Statistics
Random Walk
Give thousands of particles no goal at all, just a coin flip each step, and a shape still emerges. The crowd spreads into a bell whose width grows like the square root of time: the particle's-eye view of diffusion.
Steps t0
Measured spread σ0.0
Predicted √t0.0
Top: time runs downward from the shared start. A few walkers are traced so you can see the jagged paths, and the curved cone is the ±√t spread. Bottom: the histogram right now, with the predicted bell.
What to observe
- No walker has any preferred direction, yet the crowd reliably fans out into a bell curve. Same mechanism as the Central Limit Theorem: a sum of independent steps.
- Watch the readouts: the spread σ tracks √t, not t. To reach twice as far, the cloud needs four times the steps. That square-root law is the fingerprint of diffusion.
- This is the heat equation seen from the particles. The bell widening here is the very same profile that heat follows as it spreads.
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