Zero average is not no movement
Signed displacements can cancel. Their squares still record movement.
Read · Brownian motion: from wandering to a measurable law · Results
Brownian motion: from wandering to a measurable law
Read the displacement argument, ask for its missing steps, and investigate the same relationships in three working laboratories.
This is newly authored explanatory text in modern notation, with editorial review pending. It is not the German source, an English translation, or a complete edition of the paper. The reviewed source faces and pinned facsimile remain in preparation. Section headings identify the argument being discussed, not a completed source inventory.
Why the square grows with time
Independent centred steps add mean squares; the root therefore grows as the square root of time.
From a step law to a density law
Normalization preserves probability; symmetry removes drift; the second moment supplies diffusion.
What the spreading curve predicts
For an unbounded point source, the Gaussian has mean square 2Dt and coordinate RMS √(2Dt).
Why viscosity changes the spread
Combining ideal osmotic pressure with mobility and diffusive balance relates diffusivity to temperature, viscosity and radius.
What would let us count molecules?
Displacement constrains diffusivity; independent radius, viscosity, temperature and gas-constant information are needed to infer molecular number.