Bacteria like E. coli swim in straight runs then randomly reorient (tumble). High density leads to motility-induced phase separation (MIPS) — clustering without attractive interactions.
Run-and-tumble particles (RTPs) slow down in dense regions (excluded volume). This positive feedback — slow regions accumulate more particles — drives MIPS: a phase transition into coexisting dense and dilute phases with no attractive forces. The Peclet number Pe = v₀/(Dᵣ·σ) controls the transition; above a critical Pe, macroscopic clusters emerge.