Matter drifts down its own gradient. Where a substance is crowded it thins; where it is sparse it fills — and the flux is exactly proportional to the negative slope of concentration, J = −D ∂c/∂x. Feed that back into a balance and the profile spreads as a Gaussian whose width grows not with time but with the square root of time. Down the center, data flows: gradient goes in, the engine computes flux and spread, the proven law comes out. The blue team builds and defends it; the red team tries to break it.
source A. Fick, Ueber Diffusion, Ann. Phys. 170 (1855) 59–86 — doi:10.1002/andp.18551700105. Rendered, not quoted.
Nothing here is looked up; it all falls out of one balance:
First law — flux is proportional to the negative gradient: J = −D ∂c/∂x. Matter flows from high to low. A flat profile (∂c/∂x = 0) carries zero flux.
Second law — conservation (∂c/∂t = −∂J/∂x) turns the first into the diffusion equation ∂c/∂t = D ∂²c/∂x², solved by the spreading Gaussian c = (4πDt)−½ e−x²/4Dt.
For the current settings, the live readout of the point flux and the spread:
| quantity | value | law |
|---|
Fick lifted his form straight from Fourier: swap temperature for concentration and heat-flux for mass-flux and ∂c/∂t = D ∂²c/∂x² is the-heat-equation. Same operator, same Gaussian kernel, same √(Dt) spread.
Downstream it becomes the-diffusion of carbon into iron — the atoms that harden steel walk exactly this random-walk profile. Each sphere is the next one's premise.
The blue team's live check: recompute the sign of the flux, the PDE residual, the √t spread and the conserved mass — and confirm them against the known law. If red drops the minus sign, this badge is where it shows.
Diffusion needs exactly one thing to push it: a concentration gradient ∂c/∂x — how steeply the substance piles up across space. Everything else the engine derives.
| you feed | symbol | meaning |
|---|---|---|
| diffusivity | D | how fast it wanders (m²/s) |
| time | t | how long it has spread |
| position | x | where you probe the flux |
Give it D, t and a probe point x on the spreading profile — that is the whole input. The engine returns the flux there and the width of the cloud.
The curve is the concentration c(x,t); the marker is your probe. Flux points down the slope — outward from the crowded center.
Move any slider — the flux and the spread are computed from J = −D ∂c/∂x and the Gaussian solution on the spot, never looked up.
What the machine produces, proven: flux runs opposite the gradient (high→low), a flat profile carries zero flux, the Gaussian satisfies ∂c/∂t = D ∂²c/∂x² to a finite-difference tolerance, the RMS spread grows as √(2Dt) — not linearly — and the total mass stays 1 as it spreads. The current probe's flux is above; these invariants are the output.
The blue team's witness (left) confirms these live; the red team (right) tries to make matter flow uphill.
And the "spread as √t" is the Gaussian answer; anomalous (sub/super-)diffusion in disordered or crowded media scales as tα, α≠½. The law is exact as written — but its domain is narrow.
"Diffusion means things move at a steady speed." Cut. The front advances as √(Dt), so speed ∝ 1/√t — it slows forever; nothing travels at constant velocity.
"Fick discovered the diffusion equation." Cut. He transcribed Fourier's 1822 heat equation into mass transport (1855) and backed it with salt-diffusion experiments. Rendered as lineage, not invention.
"Matter flows from low to high to even out." Kept, corrected. It evens out — but by flowing high→low. The minus sign in J = −D ∂c/∂x is the whole physics. Drop it and you get window 6.
The red team's move: delete the minus sign — set J = +D ∂c/∂x so matter flows up the gradient, piling higher where it is already crowded. Physically impossible; the witness (window 7) is watching.
Flip J to +D ∂c/∂x and the flux now points the same way as the gradient — uphill, unphysical. The witness recomputes the high-to-low sign check, disagrees with the law, and turns red. Nothing is faked; the attack is real and it is caught.