In words: one blind step at a time

A random walker's entire rulebook is one sentence: every tick, take one blind step — a coin flip picks the direction. Each grain is ignorant of calculus. It has no idea a crowd exists, let alone where the crowd is. And yet release thousands of them and the shape of the crowd — how many grains sit in each patch of space — spreads out exactly the way Course 2 says a temperature curve spreads. The smooth fields of Course 2 reappear here as crowd behavior.

Which raises the obvious question: where does the equation actually live? Not inside a walker — you can read a grain's rule all day and never find a derivative in it. The equation only shows up once you stop following individuals and start counting how many are where. It's that count, not the grains, that obeys the heat equation.

And that's the one real difference between this page and Courses 2–4. Those courses chopped space into fixed boxes and, for each box, tracked the amount inside it: the box never moves, its contents rise and fall. Every bead, every cell of the heat sheet, every cell of the reaction–diffusion dish worked that way. Here we do the opposite — forget the boxes, pick a grain, and record where it goes. The thing you're following moves along with the stuff.

Those two habits have names worth knowing, because you'll meet them in any simulation work: the fixed-box view is Eulerian, the follow-a-grain view is Lagrangian. One asks how full is this place?, the other asks where did this piece go? Same physics, two sets of books — and now the bookkeeping metaphor should land: they're two ways of accounting for identical events.

They don't agree perfectly, and the disagreement is the interesting part. The equation describes the average crowd — what you'd get from infinitely many walkers. Any actual crowd is one sample of that average, so its count in each patch jiggles a little above and below the equation's answer. How much? Roughly one over the square root of the number of grains (written 1/√n): with about 100 grains in a patch the count wobbles by around 10%, with 10,000 by around 1%. Quadruple the crowd and you halve the wobble — but it never reaches zero.

So to be precise about what the smooth field leaves out: it isn't ignoring the wobble, it has no way to express it. A field stores one number per place — an average — and an average has no room in it for "and it jitters by this much." That's not a flaw; it's the trade you make for a law you can solve.

And the wobble is real physics, not a rendering artifact. When a patch of space holds only a handful of particles, 1/√n is large and the graininess is genuinely there and genuinely measurable — that's precisely what Robert Brown saw jittering in his pollen, and what Perrin later measured to weigh an atom. At everyday concentrations the count per patch is astronomically large, the wobble becomes invisible, and the smooth field is indistinguishable from the truth. In short: diffusion really is grainy — you just have to be looking at few enough particles to see it.

In math: count, don't integrate

Four short equations take us from a coin flip to the heat equation. Every symbol in them is defined — hover any colored term to read what it is, or use the full list at the end of this section.

Start by counting. To be standing at position x a moment from now, a walker must be exactly one stride away right now — and half of those neighbors happen to flip our way. So:

Now subtract the crowd that's already at x from both sides, so we're looking at the change rather than the total:

That bracket is worth slowing down on, because it's the whole bridge to calculus. Read it as a curvature meter. Suppose your two neighbors average out to exactly your own height: then you sit on a straight line, the bracket is zero, and nothing changes — even if that line is steep. If they average higher than you, you're sitting in a dip, and the bracket is positive, so the crowd here grows. Lower, and you're on a bump, so it drains. Course 2, Step 6 named exactly this quantity: comparing a value to the average of its neighbors is what a second derivative measures. That's the connection — not an analogy, the same arithmetic.

So now make the walk finer: smaller strides, faster ticks. Two things happen at once. The bracket shrinks — halve the stride and it gets about four times smaller, because it scales with h² — and the ticks come faster, so more of them land per second. Those two effects fight, and if you shrink stride and tick together at the right rate they balance exactly, leaving a finite answer. Divide the change by the tick, divide the bracket by the stride squared, and every trace of h and Δt collapses into a single constant:

About that D, since it appears out of nowhere: it's the diffusion coefficient — one number saying how vigorously this particular stuff spreads. And notice it isn't a new ingredient smuggled in. It's built entirely from the walk we already had: the stride squared, divided by twice the tick. A jumpier grain means a bigger D means a faster-spreading crowd. This is the same role played by α in Course 2's heat equation and by Dᵤ and Dᵥ in Course 4's dish — and it's the D on the lab's jitter slider below.

One last equation, and it's the one the simulator actually runs. Everything above described the crowd; this describes a single grain's move in a single tick:

Both terms are knobs in the lab. The push, v, is advection — a current the whole crowd rides, the same knob the reaction–diffusion dish just grew on its tank walls. The kick's size is set by D: that's diffusion.

And here's the fun part. If D changes from place to place, a grain just uses whatever value holds where it's standing — no extra rule needed. That has a lovely side effect: grains dawdle wherever the kicks are weak, so the crowd quietly piles up on slow ground. One species, zero interactions between grains, and already a pattern-maker. (Serious transport codes add a correction to cancel that pile-up when they don't want it. Our lab keeps the naive version, because here the pile-up is the whole show — go paint some.)

Every symbol, once

In code: one line of physics per axis

This is the entire simulator below — no grid for the physics, no Laplacian, no stencil. Compare it with the bead loop of Course 2: dT[i] = α·((T[i−1]+T[i+1])/2 − T[i]). Same equation; the beads update amounts at fixed places, the walkers update places of fixed grains.

The lab

24 000 grains, live. Two regimes. Sealed dish: walls reflect, nothing is born or dies — a drop of ink spreading; add drift and the floor turns it into an atmosphere, dense below, thin above. Flow-through: grains rain in at a slot up top, ride the drift down, and die at rate k — a steady plume, the textbook picture of a contaminant leaking into groundwater. In both, drag on the canvas to paint slow ground and watch the crowd gather where it dawdles.