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
§4 · From random displacement to diffusion
Read the displacement argument, ask for its missing steps, and investigate the same relationships in three working laboratories.
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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).