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This paper is profoundly innovative for the Evo-SETI (Evolution and SETI) mathematical theory. While this author's previous papers were all based on the notion of a b-lognormal, that is a probability density function in the time describing one's life between birth and ‘senility’ (the descending inflexion point), in this paper the b-lognormals range between birth and peak only, while a descending parabola covers the lifespan after the peak and down to death. The resulting finite curve in time is called a LOGPAR, a nickname for ‘b-LOGnormal and PARabola’. The advantage of such a formulation is that three variables only (birth, peak and death) are sufficient to describe the whole Evo-SETI theory and the senility is discarded forever and so is the normalization condition of b-lognormals: only the shape of the b-lognormals is kept between birth and peak, but not its normalization condition.
In addition, further advantages exist:
1) The notion of ENERGY becomes part of Evo-SETI theory. This is in addition to the notion of ENTROPY already contained in the theory as the Shannon Information Entropy of b-lognormals, as it was explored in this author's previous papers. Actually, the LOGPAR may now be regarded as a POWER CURVE, i.e. a curve expressing the power of the living being to which it refers. And this power is to be understood both in the strict sense of physics (i.e. a curve measured in Watts) and in the loose sense of ‘political power’ if the logpar refers to a Civilization.
Then the integral in the time of this power curve is, of course, the ENERGY either absorbed or produced by the physical phenomenon that the LOGPAR is describing in the time. For instance, if the logpar shows the time evolution of the Sun over about 10 billion years, the integral of such a curve is the energy produced by the Sun over the whole of its lifetime. Or, if the logpar describes the life of a man, the integral is the energy that this man must use in order to live.
2) The PRINCIPLE OF LEAST ENERGY, reminiscent of the Principle of Least Action, i.e. the key stone to all Physics, also enters now into the Evo-SETI Theory by virtue of the so-called LOGPAR HISTORY FORMULAE, expressing the b-lognormal's mu and sigma directly in terms of the three only inputs b, p, d. The optimization of the lifetime of a living creature, or of a Civilization, or of a star, is obtained by setting to zero the first derivative of the area under the logpar power curve with respect to sigma. That yields the best value of both mu and sigma fulfilling the Principle of Least Energy for Evo-SETI Theory.
3) We also derive for the first time a few more mathematical equations related to the ‘adolescence’ (or ‘puberty’) time, i.e. the time when the living organism acquires the capability of producing offsprings. This time is defined as the abscissa of ascending inflection point of the b-lognormal between birth and peak. In addition, we prove that the straight line parallel to the time axis and departing from the puberty time comes to mean the ‘Fertility Span’ in between puberty and the EOF (End-Of-Fertility time), which is where the above straight line intersects the descending parabola. All these new results apply well to the description of Man as the living creature to which our Evo-SETI mathematical theory perfectly applies.
In conclusion, this paper really breaks new mathematical ground in Evo-SETI Theory, thus paving the way to further applications of the theory to Astrobiology and SETI.
In two recent papers (Maccone 2013, 2014) as well as in the book (Maccone 2012), this author described the Evolution of life on Earth over the last 3.5 billion years as a lognormal stochastic process in the increasing number of living Species. In (Maccone 2012, 2013), the process used was ‘Geometric Brownian Motion’ (GBM), largely used in Financial Mathematics (Black-Sholes models). The GBM mean value, also called ‘the trend’, always is an exponential in time and this fact corresponds to the so-called ‘Malthusian growth’ typical of population genetics. In (Maccone 2014), the author made an important generalization of his theory by extending it to lognormal stochastic processes having an arbitrary trend m L (t), rather than just a simple exponential trend as the GBM have.
The author named ‘Evo-SETI’ (Evolution and SETI) his theory inasmuch as it may be used not only to describe the full evolution of life on Earth from RNA to modern human societies, but also the possible evolution of life on exoplanets, thus leading to SETI, the current Search for ExtraTerrestrial Intelligence. In the Evo-SETI Theory, the life of a living being (let it be a cell or an animal or a human or a Civilization of humans or even an ET Civilization) is represented by a b-lognormal, i.e. a lognormal probability density function starting at a precise instant b (‘birth’) then increasing up to a peak-time p, then decreasing to a senility-time s (the descending inflexion point) and then continuing as a straight line down to the death-time d (‘finite b-lognormal’).
Having so said, the present paper describes the further mathematical advances made by this author in 2014–2015, and is divided in two halves: Part One, devoted to new mathematical results about the History of Civilizations as b-lognormals, and
Part Two, about the applications of the Evo-SETI Theory to the Molecular Clock, well known to evolutionary geneticists since 50 years: the idea is that our EvoEntropy grows linearly in time just as the molecular clock.
Summarizing the new results contained in this paper: In Part One, we start from the History Formulae already given in (Maccone 2012, 2013) and improve them by showing that it is possible to determine the b-lognormal not only by assigning its birth, senility and death, but rather by assigning birth, peak and death (BPD Theorem: no assigned senility). This is precisely what usually happens in History, when the life of a VIP is summarized by giving birth time, death time, and the date of the peak of activity in between them, from which the senility may then be calculated (approximately only, not exactly). One might even conceive a b-scalene (triangle) probability density just centred on these three points (b, p, d) and we derive the relevant equations. As for the uniform distribution between birth and death only, that is clearly the minimal description of someone's life, we compare it with both the b-lognormal and the b-scalene by comparing the Shannon Entropy of each, which is the measure of how much information each of them conveys. Finally we prove that the Central Limit Theorem (CLT) of Statistics becomes a new ‘E-Pluribus-Unum’ Theorem of the Evo-SETI Theory, giving formulae by which it is possible to find the b-lognormal of the History of a Civilization C if the lives of its Citizens C i are known, even if only in the form of birth and death for the vast majority of the Citizens.
In Part Two, we firstly prove the crucial Peak-Locus Theorem for any given trend m L (t) and not just for the GBM exponential. Then we show that the resulting Evo-Entropy grows exactly linearly in time if the trend is the exponential GMB trend.
In addition, three Appendixes (online) with all the relevant mathematical proofs are attached to this paper. They are written in the Maxima language, and Maxima is a symbolic manipulator that may be downloaded for free from the web.
In conclusion, this paper further increases the huge mathematical spectrum of applications of the Evo-SETI Theory to prepare Humans for the first Contact with an Extra-Terrestrial Civilization.
In a series of recent papers and in a book, this author put forward a mathematical model capable of embracing the search for extra-terrestrial intelligence (SETI), Darwinian Evolution and Human History into a single, unified statistical picture, concisely called Evo-SETI. The relevant mathematical tools are:
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 (Black–Sholes 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.
The probability distributions known as b-lognormals, i.e. lognormals starting at a certain positive instant b>0 rather than at the origin. These b-lognormals were then forced by us to have their peak value located on the exponential mean-value curve of the GBM (Peak-Locus theorem). In the framework of Darwinian Evolution, the resulting mathematical construction was shown to be what evolutionary biologists call Cladistics.
The (Shannon) entropy of such b-lognormals 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 b-lognormals 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.
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 extra-terrestrial civilizations existing in the Galaxy (as a consequence of the central limit theorem of statistics).
But the most striking new result is that the well-known ‘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 Evo-SETI model, because they both grew linearly in time since the origin of life.
Furthermore, we apply our Evo-SETI 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.
Finally, we show that the Markov & Korotayev (2007, 2008) model for Darwinian Evolution identifies with an Evo-SETI model for which the mean value of the underlying lognormal stochastic process is a cubic function of the time.
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 number N 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.
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