If you have the data (time since first settlement, and number of languages currently spoken in the location, and whatever other confounding variables you think there might be) you can simply put them into a multiple regression.

Agree Stephen. But there is a problem with that of "languages spoken CURRENTLY".

I search a natural correlation of linguistic diversification where no linguistic repressive policies involved (in which a dominant language removes to other minority languages).

Thanks for your answer.

I'm not sure that is possible to know--it would require guessing how many languages there would be somewhere if certain policies or historical events had not happened.

However, if you can think of a way to operationalize "repressive policies" or historical events like that as a variable, you could include that in the model to account for it.

It will always be a hypothetical construct open to questions. Moreover, this hypothesis is being constructed  on imaginary situation which is only possible if there are clear laws on language change available. Unfortunately in Historical Linguistics all such changes are reconstructed in reverse and have ever taken the status of laws.

That's true, dear Ahsan, but the problem is that language change follows a non-linear behavior (F (u) = 0). At least, I have no knowledge to formulate a nonlinear equation. On the other hand, if we just take care of the process of grammaticalization, which is a purely structural phenomenon of language, then how do I determine an equation that encompasses this variable with time diversification?

I think the correlation of diversification depends more on a non-linguistic variable. But what of all?

There are many variables that may be considered in this regard, for example how many people get detached through a river or a mountain range in between them that separates and make communication distinct, how many travel far away and settle there to allow their own variation to develop or interact with other communities to let Pidgin and creoles develop? and many other of the like.

What we do know is that an isolated community can create a language in a very short time. Idioma de Señas de Nicaragua developed between 1977 and 1986 within a community that ultimately grew to a bit over 400 people in Managua (growth from about 50 to 100 between 1977 and 1979, from 100 to 400 by 1983).

In this case the functional separation of populations was that these 400 people were deaf teenagers and children attending schools where the teachers had no ability to do anything but attempt to train the students to lipread Spanish and spell Spanish words letter-by-letter with their fingers in the air.

We also know that Tok Pisen developed from a crude mechanism for New Guineans to communicate with Americans during World War 2 into a functional language by the late 1960s. It is currently a national language of Papua New Guinea and the most widely used language in that country with 4-5 million speakers.

This is a case of the opposite phenomenon compared to Idioma de Señas de Nicaragua. Tok Pisin is used to communicate in what is probably the most linguistically varied place on earth. Tok Pisin functions as a language to unify the nation across a geography that leads to isolated development of cultures and language. It does this partly because it is based on no native New Guinean language, resulting in no ethno-linguistic bias toward any existing group, though there is a prestige element in adopting an English-derived language, especially as an intermediate to learning a standard English dialect.

Not all historical linguistics happens in reverse. And, to be fair, historical linguistics is mostly oriented toward the past because there was no established community focused on the systematic study of the relationships between languages until the19th century.

To put things in perspective, at that point doctors believed that tending to patients while their hands, instruments, and clothing were still wet with the blood and other bodily fluids was a comforting sign to the patient that the doctor was in high demand and experienced. The germ theory of disease transmission was still rejected by many medical professionals in the late 19th century.

And like medicine, the activity is focused on the area where the techniques are developed first before moving to other parts of the world. Like medicine as well, it is much easier to develop and transmit knowledge in literate societies. The history of languages that have no significant literary tradition can only be determined by reasoning backwards in time from the existing body of speakers.

If you don't like "laws", feel free to use the modern refined terminology that focuses on the phenomenon rather than the discoverer. Instead of "Grimm's Law" (which is where historical linguistics was born), call it the "First Germanic Sound Shift".

But if what you are truly looking for is an equation and/or theorem, you should stop. There are none.

You can try looking into the method of glottochronology, but it is mostly used to determine the time of divergence of languages in a pair-by-pair basis based on calculating differences between the languages and comparing that with a number expected to represent the average rate of change of language (we are usually dealing with phonetic differences). It is not to intended to provide estimates of how many languages should be present in any geographical area at any particular time.It also has many flaws.  A severely flawed equation can be worse than no equation.

That should be phonemic differences with regard to the focus of glottochronology, not phonetic differences. Often, as well, the method used is to measure the phonemic differences between languages for a standardized set of words. The Swadesh List or some modification of it (Polynesian islanders have little need for words for wheels, central Asians don't use boats much).