Unseen · flux ↓ down the gradient

Fickian — the slow spread of everything, dye diffusing into still water

Drop dye into still water and it spreads — not pushed, not stirred, only wandering. Each molecule takes a blind random walk down the concentration gradient, and the sharp edge you released softens, inevitably, into a smooth bell.

The tank is drawn live in your browser. Enable JavaScript to release dye and watch it diffuse into Gaussian contours.

Click the still water · release a drop · watch two plumes overlap and sum

0.00 Peak concentration

0px Spread σ = √2Dt

1 Drops released

— c under cursor

Cross-section through the peak

Nothing pushes it. It wanders there.

There is no current in the tank, no hand to guide the colour outward. Each dye molecule is jostled at random by the water around it, stepping this way and that with no memory and no destination. Yet from that blindness a direction emerges: where the dye is crowded, more of it steps out than steps back in.

That statistical drift is diffusion. Adolf Fick wrote it down in 1855 as a debt paid to the gradient — matter flows from where there is much to where there is little, at a rate set only by how steeply the crowd thins. The steep front you release cannot last: the peak sinks, the skirt fills, and the profile relaxes toward the one shape diffusion allows.

The contours you see are lines of equal concentration — isopleths pulled straight from the field by marching squares.

Watch them bloom outward and space themselves evenly: the mark of a spreading Gaussian, whose width grows as the square root of time.

Release a second drop and the fields simply add. Diffusion is linear — two plumes never collide, they sum.

What the plume already knows

01 · first law Flux follows the slope J = −D ∇c Matter drifts from crowded to empty, at a rate set only by how steep the crowd is — and always downhill.

02 · second law Curvature flattens ∂c/∂t = D ∇²c Where the profile is curved, it relaxes. Sharp peaks sink, hollows fill, and every edge is rounded off in time.

03 · the solution A bell that only thins σ = √2Dt A point release becomes a Gaussian that never stops widening — it does not travel, it only ever spreads and fades.