The physics of randomness and regularities for languages (mother tongues) and their lifetimes and family trees and for the second languages are studied in terms of two opposite processes; random multiplicative noise [1], and fragmentation [2], where the original model is given in the matrix format. We start with a random initial world, and come out with the regularities, which mimic various empirical data [3] for the present languages.
Introduction: In a recent work [4] the cities and the languages are treated differently (and as connected; languages split since cities split, etc.); thus two distributions are obtained in the same computation at the same time. In the present contribution we consider only the languages, where our essential aim is to show that the several regularities about the languages may be evolving out of randomness; and, the present processes may be shaping the evolution. Following section is the original model in matrix format; the next section is for application and Section 4 is for the results. Discussions and conclusion is the last section.
Model:
At the beginning of our time (t), we have m(t=0) languages, where the number of speakers are (L(t=0) i , i≤m(0)) defined randomly; and, each of the ancestor language belongs to an ancestor family f(t=0) (≤m(0)). Adults choose a language as a second language if it is bigger than the mother language in size.
At each time step t, the languages split (fragment) with the probability h, where the splitting ratio is s. The fragmentation (splitting, mutation) means that, if the current population of a language (i) is L(t) i , sL(t) i many members form another population and (1-s)L(t) i many continue to speak the same language. It is obvious that, the results do not change if the mutated and surviving members are interchanged, i.e., if 1 -s is substituted for s. The number of the languages m(t) increases by one if one language splits; if any two of them split at t, then m(t) increases by two, etc.
The languages (L(t+1)) at a time t+1 evolve out of these (L(t)) at t;
where L(t) is a (1xm(t)) matrix, and E(t) is a rectangular (m(t+1)xm(t)) matrix, which represents the evolution of the languages in a time step from t to t+1 (evolutor). The number of the current languages m(t) increases randomly, where one may have
for a random number r m which is uniformly distributed between zero and one, (0≤r m <1) and for a given constant h, which is small. It is obvious that, m(t) increases exponentially (up to randomness) with time, where the exponent is proportional to h. We may have a similar relation for the families f(t) (f(t) m(t)), in terms of h fam (h fam h); where, h fam is the fragmentation rate for the families, and the number of the families (f(t)) increases exponentially (with the exponent, which is proportional to h fam ) in time.
In Eq. ( 1) we define E(t) as:
where, δ ji is the Kronecker delta (δ ji =0 if j≠i, and δ ji =1 if j=i), and r ji is a random is uniformly distributed between zero and one, (0≤r m <1) and r is a real constant, which is small. In Eq. (3) r ji may be considered as the number of the adults who change their language from their mother tongue (i) to another language (j) at t. (These individuals may have immigrated to a city where the language (j) is spoken, or another reason may be decisive.) For simplicity, we take r ji =r j δ ji , (It is obvious that, r ji may be utilized without the present approximation for computations using super computers.) where r j is a random number which is uniformly distributed between zero and one so that the multiplication r r j (Eq. ( 3)) may represent the net (positive) change in the number of the speakers of the language (j).
On the other hand, all of the speakers of a language (which emerged newly in terms of fragmentation, for example) may change their language, or they may be colonized, etc. We consider these cases, with a probability x. Secondly, when a new language is formed (at t), she may belong to the same family as the home language or a new family may be started in the meantime; which means that, the families fragment (by the probability h fam ). Please note that, the probability of forming a language (belonging to a new family) defines the parameter h fam , and the families grow exponentially in number (up to randomness) with time, as the languages do, Eq. ( 2)).
For the lifetimes we simply subtract the number of the time step when a language is emerged, from the total number of time steps for the ages of the living languages, and we ignore the lifetimes of the extinct languages. The following remark finishes the definition of our model: Whenever a language (family) becomes less then unity in size, we consider her as extinct. (We do not consider punctuation of the languages or of the families with their population, since it is not historically real.)
Application: We have m(0) ancestor languages, and f(0) ancestor families. The evolution of the size of any (the j th ) language maybe given as; L(t+1
, where W(t) is the current world population, i.e.,
Please note that, the growth rate of a language is proportional to this of the world (up to randomness), since we assume that each human speaks one (mother) language; and, if the languages grow by a random probability between 0 and r at each time step, then W(t) grows by the same random probability between 0 and r up to a constant (and, vice versa). It is obviou
This content is AI-processed based on open access ArXiv data.
