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.

A crowd that slows itself down

Let them gather

Free swimming speed

v_0

7

Direction memory time

\tau

4

Crowding slows swimming

\lambda

2.6

Motion model

Drag a particle out of a dense patch and watch it escape. Arrow keys move the selected particle.

Sampling and system size

Translational diffusion

D_t

0.008

Particle count

N

256

Periodic box side

L

18

Random seed

s

1729

Changing size, count, seed or motion starts a fresh uniform preparation and advances it by thirty model time units. Other sliders change the running model immediately.

What these dots model

Dimensionless dry particles sense their neighbors within a smooth disk. Their drawing radius has no physical size. There is no solvent flow, hard-core contact or orientational alignment.

v(\bar\rho)=v_0[0.01+0.99e^{-\lambda\bar\rho/\rho_0}]

The attraction preset uses a separate soft pair potential. The density-feedback runs contain no pair force.

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

Read the active matter course: from persistent motion to defects and flow

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 misconceptionOne bright cluster proves a bulk phase transition.

Better mental modelThe 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 misconceptionThe displayed particles are hard disks colliding in a liquid.

Better mental modelThis 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

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.
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.
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.

Active Particles Gather Without Attraction

A crowd can grow by slowing down

Core mathematical model

A persistent swimmer

The prescribed speed response

The local instability criterion

Common difficulties

A cluster is a finite observation

Speed sensing is a chosen interaction

Run the experiment

Remove the feedback

Watch a direction change

Compare the passive case

Related experiments and articles

Active matter from swimming particles to moving defects