SETI, Evolution and Human History Merged into a Mathematical Model

In this paper we propose a new mathematical model capable of merging Darwinian Evolution, Human History and SETI into a single mathematical scheme: (1) Darwinian Evolution over the last 3.5 billion years is defined as one particular realization of a certain stochastic process called Geometric Brownian Motion (GBM). This GBM yields the fluctuations in time of the number of species living on Earth. Its mean value curve is an increasing exponential curve, i.e. the exponential growth of Evolution. (2) In 2008 this author provided the statistical generalization of the Drake equation yielding the numberN of communicating ET civilizations in the Galaxy. N was shown to follow the lognormal probability distribution. (3) We call “b-lognormals” those lognormals starting at any positive time b (“birth”) larger than zero. Then the exponential growth curve becomes the geometric locus of the peaks of a one-parameter family of b-lognormals: this is our way to re-define Cladistics. (4) b-lognormals may be also be interpreted as the lifespan of any living being (a cell, or an animal, a plant, a human, or even the historic lifetime of any civilization). Applying this new mathematical apparatus to Human History, leads to the discovery of the exponential progress between Ancient Greece and the current USA as the envelope of all b-lognormals of Western Civilizations over a period of 2500 years. (5) We then invoke Shannon’s Information Theory. The b-lognormals’ entropy turns out to be the index of “development level” reached by each historic civilization. We thus get a numerical estimate of the entropy difference between any two civilizations, like the Aztec-Spaniard difference in 1519. (6) In conclusion, we have derived a mathematical scheme capable of estimating how much more advanced than Humans an Alien Civilization will be when the SETI scientists will detect the first hints about ETs. Received 21 November 2012, accepted 25 February 2013, first published online 23 April 2013