Foundation lesson
Ratios and scaling
Many laws say how one quantity scales with another: in proportion, inversely, as a square, or as a square root. Knowing which lets you predict the effect of a change without working out any absolute number.
Written for this edition, not translated from Einstein. Editorial review pending.
If one quantity doubles, what happens to the others?
Double the side of a square and its area grows four times. Double the side of a cube and its volume grows eight times. Length, area and volume scale as the first, second and third power of the side.
Physical laws scale in the same ways. Stokes's drag on a sphere is in proportion to its radius: double the radius, double the drag. So a particle's diffusion coefficient D is inversely proportional to its radius: double the radius, and D halves.
lambda x is the square root of 2 D t, and D is k B T over six pi eta a.
The spread of a diffusing particle grows as the square root of D times the time. So halving D does not halve the spread: it divides it by √2, about 1.41. The tempting first thought, that twice the radius means half as far, is out by that factor, and so is the same thought about twice the viscosity.
Scaling can also run away. Before 1905, giving every mode of radiation in a box its classical share of heat energy made the total grow as the cube of the highest frequency allowed: widen the range tenfold and the energy grows a thousandfold, without limit. That is the difficulty §1 of the light-quanta paper sets out.
Worked example: Double the radius: how far does the particle get?
- A sphere of radius 0.5 μm in the paper's water at 17 °C spreads 0.7948 μm in 1 s (the paper's constants, set einstein-1905-brownian-printed).
- Double the radius to 1 μm. halves.
- λx = √(2Dt) is multiplied by √½ = 0.7071, so it becomes 0.7948 × 0.7071 = 0.5620 μm, not half of 0.7948.
- Doubling the viscosity instead does exactly the same: D halves and λx is multiplied by 0.7071.
Try it: scaling by a factor
Choose a factor k and multiply a length by it. The readout shows what happens to an area, a volume and the spread of a diffusing particle, with the Brownian paper's 0.8 micrometres worked through.
Type a factor and press Enter, or drag the slider from 0.5 to 5.
- Length
- × k = 2
- Area
- × k² = 4
- Volume
- × k³ = 8
- Spread, k times as long
- × √k = 1.414: 0.7948 μm in 1 s becomes 1.124 μm in 2 s
- Spread, k times the radius
- × 1/√k = 0.7071: 0.7948 μm becomes 0.5620 μm, and k times the viscosity does the same
What it shows, in words
Multiplying a length by k multiplies an area by k² and a volume by k³. A diffusing particle's spread grows as the square root of the time, so k times as long gives √k times the spread. Multiplying the radius or the viscosity by k divides the diffusion coefficient by k and the spread by √k. At k = 2 the spread changes by 1.414 or by 0.7071, not by 2 or by one half.
Where this lesson stops
This lesson stops at how a change scales. Why D depends on the radius and the viscosity in the first place is the lesson on viscosity and Stokes drag.
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