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.