Wednesday, December 10, 2014

The Impacts of Mass Extinctions on SETI Considered

Lognormals for SETI, Evolution and Mass Extinctions




In a series of recent papers (Refs. [1], [2], [3], [4], [5], [7] and [8]) and in a book (Ref. [6]), this author suggested a new mathematical theory capable of merging Darwinian Evolution and SETI into a unified statistical framework. In this new vision, Darwinian Evolution, as it unfolded on Earth over the last 3.5 billion years, is defined as just one particular realization of a certain lognormal stochastic process in the number of living species on Earth, whose mean value increased in time exponentially. SETI also may be brought into this vision since the number of communicating civilizations in the Galaxy is given by a lognormal distribution (Statistical Drake Equation).

Now, in this paper we further elaborate on all that particularly with regard to two important topics:

1)  The introduction of the general lognormal stochastic process L (t ) whose mean value may be an arbitrary continuous function of the time, m (t ), rather than just the exponential View the MathML sourcemGBM(t)=N0eμt typical of the Geometric Brownian Motion (GBM). This is a considerable generalization of the GBM-based theory used in Refs. [1], [2], [3], [4], [5], [6], [7] and [8].
2) The particular application of the general stochastic process L(t) to the understanding of Mass Extinctions like the K-Pg one that marked the dinosaurs׳ end 65 million years ago. We first model this Mass Extinction as a decreasing Geometric Brownian Motion (GBM) extending from the asteroid׳s impact time all through the ensuing “nuclear winter”. However, this model has a flaw: the “final value” of the GBM cannot have a horizontal tangent, as requested to enable the recovery of life again after this “final extinction value”.
3)  That flaw, however, is removed if the rapidly decreasing mean value function of L(t) is the left branch of a parabola extending from the asteroid׳s impact time all through the ensuing “nuclear winter” and up to the time when the number of living species on Earth started growing up again, as we show mathematically in Section 3.

In conclusion, we have uncovered an important generalization of the GBM into the general lognormal stochastic process L(t), paving the way to a better, future understanding the evolution of life on Exoplanets on the basis of what Evolution unfolded on Earth in the last 3.5 billion years. That will be the goal of further research papers in the future.

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