A relation between the second moment of a distribution and one of its particular probability I had recently posted a question here: To prove a relation involving a probability distribution
The relations quoted in the above question are used extensively in fluid mechanics and many other fields, but no one has provided any proof for that. The relations (as stated in my previous question) are:
\begin{equation}
\alpha + p_6 = 1, \quad \quad \quad 0.66 < p_6 < 1,
\end{equation}
\begin{equation}
\alpha p_6^2 = 1 / 2 \pi, \quad \quad \quad 0.34 < p_6 < 0.66,
\end{equation}
where $$p_n = \frac{e^{-c_1 n - c_2/n}}{Z},$$
and $c_1$ and $c_2$ are constants and Z is a normalization factor, also $n \geq 3$. $\alpha$ (the second moment) is defined as $\alpha = \sum_{n = 3}^{\infty} p_n (n - 6)^2$, where it is assumed that $\langle n \rangle = 6$.
The answer to my previous question shows that these relations are not correct exactly and the use of equality sign in some papers in the literature is not quite correct. My question is: Is it possible to obtain these formulas as some sort of approximate relations? The main problem is that how one could get rid of the constants in $p_n$ and obtain such a general law? (If it is possible at all!)
EDIT
As mentioned in the comments, the same relations are obtained here, see Eq.(16); however, their probability function, Eq.(8), is different from the one which is mentioned in the question. Since in the most papers, the authors have used the $p_n$ as mentioned in my question, one guesses that these relations are also derivable from the above $p_n$. The approach of the mentioned book is to construct a generating function and recast the problem into a Mayer series. Then by keeping the first few terms one can obtain the above relations; however, this approach, as far as I can see, is not applicable to the above $p_n$.
 A: Simple argument for Lemaître's law.
Let me first give a simple argument for the relation $2\pi\alpha p_6^2=1$ between the variance $\alpha$ and the probability $p_6$ (a relation known as Lemaître's law).
I approximate the distribution $p_n$ by a Gaussian $P(n)$ centered at $n=6$ and ignore the discreteness of $n$. I also ignore the restriction to $n\geq 3$ and allow $n$ to range from $-\infty$ to $+\infty$. This may be justified by the expectation that only values of $n$ close to the peak at 6 will contribute to the first few moments, provided that $p_6$ is not too small.
So I replace $p_n$ by
$$P(n)=\frac{1}{\sqrt{2\pi\alpha}}e^{-(n-6)^2/2\alpha},$$
and the relation $2\pi\alpha P(6)^2=1$ follows.
The reason that this relation fails if $p_6$ becomes close to unity, is that then the discreteness of $n$ cannot be neglected -- since only three values $n=5,6,7$ contribute. For $p_6$ near 1 that then gives the relation $\alpha+p_6=1$ derived by Iosif Pinelis.

Numerical test of Lemaître's law.
The simple argument given above is crude and needs to be justified. For that purpose I have tested it numerically, for the distribution
$$p_n = Z^{-1}e^{-c_1 n - c_2/n},\;\;n\geq 3,\;\;\text{with}\;\;Z=\sum_{n=3}^\infty e^{-c_1 n - c_2/n},$$
subject to the constraint that $\bar{n}=6$.
I show plots of $\alpha = \sum_{n = 3}^{\infty} p_n (n - 6)^2$ versus $p_6$ (blue data points) and compare with the two expressions
$\alpha=(2\pi p_6^2)^{-1}$ (red) and $\alpha=1-p_6$ (green). The agreement is quite remarkable, without any fitting parameter. The lower range $p_6>0.34$ mentioned in the OP can be relaxed to smaller values of $p_6$, the agreement between blue and red remains good for $p_6\gtrsim 0.2$. The crossover from red to green indeed happens near $p_6=0.66$.



Let me also compare with the $p_n$ used by Rivier,
$$p_n = Z^{-1}e^{c_1 n - c_2 n^2},\;\;n\geq 3,\;\;\text{with}\;\;Z=\sum_{n=3}^\infty e^{c_1 n - c_2 n^2},$$
again subject to the constraint that $\bar{n}=6$.
Now the deviation between blue and red indeed starts at larger $p_6$, close to the value 0.34 in the OP.



Here is a test of the relation $c_2=36 c_1$ mentioned by Iosif Pinelis (data points in blue), for the first $p_n$ (left plot), and also a plot of $p_6$ versus $c_1$ (right plot):


For the second $p_n$, the analogous relation is $c_2=c_1/12$, and this is indeed very close to the data points:


The right plot again shows $p_6$ as a function of $c_1$.
