Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI

The discovery of new exoplanets makes us wonder where each new exoplanet stands along its way to develop life as we know it on Earth. Our Evo-SETI Theory is a mathematical way to face this problem. We describe cladistics and evolution by virtue of a few statistical equations based on lognormal probability density functions (pdf) in the time. We call b-lognormal a lognormal pdf starting at instant b (birth). Then, the lifetime of any living being becomes a suitable b-lognormal in the time. Next, our “Peak-Locus Theorem” translates cladistics: each species created by evolution is a b-lognormal whose peak lies on the exponentially growing number of living species. This exponential is the mean value of a stochastic process called “Geometric Brownian Motion” (GBM). Past mass extinctions were all-lows of this GBM. In addition, the Shannon Entropy (with a reversed sign) of each b-lognormal is the measure of how evolved that species is, and we call it EvoEntropy. The “molecular clock” is re-interpreted as the EvoEntropy straight line in the time whenever the mean value is exactly the GBM exponential. We were also able to extend the Peak-Locus Theorem to any mean value other than the exponential. For example, we derive in this paper for the first time the EvoEntropy corresponding to the Markov-Korotayev (2007) “cubic” evolution: a curve of logarithmic increase.


MOLECULAR CLOCK, mathematically derived as ENTROPY of the Running b-Lognormal (RbL) of the Lognormal Process L(t) starting at t=ts and having an arbitrarily assigned mean value m(t).
This is the best result of the Evo-SETI Theory, since it may be extended to Exoplanets and SETI.
Clearing the Maxima memory from previous calculations. (%i1) assume(n>0,t>ts,mu>0,sigma>0,sL>0,te>ts,Ne>0,delta[Ne]>0); (%o1) [ n > 0 , t > ts , µ > 0 , σ > 0 , sL > 0 , te > ts , Ne > 0 , δ Ne > 0 ] Defining the probability density function (pdf) of the LOGNORMAL stochastic process L(t) starting with probability one at the initial instant t=ts, corresponding to about 3.5 billion years ago on Earth. If the value of 3.8 billion years ago was chosen, all equations would remain just the same, and only slight numeric differences would occur. In our conventions, past times are denoted by negative values, zero is nowadays, and positive times will be the future on Earth. Thus, we assume ts = -3.5*10^9*years.
(%i6) def_L_pdf:L_pdf=(%e^(-(log(n)-M(t))^2/(2*(t-ts)*sL^2)))/(sqrt(2*%pi)*n*sqrt(t-ts)*sL); This is equation (18) of the paper. The independent variable is 0<n<infinity. The time at which the stochastic process starts is ts. M(t) is an auxiliary function of the time that will soon be re-expresses in terms of the Mean Value m(t) of the General Lognormal process L(t). sL (denoted sigma sub L in the paper) is a positive parameter that we will soon determine as a function of the two initial (ts, Ns) and three final (te, Ne delta[Ne]) boundary conditions on L(t).
Checking the normalization condition on the independent variable n.
(%i7) def_normalization_condition:'integrate(rhs(def_L_pdf),n,0,inf)=radcan(integrate(rhs(def_L_pdf),n (%o7) Discovering the MEAN VALUE FORMULA, i.e. the fundamental result of this paper, yielding the mean value m(t) as a function of M(t). The fact that this integral may be found EXACTLY was a surprise to this author. It may have a much more profound meaning UNKNOWN to this author at this time.
(%i10) M_vs_m:distrib(first(solve(log(def_mean_value),M(t)))); This formula simplifies at the initial instant ts: Finding the k-th MOMENT of the General Lognormal pdf of L(t), i.e. finding ALL MOMENTS of L(t). With k=0 we get the normalization condition of L(t) again. With k=1 we get the mean value m(t) of L(t) again. With k=2 we get the mean value of the square of L(t). And so on.

Proving the PEAK-LOCUS THEOREM for an arbitrarily assigned Mean Value m(t)
Recalling the peak abscissa and ordinate of any b-lognormal.

ENTROPY of the Running b-lognormal
Shannon ENTROPY of the Running b-Lognormal in bits.