A handful of rice thrown into a glass of water scatters — the grains spread apart. Now imagine each grain propels itself forward with stubborn persistence and bounces off softly upon collision. Such particles are called active: they spend their own energy to move, like bacteria or microscopic robots. Surprisingly, they sometimes gather into dense clouds — even though there are no attractive forces between them. This phenomenon is known as motility-induced phase separation.
For a long time, it was thought that active particles form clusters because their repulsion effectively turns into attraction — somewhat like molecules in a water droplet. But precise calculations were lacking. The authors took the simplest system: two soft particles that run in straight lines, occasionally changing direction abruptly (a motion called 'run-and-tumble'), and confined them in a periodic box — where exiting on the left leads to entering on the right. Then they applied a rigorous microscopic approach, tracking each particle without averaging.
This is like two dancers in a cramped room: each stubbornly follows their own path, and the harder they push each other upon meeting, the longer they linger nearby, unable to part. Their duet creates the illusion of attraction. This perspective helps explain how bacteria form biofilms or how swarms of microrobots can self-organize without complex programs.
as written by the authors arXiv:2605.01421
Repulsive self-propelled particles tend to cluster, leading to Motility-Induced Phase Separation (MIPS). By analogy with equilibrium phase separation, the onset of MIPS has been associated with a transition to effective attraction between particles. Using an exact microscopic theory, we quantify the emergence of effective attraction in a minimal model: two soft run-and-tumble particles in a periodic domain. We show that, as repulsion increases, the leading-order behaviour is that of effective repulsion, while effective attraction emerges as a higher-order contribution to the renormalisation of the pair potential.
Escherichia coli bacteria move at about 20 micrometers per second — roughly 30 body lengths per second. If a human moved that fast relative to their height, they would sprint 100 meters in a couple of seconds.
In Stanislaw Lem's novel 'The Invincible', the crew of a starship encounters a swarm of microscopic machines that act as a single organism without central control, a pure embodiment of active matter.