Abstract

This paper presents a new mathematical model for the ENERGY that a living being needs in order to live its whole life between birth and death. This also applies to a civilization made up by many living beings. The model is based on a LOGELL POWER CURVE, which is a curve in time made up by a lognormal probability density between birth and peak, followed by an ellipse between peak and death (LOGELL means LOGnormal plus ELLipse). We derive analytic equations yielding the ENERGY in terms of three free parameters only: the time of birth b, the time of the power peak p, and the time of death, d. The author’s previously published papers about his so-called Evo-SETI Theory (Evo-SETI stands for Evolution and SETI) cover the biological evolution over the last 3.5 billion years described as an increase in the number of living species from one (RNA) to the current (say) 50 million. Past mass extinctions make this evolution become a stochastic process having an exponential mean value, called Geometric Brownian Motion (GBM). In those papers, a lifetime, rather than a logell, was a b-lognormal, i.e., a lognormal starting at instant b (birth) and descending straight to death at its descending inflexion point. Our mathematical discovery of the Peak-Locus Theorem showed that the GBM exponential is the geometric locus of all the peaks of the b-lognormals. Since b-lognormals are probability densities, the area under each of them always equals 1 (normalization condition); going from left to right on the time axis, the b-lognormals become more and more “peaky,” so they last less and less in time. This “level of civilization” is what physicists call (Shannon) ENTROPY of information, meaning that the higher Species have higher information content than the lower Species. This author also proved mathematically that for all GBMs, the (Shannon) Entropy of the b-lognormals grows LINEARLY in time. The Molecular Clock, 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. 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. Finally, this linearly growing entropy is just the new EvoSETI SCALE to measure the evolution of life on Exoplanets (measured in bits). In conclusion, our invention of the logell power curve, described in this paper, provides a new mathematical tool for our Evo-SETI mathematical description of Life, History and SETI.

Highlights

  • Introduction toLogell “Finite Lifetime” CurvesThe starting idea is easy: we seek to represent the lifetime of any living being by virtue of just three points in the time: birth, peak, death (BPD)

  • Given the input triplet (b, p, d ), (30) immediately yields the exact σ of the elliptical left part of the logell curve. It was discovered by this author on September 4, 2018, and led to this paper but to the introduction of the ENERGY spent in a lifetime by a living creature, or by a whole civilization whose “power-vs-time” behaviour is given by the logell curve, as we will understand better in the coming sections of this paper

  • LOGELL curves, just as LOGPAR curves, have greatly simplified the description of any finite phenomenon in time like the lifetime of a cell, or a human, or a civilization or even like an ET civilization

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Summary

Introduction to Logell “Finite Lifetime” Curves

The starting idea is easy: we seek to represent the lifetime of any living being by virtue of just three points in the time: birth, peak, death (BPD). After the peak, the ellipse plunges down until it reaches the time axis at the death time d with a perfectly vertical tangent This new definition of death time d is different from the old definition of d applying to b-lognormals alone, as we published prior to 2017. Representation of the History of the Roman civilization as a LOGPAR finite curve This logpar curve in time is made up by a b-lognormal in between birth and peak, and a parabola in between peak and death. Formulae expressing the b-lognormal’s two parameters μ (a real number) and σ (a positive number) in terms of the three assigned real numbers b , p and d , with the condition b < p < d

Finding the Ellipse Equation of the Right Part of the Logell
Area under the Ellipse on the Right Part of the Logell between Peak and Death
Area under the Full Logell Curve between Birth and Death
Exact “History Equations” for each Logell Curve
Considerations on the Logell History Equations
10. History of Rome as an Example of How to Use the Logell History Formulae
Part 2: Energy as the Area under Logell Power Curves
13. Mean Power in a Logell Lifetime
Part 5: Conclusions
16. Conclusions about Evo-SETI Theory

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