The statistical Drake equation and A.M.Lyapunov theorem

in problem of search for extraterrestrial intelligence, part II

Claudio Maccone

SETI Permanent Study Group, International Academy of Astronautics

Italy

The study is connected with problems of mathematical modelling in the theory of Search for Extraterrestrial Intelligence [1-14]. In this area at present it is known the Frank Drake equation. But it is very important to extend this model with using brilliant theorem of statistics: Central Limit Theorem in form of Alexander Lyapunov (or in form of Jarl Waldemar Lindeberg). In work the generalized model is developed in form of the statistical Drake equation. This article is prepared on materials of paper (IV IAA International Symposium, July, 2009, Italy), with recommendation of Dr.S.Santoli, Member of International Scientific Committee, for special issue of International scientific Journal. Here it is presented second part (p.II) of the article. First part (p.I) of this article was published earlier (p.I [14]). Besides the used here, in p.II, notations, numbers of paragraphs, tables, formulas, figures,: are corresponding to ones in p.I [14] and continuing of it.

The lognormal distribution is the distribution of the number N of extraterrestrial civilizations in the Galaxy

The CLT may of course be extended to products of random variables upon taking the logs of both sides, just as we did in equation (3) from [14]. It then follows that the exponent random variable, like Y in (6) from [14], tends to a normal random variable, and, as a consequence, it follows that the base random variable, like N in (6) from [14], tends to a lognormal random variable.

The "Data Enrichment Principle" as the best CLT consequence upon

the Statistical Drake equation (any number of factors allowed)

As a fitting climax to all the statistical equations developed so far, let us now state our "Data Enrichment Principle". It simply states that "The Higher the Number of Factors in the Statistical Drake equation, The Better".

Put in this simple way, it simply looks like a new way of saying that the CLT lets the random variable Y approach the normal distribution when the number of terms in the sum (4) from [14] approaches infinity. And this is the case, indeed. However, our "Data Enrichment Principle" has more profound methodological consequences that we cannot explain now, but hope to describe more precisely in one or more coming papers.

In Conclusions we note:

We have sought to extend the classical Drake equation to let it encompass Statistics and Probability, with using of A.M.Lyapunov theorem and its brilliant mathematical tool.

This approach appears to pave the way to future, more profound investigations intended not only to associate "error bars" to each factor in the Drake equation, but especially to increase the number of factors themselves. In fact, this seems to be the only way to incorporate into the Drake equation more and more new scientific information as soon as it becomes available. In the long run, the Statistical Drake equation might just become a huge computer code, growing up in size and especially in the depth of the scientific information it contained. It would thus be Humanity's first "Encyclopaedia Galactic".

Of course, to extend the Drake equation to Statistics, it was necessary to use a mentioned mathematical apparatus that is more sophisticated than just the simple product of seven numbers.

When this author had the honour and privilege to present his results at the SETI Institute on April 11^th, 2008, in front of an audience also including Professor Frank Drake, he felt he had to add these words: "My apologies, Frank, for disrupting the beautiful simplicity of your equation".

Acknowledgements. The author is grateful to Drs. Jill Tarter, Paul Davies, Seth Shostak, Doug Vakoch, Tom Pierson, Carol Oliver, Paul Shuch and Kathryn Denning for attending his first presentation ever about these topics at the "Beyond" Center of the University of Arizona at Phoenix on February 8^th, 2008. He also would like to thank Dan Werthimer and his School of SETI young experts for keeping alive the interplay between experimental and theoretical SETI. But the greatest "thanks" goes of course to the Teacher to all of us: Professor Frank D.Drake, whose equation opened a new way of thinking about the past and the future of Humans in the Galaxy.

References

1.             http://en.wikipedia.org/wiki/Drake_equation

2.             http://en.wikipedia.org/wiki/SETI

3.             http://en.wikipedia.org/wiki/Astrobiology

4.             http://en.wikipedia.org/wiki/Frank_Drake

5.             Athanasios Papoulis and S. Unnikrishna Pillai, "Probability, Random Variables and Stochastic Processes", Fourth Edition, Tata McGraw-Hill, New Delhi, 2002, ISBN 0-07-048658-1.

6.             http://en.wikipedia.org/wiki/Gamma_distribution

7.             http://en.wikipedia.org/wiki/Central_limit_theorem

8.             http://en.wikipedia.org/wiki/Cumulants

9.             http://en.wikipedia.org/wiki/Median

10.         Jeffrey Bennett and Seth Shostak, "Life in the Universe", Second Edition, Pearson - Addison-Wesley, San Francisco, 2007, ISBN 0-8053-4753-4. See in particular page 404.

11.         F.Drake. Intelligent life in space. Macmillan, New York, 1962.

12.         P.C.W.Davies. Space and time in the modern universe. Cambridge Univ. Press, Cambridge, Mass., 1977.

13.         C.Sagan,: Communication with extraterrestrial intelligence. The M.I.T. Press Cambridge, Mass., 1973.

14.         C.Maccone. The statistical Drake equation and A.M.Lyapunov theorem in problem of search for ExtraTerrestrial intelligence, part I, Int. Russian-American Journal "Actual problems of aviation and aerospace systems", No.1(32), v.16, 2011, 38-63.

 

 

 

 

 

Claudio Maccone, a mathematical physicist and a space scientist (Turin, Italy). SETI was his main interest over the last 30 years and he is Co-Chair of the SETI Permanent Study Group of the IAA. He proposed to replace the FFT by the KLT, suggested crater Daedalus (Moon Farside) as the best RFI-free observing site for radio astronomy, wrote a book about the FOCAL space missions to 550 AU to exploit the radio magnification provided by the Sun as a gravitational lens, and discovered the statistical extension of the Drake equation described in this paper.šš URL: http://www.maccone.com/