## In 1961 Frank Drake introduced his famous “Drake equation” described at the web site http://en.wikipedia.org/wiki/Drake_equation. It yields the number *N* of communicating civilizations in the Galaxy: ## In 1961 Frank Drake introduced his famous “Drake equation” described at the web site http://en.wikipedia.org/wiki/Drake_equation. It yields the number *N* of communicating civilizations in the Galaxy: -
## Frank Donald Drake (b. 1930)
## The meaning of the seven factors in the Drake equation is well-known. ## The meaning of the seven factors in the Drake equation is well-known. ## The middle factor **fl** is Darwinian Evolution. ## In the classical Drake equation the seven factors are just POSITIVE NUMBERS. And the equation simply is the PRODUCT of these seven positive numbers. ## It is claimed here that Drake’s approach is too “simple-minded”, since it does NOT yield the ERROR BAR associated to each factor!
## If we want to associate an ERROR BAR to each factor of the Drake equation then… ## If we want to associate an ERROR BAR to each factor of the Drake equation then… ## … we must regard each factor in the Drake equation as a RANDOM VARIABLE. ## Then the number *N* of communicating civilizations also becomes a random variable. ## This we call the STATISTICAL DRAKE EQUATION and studied in our mentioned reference paper of 2010 (Acta Astronautica, Vol. 67 (2010), pages 1366-1383)
## Denoting each random variable by capitals, the STATISTICAL DRAKE EQUATION reads ## Where the D sub i (“D from Drake”) are the 7 random variables, and *N* is a random variable too (“to be determined”).
## Consider the statistical equation ## Consider the statistical equation ## This is the generalization of our Statistical Drake Equation to the product of ANY finite NUMBER of positive random variables. ## Is it possible to determine the statistics of *N* ? ## Rather surprisingly, the answer is “yes” !
## First, you obviously take the natural log of both sides to change the finite product into a finite sum ## First, you obviously take the natural log of both sides to change the finite product into a finite sum ## Second, to this finite sum one can apply the CENTRAL LIMIT THEOREM OF STATISTICS. It states that, in the limit for an infinite sum, the distribution of the left-hand-side is NORMAL. ## This is true WHATEVER the distributions of the random variables in the sum MAY BE.
## So, the random variable on the left is NORMAL, i.e. ## So, the random variable on the left is NORMAL, i.e. ## Thus, the random variable *N* under the log must be LOG-NORMAL and its distribution is determined! ## One must, however, determine the mean value and variance of this log-normal distribution in terms of the mean values and variances of the factor random variables. This is DIFFICULT. But it can be done, for example, by a suitable numeric code that this author wrote in MathCad language.
## Our Statistical Drake Equation, now Generalized to any number of factors, embodies as a special case the Statistical Drake Equation with just 7 factors. ## Our Statistical Drake Equation, now Generalized to any number of factors, embodies as a special case the Statistical Drake Equation with just 7 factors. ## The conclusion is that the random variable *N* (the number of communicating ET Civilizations in the Galaxy) is LOG-NORMALLY distributed. ## The classical “old pure-number Drake value” of *N* is now replaced by the MEAN VALUE of such a log-normal distribution. ## But we now also have an ERROR BAR around it !
## The Statistical Drake Equation ## Acta Astronautica, V. 67 (2010), p. 1366-1383.
## Acta Astronautica, Vol.101 (2014), pag. 67-80.
## (OLEB), Vol. 41 (2011), pages 609-619.
## Life on Earth evolved since 3.5 billion years ago. ## The number of Species GROWS EXPONENTIALLY: assume that today 50 million species live on Earth ## Then:
## Life on Earth evolved since 3.5 billion years ago. ## The number of Species GROWS EXPONENTIALLY: assume that today 50 million species live on Earth ## Then: ## with:
## Each b-lognormal has its peak on the exponential. ## PRACTICALLY an “Envelope”, though not so formally.
## A Mathematical Model for Evolution and SETI ## A Mathematical Model for Evolution and SETI ## Origins of Life and Evolution of Biospheres ## (OLEB), Vol. 41 (2011), pages 609-619.
## SETI, Evolution and Human History Merged into a Mathematical Model. ## SETI, Evolution and Human History Merged into a Mathematical Model. ## International Journal of ASTROBIOLOGY, ## Vol. 12, issue 3 (2013), pages 218-245.
## Evolution and History in a new ## Evolution and History in a new ## “Mathematical SETI” model. ## ACTA ASTRONAUTICA, Vol. 93 (2014), pages 317-344. Online August 13, 2013.
## A Mathematical Model for Evolution and SETI ## A Mathematical Model for Evolution and SETI ## Origins of Life and Evolution of Biospheres ## (OLEB), Vol. 41 (2011), pages 609-619.
## SETI, Evolution and Human History Merged into a Mathematical Model. ## SETI, Evolution and Human History Merged into a Mathematical Model. ## International Journal of ASTROBIOLOGY, ## Vol. 12, issue 3 (2013), pages 218-245.
## Evolution and History in a new ## Evolution and History in a new ## “Mathematical SETI” model. ## ACTA ASTRONAUTICA, Vol. 93 (2014), pages 317-344. Online August 13, 2013.
## New Evo-SETI Results about Civilizations and ## New Evo-SETI Results about Civilizations and ## Molecular Clock. ## International Journal of Astrobiology, in press (2016), available on line on March 28, 2016.
## New Evo-SETI Results about Civilizations and ## New Evo-SETI Results about Civilizations and ## Molecular Clock. ## International Journal of Astrobiology, in press (2016), available on line on March 28, 2016.
**LOGNORMAL PROCESSES WITH ARBITRARY MEAN** **LOGNORMAL PROCESSES WITH ARBITRARY MEAN**
**PEAK-LOCUS THEOREM WITH ARBITRARY MEAN**
**EVO-ENTROPY WITH ARBITRARY MEAN** **EVO-ENTROPY WITH ARBITRARY MEAN**
## Alexander V. Markov and Andrey V. Korotayev have demonstrated that changes in biodiversity through the Phanerozoic correlate much better with **hyperbolic model **than with exponential and logistic models (traditionally used in population biology). The latter models imply that changes in diversity are guided by a first-order positive feedback (more ancestors, more descendants) and/or a negative feedback arising from resource limitation. ## Alexander V. Markov and Andrey V. Korotayev have demonstrated that changes in biodiversity through the Phanerozoic correlate much better with **hyperbolic model **than with exponential and logistic models (traditionally used in population biology). The latter models imply that changes in diversity are guided by a first-order positive feedback (more ancestors, more descendants) and/or a negative feedback arising from resource limitation. **Hyperbolic model implies a second-order positive feedback. **The hyperbolic pattern of the world population growth has been demonstrated by Korotayev to arise from a second-order positive feedback between the population size and the rate of technological growth.
## According to Korotayev and Markov, the hyperbolic character of biodiversity growth can be similarly accounted for by a feedback between the diversity and community structure complexity. They suggest that the similarity between the curves of biodiversity and human population probably comes from the fact that both are derived from the interference of the hyperbolic trend with cyclical and stochastic dynamics.
**Dostları ilə paylaş:** |