Foundation lesson
Functions and graphs
A function gives exactly one output for each input. Its graph lays those pairs out on two axes, and every slope or area read off it carries units made from the two axes' units.
Written for this edition, not translated from Einstein.
How does a graph record the way one quantity depends on another?
A bath fills from a tap. After one minute it holds 10 litres, after two 20, after three 30. For each time there is exactly one amount of water, and that pairing, one output for each input, is a function. Plot the pairs with time along the bottom and litres up the side and they fall on a straight line that climbs 10 litres for every minute. The slope, in litres per minute, is how fast the bath fills.
In the papers the pairs are physical: the mean square displacement of a Brownian particle for each elapsed time, or the concentration for each position along a tube. A function says how one thing responds when another changes.
A graph puts the input along the horizontal axis and the output up the vertical one, so each point is one pair. Both axes carry units, and so does everything read off the graph: the slope of mean square displacement against time is in square micrometres per second, and the area under a probability density is a plain probability.
A formula such as ⟨x²⟩ = 2Dt describes an average over many particles. One particle's squared displacement scatters widely around the line; the average over many lies close to it. The Brownian paper predicts the average, and ends by hoping that a researcher will soon decide the question by observation.
Worked example: Mean square displacement plotted against time
The mean square displacement equals two times the diffusion coefficient times the time.
- Take D = 0.5 square micrometres per second, so 2D = 1 square micrometre per second.
- At t = 1 s the mean square displacement is 1 µm²; at t = 4 s it is 4 µm²; at t = 9 s, 9 µm².
- Plotted against time, these points lie on a straight line through the origin whose slope is 2D, 1 µm² per second.
- Their square roots, the root-mean-square displacements, are 1, 2 and 3 µm: four times the time gives twice the distance.
Try it: from a table to a curve
In §4 of the Brownian paper, a particle's root-mean-square displacement in a time t is λx = √(2Dt). The mean square grows in proportion to the time, so the typical displacement grows only as the square root of the time. The table below is worked out from that formula for D = 2 µm²/s. Plot it point by point to see what square-root growth looks like.
| Time t (s) | ⟨x²⟩ (µm²) | √⟨x²⟩ (µm) | Status |
|---|---|---|---|
| 0 | 0 | 0.0 | Plotted |
| 1 | 4 | 2.0 | Plotted |
| 4 | 16 | 4.0 | Plotted |
| 9 | 36 | 6.0 | Plotted |
| 16 | 64 | 8.0 | Plotted |
What it shows, in words
Each row pairs a time in seconds with the mean square displacement in square micrometres and its square root. Four times the time gives twice the displacement, not four times: the displacement grows as the square root of time. A particle drifting at a steady speed would go four times as far.
Where this lesson stops
A graph is a record of paired quantities, each with its units. A slope read off it has units too, and a slope stated without them explains nothing.
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