Energy of extra-terrestrial civilizations according to evo-seti theory

Abstract

Consider two great scientists of the past: Kepler (1571-1630) and Newton (1642-1727). Kepler discovered his three laws of planetary motion by observing Mars: he knew experimentally that his three laws were correct, but he didn't even suspect that all three mathematical laws could be derived as purely mathematical consequences by a "superior" mathematical law. The latter was the Law of Gravitation that Newton gave the world together with his supreme mathematical discovery of the Calculus, necessary for that mathematical derivation. Dare we say that we did the same for the MOLECULAR CLOCK, the experimental law of genetics discovered in 1962 by Émile Zuckerkandl (1922-2013) and Linus Pauling (1901-1994) and derived by us as a purely mathematical consequence of our mathematical Evo-SETI Theory? Let us explain how this mathematical derivation was achieved, please. Darwinian evolution over the last 3.5 billion years was an increase in the number of living species from one (RNA?) to the current (say) 50 million. This increasing trend in time looks like being exponential, but one may not assume an exactly exponential curve since many species went extinct in the past, especially in the five, big mass extinctions. Thus, the simple exponential curve must be replaced by a stochastic process having an exponential mean value. Borrowing from financial mathematics (the "Black-Sholes models"), this "exponential" stochastic process is called Geometric Brownian Motion (GBM). Its probability density function (pdf) is a lognormal (and not a Gaussian) (Proof: see ref. [3], Chapter 30, and ref. [4]). Lognormal also is the pdf of the statistical number of communicating ExtraTerrestrial (ET) civilizations in the Galaxy at a certain fixed time, like a snapshot: this result was obtained in 2008 by this author as his solution to the Statistical Drake Equation of SETI (Proof: see ref. [1]). Thus, the GBM of Darwinian evolution may also be regarded as the extension in time of the Statistical Drake equation (Proof: see ref. [4]). But the key step ahead made by this author in his Evo-SETI (Evolution and SETI) mathematical theory was to realize that LIFE also is just a b-lognormal in time: every living organism (a cell, a human, a civilization, even an ET civilization) is born at a certain time b ("birth"), grows up to a peak p (with an ascending inflexion point in between, a for adolescence), then declines from p to s (senility, i.e. descending inflexion point) and finally declines linearly and dies at a final instant d (death). In other words, the infinite tail of the b-lognormal was cut away and replaced by just a straight line between s and d, leading to simple mathematical formulae ("History Formulae") allowing one to find this "finite b-lognormal" when the three instants b, s, and d are assigned. Next the crucial Peak-Locus Theorem comes. It means that the GBM exponential may be regarded as the geometric locus of all the peaks of a one-parameter (i.e. the peak time p) family of b-lognormals. Since b-lognormals are pdf-s, the area under each of them always equals 1 (normalization condition) and so, going from left to right on the time axis, the b-lognormals become more and more "peaky", and so they last less and less in time. This is precisely what happened in Human History: civilizations that lasted a millennium each (like Ancient Greece and Rome) lasted just a few centuries (like the Italian Renaissance and Portuguese, Spanish, French, British and USA Empires) but they were more and more advanced in the "level of civilization". This "level of civilization" is what physicists call ENTROPY. Also, in refs. [3] and [4], this author proved that, for all GBMs, the (Shannon) Entropy of the b-lognormals in his Peak-Locus Theorem grows LINEARLY in time. The Molecular Clock (refs. [6] through [11]), well known to geneticists since 1962, shows that the DNA base-substitutions occur LINEARLY in time since they are neutral with respect to Darwinian selection. This is Kimura's neutral theory of molecular evolution. The conclusion is that the Molecular Clock and the LINEAR increase of EvoEntropy in time are just the same thing! In other words, we derived the Molecular Clock mathematically as a part of our Evo-SETI Theory. In addition, our EvoEntropy, i.e. the Shannon Entropy of the b-lognormal (with the minus sign reversed and starting at zero at the the time of the origin of Life on Earth) the new EvoSETI SCALE to measure the evolution of life on Exoplanets (measured in bits). Is that all? No. That was all up to the present paper, first published in 2017. In fact, just classical thermodynamics entails both ENERGY and ENTROPY, so our Evo-SETI Theory needs entailing the ENERGY used by a living Species or Civilization along its whole lifetime in addition to its ENTROPY (i.e. Molecular Clock). In other words still, while the Molecular Clock is a measure of the advancement in evolution, the ENERGY required to get that advancement is another topic that was not faced by this author prior to 2017. But in the present paper we were able to add the consideration of ENERGY in addition to ENTROPY by replacing the b-lognormal probability densities previously used by a new curve, finite in the time, that we call LOGPAR. The logpar is made up by an ascending b-lognormal in the time between the birth and the peak of the living organism, followed by a descending parabola in the time between its peak and death. The logpar curve is not normalized to one: the area under the logpar may be any positive number since it represents the ENERGY requested by the organism to live over its entire lifetime "birth-to-death". In other words still, we mathematically demonstrate in this paper that just three instants (birth b, peak p and death d) must be assigned in order to make the mathematical logpar perfectly described. The history of the Roman Civilization fits to this description in that we not only know when Rome was funded (753 B.C.) and reached its peak (117 A.D.) but also when it collapsed (in the West), i.e. 476 A.D. Its ENERGY is then estimated in terms of money (Sestertii) and so we dare to say that our Evo-SETI Theory so extended adequately describes not only the Entropy but also the Energy of Rome. Jumping then to the life of a G2 star like the Sun, the relevant logpar power curve is described over the about 10 billion years of the Sun lifetime. And so we do for an M star lasting about 45 billion years. But we did not have the time to dig deeply into the astrophysics allowing such a production of ENERGY of the whole lifetime of these stars. That has to be differed to another paper. In conclusion, our invention of the logpar power curve provides a new, formidable mathematical tool for our Evo-SETI mathematical description of Life, History and SETI.