Editing Log-normal distribution (section)

=== Extremal principle of entropy to fix the free parameter ''σ''===

In applications, <math>\sigma</math> is a parameter to be determined. For growing processes balanced by production and dissipation, the use of an extremal principle of Shannon entropy shows that<ref name="bai">{{cite journal | last1 = Wu | first1 = Ziniu | last2 = Li | first2 = Juan | last3 = Bai | first3 = Chenyuan | title = Scaling Relations of Lognormal Type Growth Process with an Extremal Principle of Entropy | journal = Entropy | volume = 19 | issue = 56 | year = 2017 | pages = 1–14 | doi = 10.3390/e19020056 | bibcode = 2017Entrp..19...56W | doi-access = free}}</ref>
<math display="block">\sigma = \frac 1 \sqrt{6} </math>

This value can then be used to give some scaling relation between the inflexion point and maximum point of the log-normal distribution.<ref name = bai/> This relationship is determined by the base of natural logarithm, <math>e = 2.718\ldots</math>, and exhibits some geometrical similarity to the minimal surface energy principle.
These scaling relations are useful for predicting a number of growth processes (epidemic spreading, droplet splashing, population growth, swirling rate of the bathtub vortex, distribution of language characters, velocity profile of turbulences, etc.).
For example, the log-normal function with such <math>\sigma</math> fits well with the size of secondarily produced droplets during droplet impact <ref name="wu"/> and the spreading of an epidemic disease.<ref name="Wang">{{cite journal | last1 = Wang | first1 = WenBin | last2 = Wu | first2 = ZiNiu | last3 = Wang | first3 = ChunFeng | last4 = Hu | first4 = RuiFeng | title = Modelling the spreading rate of controlled communicable epidemics through an entropy-based thermodynamic model | journal = Science China Physics, Mechanics and Astronomy | volume = 56 | issue = 11 | year = 2013 | pages = 2143–2150 | issn = 1674-7348 | doi = 10.1007/s11433-013-5321-0 | pmid = 32288765 | pmc = 7111546 | arxiv = 1304.5603 | bibcode = 2013SCPMA..56.2143W}}</ref>

The value <math display="inline">\sigma = 1 \big/ \sqrt{6}</math> is used to provide a probabilistic solution for the Drake equation.<ref name="Bloetscher">{{cite journal | last1 = Bloetscher | first1 = Frederick | title = Using predictive Bayesian Monte Carlo- Markov Chain methods to provide a probabilistic solution for the Drake equation | journal = Acta Astronautica | volume = 155 | year = 2019 | pages = 118–130 | doi = 10.1016/j.actaastro.2018.11.033 | bibcode = 2019AcAau.155..118B | s2cid = 117598888}}</ref>