I am in the middle of making a model, and I am looking for an analytical expression which could resemble this evolution for one of the parameters:
In short: a sudden increase from 0 to a global maximum, followed by slower decrease. Once I have an idea of a functional form that presents the right characteristics, I can tweak it (peak height, peak position) to my liking.
The only expression I've been able to come up with so far involves rather high powers (12 and 6) of $1/x$:
(In case you wonder how I came to think of this one, it's the Boltzmann factor of a Lennard-Jones potential…)
So is this supposed to be for $r > 0$? You might try more generally $y = k e^{a/r^b - c/r^d}$, where $b < d$. The maximum is at $r = \left( \frac{cd}{ab}\right)^{1/(d-b)}$, and the limit as $r \to \infty$ is $k$.