# What the diffusion coefficient means

D sets how fast a spreading cloud widens. Along one line, the mean square distance of its particles from where they started grows by 2D every second.

Written for this edition, not translated from Einstein.

## What changes when D changes?

Put a drop of dye into a long tube of still water. At first the dye sits in one spot; an hour later it has spread along the tube, thinner and wider. The diffusion coefficient D says how fast that spreading goes.

The rule it follows is local. Look at one spot. If the dye is thicker on both sides than at the spot itself, a dip, dye drifts in from both sides and the spot fills. If the spot is a peak, thicker than its neighbours, dye drifts out and it thins. The more sharply the dip or peak is curved, the faster it changes, and D sets the pace. With p for how much dye sits per metre of tube at place x and time t:

$$
\frac{\partial p}{\partial t}=D\frac{\partial^2p}{\partial x^2}
$$

The time rate of change of density is D times its spatial curvature.

The right-hand side is D times the curvature, which is positive at a dip and negative at a peak. For one particle instead of a drop of dye, p is the probability per metre of finding it at x. Where the tube ends matters too: a closed tube keeps all its dye, and after a long time spreads it evenly, not in the ever-widening bell of an endless tube.

The same rule can be read as a flow. The amount crossing a point each second runs from thick to thin, faster where the density falls more steeply:

$$
J=-D\frac{\partial p}{\partial x}
$$

Fick’s constitutive law sends flux down the density gradient.

Combining the flux law with conservation gives the diffusion equation for constant D. The flux law itself is a model premise in this route, not something conservation alone proves.

## Worked example: Doubling the diffusion coefficient, then quadrupling the time

Suppose the dye's typical distance from where it started is 1 mm after one minute. Double D, keeping everything else the same: the mean square doubles, so the typical distance becomes about 1.4 mm, the square root of 2 times as far, not twice. Keep the original D and wait four minutes instead of one: the typical distance doubles, to 2 mm. Neither comparison says where any single particle goes.

## Where this lesson stops

D is measured in square metres per second. Before using it, say where the tube ends and what the model assumes, since both change the answer.

## This lesson builds on

- [Counting what crosses a boundary](/foundations/flux-continuity/)
- [Squares and square roots](/foundations/bridge-squaring-square-roots/)