---
title: "Active Particles Gather Without Attraction Interactive Lab | Sandbox Physics"
description: "How can particles gather when nothing pulls them together? Particles spend longer where they swim slowly."
canonical_url: "https://sandboxphysics.com/labs/active-particle-clustering/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/active-particle-clustering/"
last_modified: "2026-10-28"
---

Active matter · Persistence and crowding

# Active Particles Gather Without Attraction

**How can particles gather when nothing pulls them together?** Particles spend longer where they swim slowly. If crowding slows them enough, a small dense patch traps more swimmers and grows. Their propulsion keeps supplying energy, so this gathering can happen without an attractive force.

Follow self-propelled particles that slow down in crowded neighborhoods. Compare persistent swimming, sudden tumbles, passive diffusion and attraction, then export their simulated trajectories.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

This dimensionless two-dimensional model prescribes density-dependent speed in a periodic box. Particles sense a smooth neighborhood; they have no hard core, alignment or fluid coupling. The attraction comparison uses a separate soft pair potential. Finite clusters and a chosen density threshold describe this run; establishing bulk phase coexistence requires larger systems and longer sampling.

In dense neighborhoods

$64.5\%$

Mean swimming speed

$0.316$

Free persistence length

$28$

Simulation time

$30$

Gold marks particles whose neighbor density exceeds twice the uniform mean. The trace and histogram use these simulated positions.

Save the run and read the model

The JSON contains the seed, version, fixed timestep, parameter changes, sampled positions and statistics. Position frames retain the latest two hundred samples; their times are explicit. Dragging is saved as an external position change.

The mechanism follows the density-dependent motility model in Cates and Tailleur, sections 2–4. A finite crowded patch is a useful observation; bulk phase coexistence requires size and time scaling. [Read the source](https://arxiv.org/html/1406.3533v1)

[Read the active matter course: from persistent motion to defects and flow](https://sandboxphysics.com/stories/active-matter/)

Physics tutorial

## A crowd can grow by slowing down

**Background** A swimmer keeps its direction for a while. When it enters a slow patch, it takes longer to leave. More swimmers arrive in the meantime.

**Why it matters** This offers a route to dense and dilute regions even when particles exert no attraction. The crucial question is how strongly density changes their speed.

Start with the essentials

- **Focus question**: How can particles gather when nothing pulls them together?
- **One-sentence intuition**: A slow patch collects swimmers; sufficiently strong crowding feedback makes that patch slow down further.

### Core mathematical model

#### A persistent swimmer

$$
d\mathbf r_i=v(\bar\rho_i)\mathbf n_i\,dt+\sqrt{2D_t}\,d\mathbf W_i,\qquad d\theta_i=\sqrt{2/\tau}\,dB_i
$$

Position changes through self-propulsion and translational Brownian noise. The angle diffuses; its correlation decays over the direction memory time. The tumble option instead redraws an isotropic direction at random times with that same mean waiting time.

#### The prescribed speed response

$$
v(\bar\rho)=v_0\left[0.01+0.99e^{-\lambda\bar\rho/\rho_0}\right],\qquad\ell_0=v_0\tau
$$

The local density averages nearby particles with a smooth disk kernel and excludes the particle itself. The density scale is the uniform mean of all other particles. The displayed free persistence length uses the free swimming speed; it is not the distance traveled inside a dense patch.

#### The local instability criterion

$$
1+\rho\,\partial_\rho\ln v(\rho)<0
$$

With negligible translational diffusion, slowly varying density and isotropic orientation relaxation, crowding amplifies when speed drops faster than the inverse density. This leading-gradient criterion helps explain the mechanism; a finite simulation needs separate size and time checks.

### Common difficulties

#### A cluster is a finite observation

**Typical misconception** One bright cluster proves a bulk phase transition.

**Better mental model** The dense fraction counts particles above twice the uniform neighbor density. Change system size, particle count, threshold and observation time before inferring bulk coexistence.

#### Speed sensing is a chosen interaction

**Typical misconception** The displayed particles are hard disks colliding in a liquid.

**Better mental model** This dimensionless two-dimensional model prescribes density-dependent speed in a periodic box. Particles sense a smooth neighborhood; they have no hard core, alignment or fluid coupling. The attraction comparison uses a separate soft pair potential. Finite clusters and a chosen density threshold describe this run; establishing bulk phase coexistence requires larger systems and longer sampling.

### Run the experiment

1. 01

   #### Remove the feedback

   Choose Slow down and gather, then move Crowding slows swimming to zero. Predict whether the dense patches will remain.

   **What to observe:** The swimmers keep moving at the same speed in both crowded and sparse areas. Dense patches disperse; the gold fraction falls.

2. 02

   #### Watch a direction change

   Compare Persistent roaming with Runs and sudden turns. Follow a single particle and its trail.

   **What to observe:** Rotational diffusion bends paths continuously; a tumble changes direction abruptly. Both can look diffusive after many direction-memory times.

3. 03

   #### Compare the passive case

   Choose A passive cloud, then Attraction gathers particles. Export a JSON record to inspect the sampled positions and parameters.

   **What to observe:** Passive diffusion spreads the cloud. A soft attractive pair potential can gather passive particles by a different mechanism.

Physical conclusion

Ask whether a dense patch makes particles leave more slowly. Then compare that mechanism across enough sizes and times to know what the observed clusters mean.

## Related experiments and articles

Active matter · Reading path

### Active matter from swimming particles to moving defects

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **A swimming particle remembers its direction**](https://sandboxphysics.com/stories/active-matter/persistent-random-walks/)
2. [Article **Two ways for a swimmer to lose its heading**](https://sandboxphysics.com/stories/active-matter/swimming-and-tumbling/)
3. [Article **A crowd can grow without anyone pulling**](https://sandboxphysics.com/stories/active-matter/crowding-without-attraction/)
4. [Article **Calculate when a density ripple grows**](https://sandboxphysics.com/stories/active-matter/density-instability/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)