Chiral active matter — self-propelled particles with torque
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Active Brownian particles (ABPs) self-propel at speed v₀ in their current orientation θ. Circle swimmers add a constant angular velocity ω (torque), causing circular trajectories of radius R = v₀/ω. Rotational diffusion Dr randomizes orientation over time τ_R = 1/Dr. Collective behavior: at low Dr, particles trace persistent circles; clustering occurs from motility-induced phase separation. Real examples: bacteria near surfaces, sperm cells, asymmetric self-phoretic colloids. The competition between ω and Dr determines the persistence length ℓ = v₀/Dr.