I've recently come across a particular problem whose solution turns out to be a probability distribution given by $f(x) = \alpha \|x\|^{-2/3}$ in the unit disk in $\mathbb{R}^2$ and zero elsewhere (I alluded to this in a previous question
http://mathoverflow.net/questions/85624/fitting-a-mesh-to-a-density-function
which was very helpfully answered by Anton Petrunin). Does this distribution appear in any other contexts? I've seen a $-2/3$ 2/3$ power law in reference to metabolic rates of animals:
http://www.ncbi.nlm.nih.gov/pubmed/19906667
and in kinematics:
http://www.ncbi.nlm.nih.gov/pubmed/9844558
but both of the preceding cases appear to be looking at rather one-dimensional quantities ; (and they're looking at positive powers rather than negative in my case, not an important distinction); they have $f(t) = \alpha t^{-2/3}$t^{2/3}$, where $t$ represents mass in the first case and angular velocity of the tip of a pen in the second. This seems different from the situation that I'm describing. To put it succinctly,
"Are there natural quantities that are proportional to the distance to some point, raised to the $-2/3$ power?"
(This may be more appropriate for another forum; if so, I welcome any suggestions)
