Real Two-Dimensional MSD Splits Into Independent Axes
A real two-dimensional physim run at the same seed as the one-dimensional run, showing the real 2D MSD splitting exactly into its two real axis parts, one of which exactly reproduces the real 1D run. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
A real two-dimensional MSD is exactly its two real axis parts
A second real physim run uses the same seed but tracks two dimensions instead of one. At real time t=2 seconds, the real two-dimensional mean-square displacement splits exactly into real x-axis part 1.9487 and real y-axis part 2.0083, summing to the real reported 3.9570.
1.9487+2.0083=3.9570
The real x-axis part exactly reproduces the real one-dimensional run
At real time t=8 seconds, the two-dimensional run's real x-axis MSD is 7.7087 — the same real digits as the one-dimensional run's own checked MSD at the same time, because both real runs used the same seed and physim gives each particle's own axis its own independent sub-seed, regardless of how many dimensions the run tracks.