Evolution and History in a new “Mathematical SETI” model

In a recent paper (Maccone, 2011 [15]) and in a recent book (Maccone, 2012 [17]), this author proposed a new mathematical model capable of merging SETI and Darwinian Evolution into a single mathematical scheme. This model is based on exponentials and lognormal probability distributions, called “b-lognormals” if they start at any positive time b (“birth”) larger than zero. Indeed:

• 1.Darwinian evolution theory may be regarded as a part of SETI theory in that the factor f_l in the Drake equation represents the fraction of planets suitable for life on which life actually arose, as it happened on Earth.
• 2.In 2008 (Maccone, 2008 [9]) this author firstly provided a statistical generalization of the Drake equation where the number N of communicating ET civilizations in the Galaxy was shown to follow the lognormal probability distribution. This fact is a consequence of the Central Limit Theorem (CLT) of Statistics, stating that the product of a number of independent random variables whose probability densities are unknown and independent of each other approached the lognormal distribution if the number of factors is increased at will, i.e. it approaches infinity.
• 3.Also, in Maccone (2011 [15]), it was shown that the exponential growth of the number of species typical of Darwinian Evolution may be regarded as the geometric locus of the peaks of a one-parameter family of b-lognormal distributions constrained between the time axis and the exponential growth curve. This was a brand-new result. And one more new and far-reaching idea was to define Darwinian Evolution as a particular realization of a stochastic process called Geometric Brownian Motion (GBM) having the above exponential as its own mean value curve.
• 4.The b-lognormals may be also be interpreted as the lifespan of any living being, let it be a cell, or an animal, a plant, a human, or even the historic lifetime of any civilization. In Maccone, (2012 [17, Chapters 6, 7, 8 and 11]), as well as in the present paper, we give important exact equations yielding the b-lognormal when its birth time, senility-time (descending inflexion point) and death time (where the tangent at senility intercepts the time axis) are known. These also are brand-new results. In particular, the σ=1 b-lognormals are shown to be related to the golden ratio, so famous in the arts and in architecture, and these special b-lognormals we call “golden b-lognormals”.
• 5.Applying this new mathematical apparatus to Human History leads to the discovery of the exponential trend of progress between Ancient Greece and the current USA Empire as the envelope of the b-lognormals of all Western Civilizations over a period of 2500 years.
• 6.We then invoke Shannon's Information Theory. The entropy of the obtained b-lognormals turns out to be the index of “development level” reached by each historic civilization. As a consequence, we get a numerical estimate of the entropy difference (i.e. the difference in the evolution levels) between any two civilizations. In particular, this was the case when Spaniards first met with Aztecs in 1519, and we find the relevant entropy difference between Spaniards an Aztecs to be 3.84 bits/individual over a period of about 50 centuries of technological difference. In a similar calculation, the entropy difference between the first living organism on Earth (RNA?) and Humans turns out to equal 25.57 bits/individual over a period of 3.5 billion years of Darwinian Evolution.
• 7.Finally, we extrapolate our exponentials into the future, which is of course arbitrary, but is the best Humans can do before they get in touch with any alien civilization. The results are appalling: the entropy difference between aliens 1 million years more advanced than Humans is of the order of 1000 bits/individual, while 10,000 bits/individual would be requested to any Civilization wishing to colonize the whole Galaxy (Fermi Paradox).
• 8.In conclusion, we have derived a mathematical model capable of estimating how much more advanced than humans an alien civilization will be when SETI succeeds.