# What kind of analytical function could resemble that plot?

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…)

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Maybe the Laplace transform of your data could show some structure inside of this impulse... – user31373 Jun 24 '12 at 19:40
@LeonidKovalev well, actually, I am not exactly “fitting experimental data”, but generalizing the expected evolution of one model parameter. I know it starts at zero, jumps and goes down… I've edited the question to reflect that better. – F'x Jun 24 '12 at 19:44

## 1 Answer

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 $k$.

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