None of the pieces so far had any time in them. Count the states, pick the arrangement that happens most often, and temperature and pressure both came out. From here we look at the motion that gets a system there.
We start from the simplest motion there is. At each step, go left or right, each as likely as the other. That is the whole rule. The broken line in the figure is one such walk, steps across and position up the side.
Watching one walk, nothing is settled. Run it again under the same rule and it ends somewhere else. And yet something is settled once there are many of them. That argument is exactly the one that let us talk about a gas statistically without following each particle.
The figure runs many walks under the same rule. All of them start at the origin and scatter apart.
The scattering has a shape. It is densest in the middle and thinner further out, and the whole of it widens as the number of steps grows.
Two questions follow. How fast does the width grow? And what shape does the scattering settle into? Both come out of counting.
The average position stays 0. Left and right being equally likely, there is no direction it tends to go. It moves and yet its average does not, so to see the width we take the square.
The direction of each step is independent of the others, so the cross terms average away. What is left is one step squared, added N times: \langle x^2 \rangle = Na^2.
So the width goes as \sqrt{N}. The two curves in the figure are how it wraps the walks, which mostly stay inside. Take 4 times as many steps and the width only doubles.
How many walks go right n times out of N steps is a binomial coefficient. It is exactly the count made in the piece on many particles fixing the outcome.
The figure is that shape. Raise N and the peak smooths out and draws towards a normal distribution. Its width grows as \sqrt{N}, so the peak grows lower and broader.
Splitting particles between the halves of a box and splitting steps between left and right gave the same count. The subject can differ; if the counting is the same, the shape of the answer is the same. This is the strongest thing statistical mechanics has.
If one step takes \tau, then in t seconds the number of steps is N = t/\tau. Putting that into \langle x^2 \rangle = Na^2 gives \langle x^2 \rangle = 2Dt, with D = a^2/2\tau the diffusion constant.
The figure is the spreading of a concentration. It grows lower and broader as time passes, while the area under the curve does not change, because no particle is created or lost.
It spreads as the square root of the time, not as the time. That is the point about diffusion. Carrying something twice as far takes 4 times as long, which is why diffusion is no way to move anything far. In the next piece we look at where these steps come from.