Each disc is a Janus particle viewed edge-on in a thin quasi-2D bath, exactly as real Pt/SiO₂ micromotors are tracked under a microscope (Howse et al., 2007). The Pt cap decomposes fuel and builds a local concentration gradient that self-diffusiophoretically pushes the particle along its cap→body axis:
v₀(C) = v_max · C / (K_M + C) (Michaelis–Menten catalytic turnover)
dx/dt = v₀ cosθ + √(2D_t)·ξ_x (2D Langevin equations of motion)
dy/dt = v₀ sinθ + √(2D_t)·ξ_y
dθ/dt = √(2D_r)·ξ_θ (heading itself diffuses)
In a genuine 2D plane the heading angle θ has only one rotational degree of freedom, so its autocorrelation decays as e^(−D_r t) — not e^(−2D_r t) as for a freely-tumbling 3D orientation vector. That changes both the persistence time and the long-time effective diffusion coefficient relative to the 3D case:
D_t = k_BT / (6πηr) translational diffusion (Stokes–Einstein)
D_r = k_BT / (8πηr³) rotational diffusion of the sphere
τ_r = 1 / D_r 2D orientation persistence time
D_eff = D_t + v₀²τ_r / 2 2D long-time effective diffusion (Howse 2007)
This D_eff/2 relation (as opposed to the 3D sim's D_eff = D_t + v₀²τ_r/6 with a 3D-convention τ_r) was verified here by direct numerical integration of the Langevin equations above at long time: an ensemble of independently simulated swimmers, run out to ~60 persistence times, reproduces the D_t + v₀²τ_r/2 prediction to within ≈2% — confirming the formula rather than assuming it. The live "measured" readout on this page runs the same check continuously, on a handful of tracer particles inside the swarm, via a sliding-window mean-squared-displacement estimator (⟨Δr²⟩ / 4t).
- Fuel concentration — sets catalytic turnover via Michaelis–Menten saturation; more fuel → faster propulsion, up to a plateau.
- Particle radius — bigger particles rotate more slowly (D_r ∝ 1/r³), so they hold a heading longer and travel further ballistically before randomizing.
- Bath temperature — raises k_BT and lowers water's viscosity η (Vogel equation), boosting both translational and rotational diffusion.
- Catalytic cap toggle — sets v₀ to zero, turning every swimmer into a plain passive Brownian disc for comparison — the measured D_eff should then collapse onto D_t alone.
Simulated time is accelerated ~40× relative to a real H₂O₂ bath, matching the 3D companion simulation of this same nanomotor, so the crossover from ballistic to diffusive spreading is visible within seconds.