Separated Depth Cross Scans

Diffusion Time Three Row Scan

mean-square spread grows by time

A diffusion formula is clearer when time is scanned through three exact rows. 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 second gives two square metres of mean-square spread

Hold diffusion at 1 square metre per second. Time 1 s gives mean-square spread 2 square metres. This is a spread budget, not a path length.

⟨x2⟩=2Dt=2⋅1⋅1=2\langle x^{2}\rangle=2Dt=2\cdot1\cdot1=2
Mean square spreadThe spread claim is an x-squared budget, not a distance marker.packetmean square

Two seconds doubles the spread budget

The diffusion coefficient stays 1 square metre per second. Time 2 s gives mean-square spread 4 square metres, so the diagram bar doubles.

⟨x2⟩=2Dt=2⋅1⋅2=4\langle x^{2}\rangle=2Dt=2\cdot1\cdot2=4
Mean square spreadThe spread claim is an x-squared budget, not a distance marker.packetmean square

Three seconds makes the linear time relation scannable

Time 3 s gives mean-square spread 6 square metres. The three rows show that time scales the mean-square quantity linearly in this training model.

⟨x2⟩=2Dt=2⋅1⋅3=6\langle x^{2}\rangle=2Dt=2\cdot1\cdot3=6
Mean square spreadThe spread claim is an x-squared budget, not a distance marker.packetmean square