Annus MirabilisEinstein in 1905
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brownian motion

The equation explained

λx=t⋅RTN13πkP.\lambda_x = \sqrt{t} \cdot \sqrt{\frac{R T}{N} \frac{1}{3 \pi k P}}.
In wordsThe displacement along the axis is the square root of the elapsed time, times the square root of this quotient: the gas constant times the absolute temperature, divided by the number of molecules in a gram-molecule and by three pi times the coefficient of friction of the liquid times the radius of a particle.OverviewEliminate the diffusion coefficient between the two, and the displacement stands in terms of the temperature, the liquid's stickiness, the particle size, and the number of molecules in a gram-molecule. Everything but the last can be measured.Full explanationPutting the first line into the second removes the diffusion coefficient. The six pi becomes three pi because the factor two of 2Dt2 D t is absorbed. What the paper says of the result is that this equation lets one see how the displacement must depend on the temperature, the viscosity and the particle radius, and those dependences are worth stating carefully, because each runs through a square root. The displacement grows as the square root of the absolute temperature. It falls as the square root of the viscosity, so a liquid twice as viscous gives a displacement smaller by a factor of about 1.41, not by half. It falls in the same way with the particle radius. A reader who turns any of these into a proportionality without the root has changed the claim.Every stepλx=2Dt\lambda_x = \sqrt{2 D t}Section 4's result, from the preceding display.2D=2⋅RTN16πkP=RTN13πkP2 D = 2 \cdot \frac{R T}{N} \frac{1}{6 \pi k P} = \frac{R T}{N} \frac{1}{3 \pi k P}Twice section 3's diffusion coefficient: the two cancels against the six and leaves three.λx=t⋅RTN13πkP\lambda_x = \sqrt{t \cdot \frac{R T}{N} \frac{1}{3 \pi k P}}Put that under the root in place of 2D2 D.λx=t⋅RTN13πkP.\lambda_x = \sqrt{t} \cdot \sqrt{\frac{R T}{N} \frac{1}{3 \pi k P}} .The root of a product is the product of the roots, so the time separates from the rest: the printed line, with everything that does not change during an observation gathered in the second factor.Also

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