How far does a molecule travel between collisions? Clausius (1858) imagined ONE molecule threading a field of frozen targets and got λ=1/(n·σ). But the targets move too — by how much does that shorten the free path?
▶ Launch the interactive simulationRun a from-scratch event-driven hard-disk gas (equal masses, elastic line-of-centres collisions, energy conserved to machine precision), started OFF equilibrium — every disk at the same speed, random direction — and let it thermalize to Maxwell–Boltzmann. Recover Maxwell's √2 TWO independent ways without ever plugging it in: kinematically as ⟨v_rel⟩/⟨v⟩ (mean relative ÷ mean speed, sampled over pairs) and dynamically as λ_naive/λ_measured (measured λ = total path travelled ÷ 2·collisions), across three densities extrapolated to the dilute limit. Decisive control: FREEZE the targets (a Poisson Lorentz gas) and re-measure. ?world=meanfreepath.
Maxwell's √2 correction (1860). The true mean free path is λ = 1/(√2·n·σ) — a factor √2 SHORTER than Clausius's stationary-target estimate — because the collision rate is set by the mean RELATIVE speed, and for a thermalized gas ⟨v_rel⟩=√2·⟨v⟩ EXACTLY (the difference of two Maxwell–Boltzmann velocities is Maxwellian with half the mass). The gas STARTS at the non-Maxwellian fixed-speed ratio 4/π=1.273 and the collisions drive it UP to √2: recovered as ⟨v_rel⟩/⟨v⟩=1.42 (0.6%) and as a density-extrapolated λ_naive/λ_measured intercept of 1.409 (0.35%). The FROZEN control collapses to ≈1 (Clausius's estimate, no √2), decisively below the moving gas's 1.43 — so the √2 exists ONLY because the targets are themselves in motion. Mean free path is temperature-INDEPENDENT (heating speeds up collisions but never lengthens the gaps) and scales as 1/(n·σ). The transport companion to ?world=maxwell (the speed distribution) and the kinetic-theory root of viscosity and diffusion.