A small grain floating in water does not sit still. This is what Brown saw through a microscope in 1827. The grain is not trying to move; water molecules are striking it without pause.
The large circle in the figure is the grain we are watching, the small dots the molecules. They strike from every side. Each push is small, and their directions are scattered.
If they cancelled exactly, the grain would not move. But the number striking from the right and from the left differs slightly at every moment, and that difference pushes the grain. What is at work here is the same many small differences as in the previous piece.
Rather than follow the strikes one at a time, we split the force on the grain in two: the part that acts against the motion whenever the grain is moving quickly, and the part with no settled direction.
The first is the drag -\gamma v, the second the fluctuating force F(t). Written down, m\dfrac{dv}{dt} = -\gamma v + F(t). This is the Langevin equation.
We split it in two, but both come from the same strikes of the same molecules. The drag and the fluctuation have only one source between them. That single line has already settled the answer of the last scene.
Suppose the direction of the fluctuating force at each moment has nothing to do with the one before. Then the memory of the velocity dies away as \langle v(0)v(t) \rangle = \dfrac{kT}{m}e^{-\gamma t/m}.
The curve in the figure is that. The height kT/m at t = 0 is equipartition itself: however large the grain, its mean kinetic energy is kT/2 for each direction. A larger grain is simply slower; the energy it carries is the same.
The time for the memory to die is m/\gamma. Watched over anything shorter, the grain appears to fly straight. Watched over anything longer, it has forgotten its direction many times over, and appears to walk at random.
Over short times the grain still remembers its direction. It goes straight, so the spread grows as the square of the time: \langle x^2 \rangle = (kT/m)t^2.
Over long times the direction has been forgotten again and again. That is the same situation as the steps of the previous piece, and \langle x^2 \rangle = 2Dt.
The heavy curve in the figure joins the two, and the two faint ones are the shapes at either end. The boundary is m/\gamma, about 10^{-7} seconds for a small grain in water. What a microscope can see is always the long side alone. The single step of the previous piece was, it turns out, all of this short time gathered into one.
Working out D on the long side gives D = \dfrac{kT}{\gamma}, the Einstein relation. The straight line in the figure is that, going as the temperature.
The left side says how far the grain spreads; the \gamma on the right says how hard it is stopped. They look like separate quantities, and yet both came from the same strikes of the same molecules. Because what pushes and what stops are one and the same, their ratio is fixed by the temperature alone. This is the fluctuation-dissipation theorem, one of the widest-reaching results in the subject.
For a spherical grain \gamma = 6\pi\eta a is known, so measuring D under a microscope gives k. With k in hand, dividing the gas constant by it gives Avogadro's number. Perrin did exactly this. Measuring a fluctuation by eye became a way of counting molecules.