Foundation lesson
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:
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:
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.
A peak empties, a dip fills
One profile of dye along the tube, thicker where the curve is higher. The arrows are what the rule does next at the two spots the lesson names.
The curvature is negative at the peak and positive at the dip. That is the whole of the sign in the rule: dye runs from thick to thin, and the two spots are one rule seen from two sides.
Later the same dye is wider and lower. The lesson's example: a typical distance of 1 mm after 1 minute.
What it shows, in words
The first drawing is one profile of dye with a peak a quarter of the way along the tube and a dip three quarters of the way along. At the peak the dye is thicker than on either side, so it leaves in both directions and the spot empties; at the dip it is thinner than on either side, so it arrives from both directions and the spot fills. Nothing in the picture treats the two spots differently: the same rule, that dye runs from thick to thin, empties one and fills the other.
The second drawing is the same dye at two times. After 1 minute its typical distance from the start is 1 mm; after four times as long it is 2 mm, twice as far, because the mean square grows with the time and the distance is its square root. Doubling the diffusion coefficient instead of the time would give about 1.4 mm, the square root of two rather than of four.
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.
If you came here from a passage, Back returns you to the exact place you left.