Two Power Curves Yielding the Energy of a Lifetime in Evo-SETI Theory

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.