Statistical Source Factor Depth Scans

Step Length Source Scan

Brownian spread follows length squared

A fixed step count and time isolate step length as the source factor for spread and diffusion. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

One-metre steps give four square metres of spread

Hold step count at 4 and step time at 2 seconds. A step length of 1 metre gives mean-square spread 4 square metres and diffusion 1/4 square metre per second.

4⋅12=44\cdot1^{2}=4
Random walk ledgerStep count and step size set mean-square spread and diffusion.spreaddiffusion

Two-metre steps give the standard sixteen-square-metre spread

The same 4 steps at length 2 metres produce mean-square spread 16 square metres and diffusion 1 square metre per second.

4⋅22=164\cdot2^{2}=16
Random walk ledgerStep count and step size set mean-square spread and diffusion.spreaddiffusion

Three-metre steps make the square dependence visible

At length 3 metres, the checked spread rises to 36 square metres. Diffusion is 9/4 square metre per second, so length is not a linear knob here.

4⋅32=364\cdot3^{2}=36
Random walk ledgerStep count and step size set mean-square spread and diffusion.spreaddiffusion