Abstract Darwinian evolution over the last 3.5 billion years was an increase in the number of living species from 1 (RNA?) to the current 50 million. This increasing trend in time looks like being exponential, but one may not assume an exactly exponential curve since many species went extinct in the past, even in mass extinctions. Thus, the simple exponential curve must be replaced by a stochastic process having an exponential mean value. Borrowing from financial mathematics (“Black-Sholes models”), this “exponential” stochastic process is called Geometric Brownian Motion (GBM), and its probability density function (pdf) is a lognormal (not a Gaussian) (Proof: see Ref. [3], Chapter 30, and Ref. [4]). Lognormal also is the pdf of the statistical number of communicating ExtraTerrestrial (ET) civilizations in the Galaxy at a certain fixed time, like a snapshot: this result was found in 2008 by this author as his solution to the Statistical Drake Equation of SETI (Proof: see Ref. [1]). Thus, the GBM of Darwinian evolution may also be regarded as the extension in time of the Statistical Drake equation (Proof: see Ref. [4]). But the key step ahead made by this author in his Evo-SETI (Evolution and SETI) mathematical model was to realize that LIFE also is just a b-lognormal in time: every living organism (a cell, a human, a civilization, even an ET civilization) is born at a certain time b (“birth”), grows up to a peak p (with an ascending inflexion point in between, a for adolescence), then declines from p to s (senility, i.e. descending inflexion point) and finally declines linearly and dies at a final instant d (death). In other words, the infinite tail of the b-lognormal was cut away and replaced by just a straight line between s and d, leading to simple mathematical formulae (“History Formulae”) allowing one to find this “finite b-lognormal” when the three instants b, s, and d are assigned. Next the crucial Peak-Locus Theorem comes. It means that the GBM exponential may be regarded as the geometric locus of all the peaks of a one-parameter (i.e. the peak time p) family of b-lognormals. Since b-lognormals are pdf-s, the area under each of them always equals 1 (normalization condition) and so, going from left to right on the time axis, the b-lognormals become more and more “peaky”, and so they last less and less in time. This is precisely what happened in Human History: civilizations that lasted millennia (like Ancient Greece and Rome) lasted just centuries (like the Italian Renaissance and Portuguese, Spanish, French, British and USA Empires) but they were more and more advanced in the “level of civilization”. This “level of civilization” is what physicists call ENTROPY. In Refs. [3] and [4], this author proved that, for all GBMs, the (Shannon) Entropy of the b-lognormals in his Peak-Locus Theorem grows LINEARLY in time. At last, we reach the new, original result justifying the publication of this paper. The Molecular Clock, well known to geneticists since 50 years, shows that the DNA base-substitutions occur LINEARLY in time since they are neutral with respect to Darwinian selection. In simple words: DNA evolved by obeying the laws of quantum physics only (microscopic laws) and not by obeying assumed “Darwinian selection laws” (macroscopic laws). This is Kimura׳s neutral theory of molecular evolution. The conclusion of this paper is that the Molecular Clock and the b-lognormal Entropy are the same thing. And, on exoplanets, molecular evolution is proceeding at about the same rate as it did proceed on Earth: rather independently of the physical conditions of the exoplanet, if the DNA had the possibility to evolve in water initially.

Highlights • Evolution and SETI (Evo-SETI) in time are a Geometric Brownian Motion (GBM). • This author published papers and a book (“Mathematical SETI”, Springer, 2012) on this. • b-lognormals are lognormal probability densities in time starting at a positive instant b>0. • Living “Species” are b-lognormals having peaks on the GBM exponential increasing in time. • Their (Shannon) Entropy increases LINEARLY in time just like the MOLECULAR CLOCK.