Table 2.

Test for two-dimensional trajectories in a free environment, and equation references for other dimensions

	Second moment (regular, MME)	Ratio (regular, MME)	Growing spheres
BM	(∝ t, ∝ t)	(2, 1.49), Eqs. 7 and 9	Pr(r ≤ r0tα/2, t) = A0
Fractals	(∝ tα, ∝ tα)	(<2, <1.49), Eqs. 7 and 9	$\text{Pr}\left(r\le r_0{t}^{\alpha /2},t\right)\propto t^{\alpha \left(2-d_{\text{f}}\right)/2}$
CTRW	(∝ tα, ∝ tα)	(>2, >1.49), Eqs. 12 and 13	Pr(r ≤ r0tα/2, t) = A0
FBM	(∝ tα, ∝ tα′), Eq. 10	(2, <1.49), Eq. 11	Pr(r ≤ r0tα/2, t) = A0

For each model (BM, diffusion on fractal, CTRW, and FBM), the second column lists the scaling behavior of the second regular and MME moments (〈r²〉 and $\langle r_{\max }^2\rangle$); the third column shows the relative values of the regular and MME ratio (〈X⁴〉/〈X²〉²); and the fourth column contains the scaling laws of the probability, at time t, to be in a sphere growing like t^α/2.