arriving each second must equal the number leaving. A region traversed at half the speed can contain twice the density. This example has a speed pattern imposed from outside; the interesting step comes when the swimmers themselves create it. ## Now let the crowd make itself slow Suppose a swimmer moves more slowly whenever it senses more neighbours. A patch that happens to become denser now also becomes slower. New arrivals linger, the patch becomes denser still, and a small fluctuation may grow. This is the central feedback behind motility-induced phase separation, usually shortened to MIPS. Self-propelled repulsive particles can also slow because they block one another. Two particles pushing head-on remain in contact until they turn enough to escape. A density-dependent speed model describes one coarse version of that slowdown. Its particles need not be literal hard disks; the local density is measured over a chosen neighbourhood. The two descriptions answer related questions at different scales. To predict a particular colloidal material, the collision law, particle shape and hydrodynamics matter. To isolate the feedback, it is useful to prescribe how speed falls with density and test what follows. Worked example ## An ordinary attraction gives a different test Prepare a crowd in three ways: passive wandering, self-propelled motion that slows in dense regions, and passive motion with an attractive force. All three can show clumps in a small box. Now switch off propulsion while keeping the attraction unchanged. The attractive particles still pull on their neighbours. The motility mechanism loses its drive when swimming stops; with positional Brownian diffusion left on, its crowd disperses. This intervention distinguishes causes more clearly than comparing two attractive-looking snapshots. There is a second useful intervention. Keep swimming on but remove the dependence of speed on density. Freely crossing particles now have no reason to linger in a crowd. Continued motion by itself is insufficient for this particular clustering mechanism. ## A cluster is a beginning, not a phase diagram A few hundred particles sometimes form one large patch because the box is small. Changing the seed may change its lifetime; changing the neighbourhood radius may change its apparent density. A bright group of dots is a useful observation, but macroscopic phase coexistence calls for more work. Measure density distributions over time, compare several box sizes at fixed mean density, and repeat with independent seeds. A local-density histogram groups sampled neighbourhoods by how crowded they are. The browser takes one sample per particle; equal-area grid cells give a different weighting. Two populations suggest dense and dilute regions. Whether those populations survive increasing size and longer observation is the next question. The method chapter turns the feedback into a stability calculation. An experiment to think through ## Predict, change, explain 1. Predict which prepared crowd survives when swimming stops and positional diffusion remains. 2. Compare passive, density-slowing and attractive presets, then reduce propulsion in the active case. 3. Explain the change in terms of arrival and escape rates. Use the exported density history to follow the growth and escape process. [Watch swimmers gather](https://sandboxphysics.com/labs/active-particle-clustering/) These Labs generate synthetic particle or continuum records. Their displayed model notes state the implemented dynamics and units. ## Try explaining it yourself ### 1. Does any small decrease of speed with density cause a growing crowd? Read an explained answer No. Spreading can still smooth the fluctuation. In the local, slowly varying limit, speed must fall steeply enough that the collective diffusion coefficient becomes negative. The next chapter calculates that threshold. ### 2. A cluster exists only for one seed in the smallest box. What should you report? Read an explained answer Report a finite-system clustering event, with its seed, lifetime and density statistics. Repeat at other sizes before drawing a phase-coexistence conclusion. ## Sources to read alongside 1. [Cates & Tailleur, Motility-Induced Phase Separation, arXiv:1406.3533v1 (2014); Annual Review of Condensed Matter Physics 6 (2015)](https://arxiv.org/abs/1406.3533v1) Sections 2–4: orientational persistence, accumulation in slow regions and the local density-instability criterion. The course derives its own worked examples from these stated assumptions. 2. [Fily & Marchetti, Athermal Phase Separation of Self-Propelled Particles with No Alignment, arXiv:1201.4847v2, Physical Review Letters 108, 235702 (2012)](https://arxiv.org/abs/1201.4847v2) Primary numerical study of repulsive self-propelled disks without alignment. Its microscopic force law differs from the browser’s density-dependent motility model. 3. [Stenhammar et al., A continuum theory of phase separation kinetics for active Brownian particles, arXiv:1307.4373v2, Physical Review Letters 111, 145702 (2013)](https://arxiv.org/abs/1307.4373v2) Source for how leading density-gradient terms extend a local motility description. The course’s weak-ripple example is a separate local-limit calculation. [Previous Two ways for a swimmer to lose its heading](https://sandboxphysics.com/stories/active-matter/swimming-and-tumbling/) [Next Calculate when a density ripple grows](https://sandboxphysics.com/stories/active-matter/density-instability/)