Every disk is an active Brownian particle: it self-propels at speed v₀ along its own heading n̂ = (cosθ, sinθ), while θ itself performs a slow random walk driven by rotational noise Dr. Disks also push each other apart with a short-range soft repulsion once they overlap — that is the only interaction; there is no alignment rule, unlike a flocking model.
dx/dt = v₀·n̂(θ) + Σ F_repel + √(2Dt)·ξ
dθ/dt = √(2Dr)·ξ
At low v₀ or high Dr, a particle's heading randomizes before it can travel far, so its trajectory looks like ordinary passive Brownian diffusion — a uniform, structureless gas. Raise v₀ and lower Dr at high packing fraction and particles move in long, persistent runs that jam into each other faster than they can steer away: dense, slow clusters nucleate and grow, surrounded by a dilute, fast-moving gas. This is motility-induced phase separation (MIPS) — a phase transition with no attractive force behind it at all, purely from self-propulsion plus excluded volume.
- Packing fraction φ — fraction of the box area covered by disks; MIPS needs enough density for clusters to nucleate.
- Self-propulsion v₀ — how far a particle coasts before its heading decorrelates; the engine of activity.
- Orientational noise Dr — how fast the heading randomizes; high noise erases persistence and pushes the system back toward passive-looking diffusion.
- Largest cluster — the fraction of particles in the single biggest connected clump of neighbours, the standard MIPS order parameter; it jumps from near-zero (gas) to a large fraction once phase separation sets in.
Colour encodes local crowding: cool blue disks are in the dilute gas, hot orange disks sit inside a dense cluster.