Evolution and mass extinctions as lognormal stochastic processes
Abstract
In a series of recent papers and in a book, this author put forward a mathematical model capable of embracing the search for extraterrestrial intelligence (SETI), Darwinian Evolution and Human History into a single, unified statistical picture, concisely called EvoSETI. The relevant mathematical tools are: (1) Geometric Brownian motion (GBM), the stochastic process representing evolution as the stochastic increase of the number of species living on Earth over the last 3.5 billion years. This GBM is well known in the mathematics of finances (BlackSholes models). Its main features are that its probability density function (pdf) is a lognormal pdf, and its mean value is either an increasing or, more rarely, decreasing exponential function of the time. (2) The probability distributions known as blognormals, i.e. lognormals starting at a certain positive instant b>0 rather than at the origin. These blognormals were then forced by us to have their peak value located on the exponential meanvalue curve of the GBM (PeakLocus theorem). In the framework of Darwinian Evolution, the resulting mathematical construction was shown to be what evolutionary biologists call Cladistics. (3) The (Shannon) entropy of such blognormals is then seen to represent the `degree of progress' reached by each living organism or by each big set of living organisms, like historic human civilizations. Having understood this fact, human history may then be cast into the language of blognormals that are more and more organized in time (i.e. having smaller and smaller entropy, or smaller and smaller `chaos'), and have their peaks on the increasing GBM exponential. This exponential is thus the `trend of progress' in human history. (4) All these results also match with SETI in that the statistical Drake equation (generalization of the ordinary Drake equation to encompass statistics) leads just to the lognormal distribution as the probability distribution for the number of extraterrestrial civilizations existing in the Galaxy (as a consequence of the central limit theorem of statistics). (5) But the most striking new result is that the wellknown `Molecular Clock of Evolution', namely the `constant rate of Evolution at the molecular level' as shown by Kimura's Neutral Theory of Molecular Evolution, identifies with growth rate of the entropy of our EvoSETI model, because they both grew linearly in time since the origin of life. (6) Furthermore, we apply our EvoSETI model to lognormal stochastic processes other than GBMs. For instance, we provide two models for the mass extinctions that occurred in the past: (a) one based on GBMs and (b) the other based on a parabolic mean value capable of covering both the extinction and the subsequent recovery of life forms. (7) Finally, we show that the Markov & Korotayev (2007, 2008) model for Darwinian Evolution identifies with an EvoSETI model for which the mean value of the underlying lognormal stochastic process is a cubic function of the time. In conclusion: we have provided a new mathematical model capable of embracing molecular evolution, SETI and entropy into a simple set of statistical equations based upon blognormals and lognormal stochastic processes with arbitrary mean, of which the GBMs are the particular case of exponential growth.
 Publication:

International Journal of Astrobiology
 Pub Date:
 October 2014
 DOI:
 10.1017/S147355041400010X
 Bibcode:
 2014IJAsB..13..290M
 Keywords:

 Darwinian Evolution;
 entropy;
 geometric Brownian motion;
 lognormal probability densities;
 molecular clock;
 statistical Drake equation