Mathematical Big History

1
Mathematical Big History Evo-SETI Theory

• Claudio Maccone
• Director for Scientific Space Exploration, Intl
  Acad. Astronautics,
• Chair of the SETI Permanent Committee of the IAA,
• Retired Director, Istituto Nazionale di
  Astrofisica (INAF), Italy
• E-mail clmaccon_at_libero.it Home Page
  www.maccone.com
• INTERNATIONAL BIG HISTORY ASSOCIATION 2016
  Conference
• University of Amsterdam, The Netherlands, July
  15-17, 2016.

2
700-pages BOOK aboutMathematical SETI
3
1959 Giuseppe Cocconi Philip Morrisons
seminal paper Searching for Interstellar
Communications,Nature 184, 844-846 (19
September 1959)  doi10.1038/184844a0
History of SETI
Cocconi
Morrison
4
1960 First search at 1420 MHz (neutral hydrogen
line) by Frank Drake, at the National Radio
Astronomy Observatory in Green Bank, West
Virginia (Project Ozma, 85 feet 25.908 meter
antenna).
History of SETI
Frank Drake
5
2009 NASAs Kepler space mission gt3000
planets.
History of SETI
6
FOCAL space mission beyond 550 AU 3.17 light
days 14 times the Sun-Pluto distance
exploiting the Sun as a GRAVITATIONAL LENS to
obtain MAGNIFIED RADIO PICTURES of an Alien
Civilization and its planet.It would allow us to
read the car plates on that planet!
Future of SETI FOCAL mission
7
Part 1 STATISTICAL DRAKE QUATIONPart 2
STATISTICAL BIG HISTORY EQUATIONPart 3 LIFE
as a b-LOGNORMAL (bbirth) Part 4 Darwinian
EXPONENTIAL GROWTH Part 5 Geometric
Brownian Motion (GBM) Part 6 Darwinian
EVOLUTION as a GBM Part 7 ENTROPY as
EVOLUTION MEASURE
TALKs SCHEME 1/2
8
Part 8 Aztecs vs. Spaniards ENTROPY in
1519Part 9 Future up to 10 million years
Part 10 Mass ExtinctionsPart 11 ARBITRARY
MEAN rather than GBMPart 12 Markov-Korotayev
CUBIC process
TALKs SCHEME 2/2
9
Part 1THE STATISTICALDRAKE EQUATION
10
Deterministic Drake Equation /1

• 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)

11
Deterministic Drake Equation /2

• 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 Drakes approach is too
  simple-minded, since it does NOT yield the
  ERROR BAR associated to each factor!

12
STATISTICAL Drake Equation /1

• 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)

13
STATISTICAL Drake Equation /2

• 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).

14
Extending the STATISTICAL Drake Equation to ANY
NUMBER OF FACTORS /1

• 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 !

15
Extending the STATISTICAL Drake Equation to ANY
NUMBER OF FACTORS /2

• 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.

16
SOLVING the STATISTICAL Drake Equation for
INFINITELY MANY FACTORS

• 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.

17
LOGNORMAL pdf
18
Conclusion about the Drake equationThe number
of Signaling Civilizations is LOGNORMALLY
distributed

• 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 !

19
REFERENCE PAPER

• The Statistical Drake Equation
• Acta Astronautica, V. 67 (2010), p. 1366-1383.

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Part 2THE BIG HISTORY EQUATION
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REFERENCE PAPER

• SETI as a Part of Big History
• Acta Astronautica, Vol.101 (2014), pag. 67-80.

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THE BIG HISTORY EQUATION is 1) The DRAKE
equation (yielding the Evolution of Life
on Earth since -3.5 billion years ago) 2) PLUS
the BEGINNING TERM i.e. the cosmological
evolution of the Universe PRIOR to -3.5 billion
years.

• In fact, the WHOLE UNIVERSE (and NOT just the
  Milky Way) evolved since the Big Bang

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BIG HISTORY EQUATION

• Where is the (unknown) number of
  Galaxies
• supposed to exist in the entire Universe.
• The probability distribution of the new random
  variable Number of Civilizations in the
  Universe is a lognormal, but we dont know when
  it started...
• at least until we will know a sufficient number
  of Alien Civilizations!

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BIG HISTORY an exponential in the growing
number of Civilizations?We wont know until...
The SETI scientists will succeedin finding the
first few Alien Civilizations
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BIG HISTORY an exponential in the growing
number of Civilizations?
26
BIG HISTORY an exponential in the growing
number of Civilizations?
27
Part 3b-LOGNORMALSas the LIFE-TIME of a
cell, of an animal, of a human, a civilization
(f sub i) even ET (f sub L)
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LIFE as a FINITE b-LOGNORMAL

• The lifetime of a cell, an animal, a human, a
  civilization can be modeled as a b-lognormal with
  tail REPLACED at senility by the descending
  TANGENT. The interception at time axis is
  DEATHd.

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LIFE as a FINITE b-LOGNORMAL

• The equation of a INFINITE b-lognormal is

• The lifetime of a cell, an animal, a human, a
  civilization can be modeled as a FINITE
  b-lognormal namely an infinite b-lognormal whose
  TAIL has been REPLACED at senility by the
  descending TANGENT STRAIGHT LINE. The
  interception of this straight line at time axis
  is DEATHd.

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LIFE as a FINITE b-LOGNORMAL

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HISTORY FORMULAE

• Let a increasing inflexion, s decreasing
  inflexion.
• Then any b-lognormal has birth time (b),
  adolescence time (a), peak time (p) and senility
  time (s).
• Romes civilization b-753, a-146, p59,
  s235.
• HISTORY FORMULAE GIVEN (b, s, d) it is always
  possible to compute the corresponding b-lognormal
  by virtue of the following two HISTORY FORMULAE

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LIFE as INFINITE b-LOGNORMAL

• Let a increasing inflexion, s decreasing
  inflexion.
• Then any b-lognormal has birth time (b),
  adolescence time (a), peak time (p) and senility
  time (s).
• Romes civilization b-753, a-146, p59,
  s235.

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LIFE as a FINITE b-LOGNORMAL

• ANY FINITE LIFE may be modeled as a b-lognormal
  with tail REPLACED at senility by the descending
  TANGENT. The interception at time axis is
  DEATHd.
• (e.g. for Rome civilization one has b-753,
  d476)

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FINITE b-LOGNORMAL Civilizations
b birth time s senility time d death time p peak time
Ancient Greece 600 BC Mediterranean Greek coastal expansion. 323 BC Alexander the Greats death. Hellenism starts. 30 BC Cleopatras death last Hellenistic queen 434 BC Pericles Age. Democracy peak. Arts and science peak.
Ancient Rome 753 BC Rome founded. Italy seized by Romans by 270 BC. 235 AD Military Anarchy starts. Rome not capital. 476 AD Western Roman Empire ends. Dark Ages start. 59 AD Christianity preached in Rome by Saints Peter and Paul against slavery
Renaissance Italy 1250 Frederick II dies Middle Ages end. Free Italian towns. 1564 Council of Trent. Tough Catholic and Spanish Rule. 1660 1600 G. Bruno executed, 1642 Galileo dies, 1667 Cimento Academy shut. 1450 Renaissance art and architecture. Science. Copernican revolution.
Portuguese Empire 1419 Madeira island discovered 1822 Brazil independent, colonies retained. 1999 Last colony Macau lost. 1716 Black slave trade to Brazil at its peak.
Spanish Empire 1492 Columbus discovers America. 1805 Spanish fleet lost at Trafalgar. 1898 Last colonies lost to the USA. 1741 California to be settled by Spain, 1759-76.
French Empire 1524 Verrazano first in New York bay. 1815 Napoleon defeated at Waterloo. 1962 Algeria lost, as most colonies. 1732 French Canada and India conquest tried.
British Empire 1588 Spanish Armada Defeated. 1914 World War One won at a high cost. 1973 The UK joins European EEC. 1868 Victorian Age. Science Faraday and Maxwell.
USA Empire 1898 Philippines, Cuba, Puerto Rico seized. 2001 9/11 terrorist attacks. 2050 ? Will the USA yield to China ? 1973 Moon landings, 1969-72
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ANCIENT GREECE (600 BC - 323 BC - 30 BC)
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ANCIENT ROME (753 BC 235 AD - 476 AD)
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ITALIAN RENAISSANCE (1250 - 1564 - 1660)
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PORTUGAL (1419 - 1822 - 1999)
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SPAIN (1492 - 1805 - 1898)
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FRANCE (1524 - 1870 - 1962)
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BRITAIN (1588 - 1914 - 1974)
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USA (1898 - 2001 - ? 2050 ?)
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FINITE b-LOGNORMAL Civilizations
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Two EXPONENTIAL ENVELOPES

• Given TWO POINTS with coordinates
• The EXPONENTIAL must have

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Part 4PEAK-LOCUS THEOREMDarwinian
EXPONENTIAL GROWTH as LOCUS of
b-LOGNORMAL PEAKS
46
REFERENCE PAPER

• A Mathematical Model for Evolution and SETI
• Origins of Life and Evolution of Biospheres
• (OLEB), Vol. 41 (2011), pages 609-619.

47
Darwinian EXPONENTIAL GROWTH

• 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

48
Darwinian EXPONENTIAL GROWTH

• 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

49
EXPONENTIAL as ENVELOPE of b-LOGNORMALS

• Each b-lognormal has its peak on the exponential.
• PRACTICALLY an Envelope, though not so
  formally.

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b-LOGNORMALS i.e. LOGNORMALS starting at
bbirth

• b-lognormals are just lognormals starting at any
  finite positive instant bgt0, that is supposed to
  be known.
• b-lognormals are thus a family of probability
  density functions with three real and positive
  parameters m, s, and b.

51
b-LOGNORMAL PEAK /1

• It is POSSIBLE to match the second equation (peak
  ordinate) with the EXPONENTIAL curve of the
  increasing number of Species ?
• YES, that may be done by setting

52
b-LOGNORMAL PEAK /2

• We discovered that it is POSSIBLE to MATCH these
  two equations EXACTLY just upon setting

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b-LOGNORMAL PEAK /3

• Moreover, the last two equations can be INVERTED,
  i.e. solved for m and s EXACTLY, thus yielding

• These two equations prove that, knowing the
  exponential (i.e. A and B) and peak time p, the
  b-lognormal HAVING ITS PEAK EXACTLY ON THE
  EXPONENTIAL is perfectly determined (i.e. its m
  and s are perfectly determined given A, B and p.
  This is the KEY RESULT to make further
  progress.

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Part 5 GEOMETRIC BROWNIAN MOTION (GBM)
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WARNING !!!GEOMETRIC BROWNIAN MOTIONis a
WRONG DENOMINATION
This process in NOT a Brownian Motion at all
since its probability density function is a
LOGNORMAL, and NOT A GAUSSIAN !!! So, the pdf
ranges between ZERO and INFINITY, and NOT
between minus infinity and infinity!!! Period.
56
GEOMETRIC BROWNIAN MOTION(GBM) exponential
mean value
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GEOMETRIC BROWNIAN MOTION(GBM) exponential
mean value
GEOMETRIC BROWNIAN MOTIONlognormal
probability density
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GEOMETRIC BROWNIAN MOTIONis the extension
in time of theSTATISTICAL DRAKE EQUATION
The two lognormals (of movie picture)
then COINCIDE.
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In other words still1) The CLASSICAL DRAKE
EQ.is STATIC, and is a SUBSET of
theSTATISTICAL DRAKE EQUATION.2) In turn,
the STATISTICAL DRAKE EQUATION is the STATIC
VERSION(i.e. the STILL PICTURE) of the
GEOMETRIC BROWNIAN MOTION(the MOVIE).
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Part 6Darwinian EXPONENTIAL GROWTHas GBM
in the number of LIVING SPECIES
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THREE REFERENCE PAPERS

• A Mathematical Model for Evolution and SETI
• Origins of Life and Evolution of Biospheres
• (OLEB), Vol. 41 (2011), pages 609-619.

62

• SETI, Evolution and Human History Merged into a
  Mathematical Model.
• International Journal of ASTROBIOLOGY,
• Vol. 12, issue 3 (2013), pages 218-245.

63

• Evolution and History in a new
• Mathematical SETI model.
• ACTA ASTRONAUTICA, Vol. 93 (2014), pages
  317-344. Online August 13, 2013.

64
BIG HISTORY is a GBM in the increasing number
of CIVILIZATIONS

• Life in the WHOLE UNIVERSE (and NOT just in the
  Milky Way) evolved since roughly 10 billion years
  ago.
• Mean value
• Two GBM parameters m and s

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BIG HISTORY a GBM in theincreasing number of
Civilizations
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BIG HISTORY a GBM in theincreasing number of
Civilizations
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Part 7ENTROPY as the EVOLUTION MEASURE
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b-LOGNORMAL ENTROPY

• Shannon ENTROPY for a probability density (in
  bits)
• Shannon ENTROPY for b-lognormals (in bits)

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b-LOGNORMAL ENTROPY for GBM

• But m ONLY is a function of the peak abscissa p
  
• Shannon ENTROPY for the b-lognormal in bits
  

70
CIVILIZATION LEVEL DIFFERENCE

• The ENTROPY DIFFERENCE among any two
  Civilizations having their two peak abscissae at
  p sub 1 and p sub 2 is given by
• ENTROPY IS THUS A MEASURE OF THE LEVEL OF
  PROGRESS REACHED BY EACH CIVILIZATION.
• ENTROPY DIFFERENCE measures the DIFFERENCE in
  civilization level among any two Civilizations.
• If it is known WHEN the two Civilizations reached
  their two peaks, the above formula yields their
  CIVILIZATION LEVEL DIFFERENCE.

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EXAMPLESof CIVILIZATION DIFFERENCES

• The DIFFERENCE in Civilization Level between the
  Spaniard and Aztecs in 1519 was about 3.84 bits
  per individual, amounting to 50 CENTURIES in
  evolution !
• The DIFFERENCE in Civilization Level between a
  Victorian Briton and a Pericles Greek was about
  1.76 bits per individual.
• The DIFFERENCE in Civilization Level between
  Humanity and the first Alien Civilization we will
  find in the Galaxy is UNKNOWN, of course, but
• but now we have a Mathematical Theory to
  ESTIMATE IT on the basis of the messages we get.
  

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EXAMPLE of GBM EVOLUTION DIFFERENCE

• The DIFFERENCE in GBM Darwinian Evolution between
  two species on Earth is given by the same
  equation
• As for the DIFFERENCE in Civilization Level,
  except we must now use the different numerical
  value of B the enveloping Darwinian exponential,
  found earlier.
• The result is that the DIFFERENCE IN EVOLUTION
  LEVEL between the first living being (RNA) 3.5
  billion years ago and Humans living now is about
  25.57 bits per individual.

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E-Pluribus-Unum THEOREM /1 from the lives of
the individuals to the life of their POPULATION

• Also, the mean value of the b-lognormal (in time)
  representing a certain POPULATION formed by N
  individuals, is expressed, in terms of the birth
  time b sub i and death time d sub i of its i-th
  INDIVIDUAL,by

74
E-Pluribus-Unum THEOREM /2 from the lives of
the individuals to the life of their POPULATION

• The variance of the b-lognormal (in time) pdf
  representing a certain POPULATION Y formed by N
  individuals, is expressed, in terms of the birth
  time b sub i and death time d sub i of its i-th
  INDIVIDUAL, by

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Part 8Example in the PAST namely AZTECS
vs. SPANIARDS
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VIRTUAL AZTEC b-lognormal
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VIRTUAL AZTEC b-lognormal
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AZTECS vs. ALL EUROPEANS in 1519
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AZTECS vs. SPANIARDS 1300 - 1520
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Part 9Example in the FUTURE up to10
MILLION YEARS
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TRENDS TWO EXPONENTIALS
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ENTROPY in the PAST for US GB
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Exps EXTRAPOLATED to 10,000 AD
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FUTURE ENTROPY to 10,000 AD
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Exps EXTRAPOLATED to 100,000 AD
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FUTURE ENTROPY to 100,000 AD
87
Exps EXTRAPOL. to 1 MILLION AD
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FUTURE ENTROPY to 1 MILLION AD
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FUTURE ENTROPY to 10 MILLION AD
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FERMI PARADOX (22 M-years)An estimate of how
many bits/individual of EVOLUTIONare needed to
settle the Galaxy.
Humans would be able to colonize the whole Galaxy
only if they could improve themselves by some
10,000 bits/individual. This is about 400 times
the 25 bits/individual improvement that Nature
took on Earth to evolve over 3.5 billion years.
But no other Alien Civilization would have to
interfere, which is highly unlikely! Thus, our
mathematical theory is crucial to estimate how
much Aliens will be more advanced than Humans,
when SETI scientists find them.
91
Darwinian EVOLUTION as a GBM
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MOLECULAR CLOCK ( EvoEntropy)
93
Motoo Kimura (1924-1994) Discoverer of the
NEUTRAL THEORY OF EVOLUTION at molecular level
(1968). Thus confirming the MOLECULAR CLOCK.
MOLECULAR CLOCK ENTROPY of b-lognormals
evolution at the molecular Level is
INDEPENDENT of the Environment where it occurs!
94
FOUR REFERENCE PAPERS

• A Mathematical Model for Evolution and SETI
• Origins of Life and Evolution of Biospheres
• (OLEB), Vol. 41 (2011), pages 609-619.

95

• SETI, Evolution and Human History Merged into a
  Mathematical Model.
• International Journal of ASTROBIOLOGY,
• Vol. 12, issue 3 (2013), pages 218-245.

96

• Evolution and History in a new
• Mathematical SETI model.
• ACTA ASTRONAUTICA, Vol. 93 (2014), pages
  317-344. Online August 13, 2013.

97

• New Evo-SETI Results about Civilizations and
• Molecular Clock.
• International Journal of Astrobiology, in press
  (2016), available on line on March 28, 2016.

98
Part 10 MASS EXTINCTIONS GBMsIN THE
DECREASINGNUMBER OF LIVING SPECIES
99
MASS EXTINCTION a GBM in thedecreasing
number of living Species
100
MASS EXCTINCTION a GBM in the decreasing
number of SPECIES

• The K-Pg IMPACT was 64 billion years ago. Suppose
  that its NUCLEAR WINTER lasted 1000 years
• In addition to the above two inputs, we must
  assign the following three more inputs

101
MASS EXCTINCTION a GBM in the decreasing
number of SPECIES

• Then, the GBM mean value is given by
• While the GBM lognormals m and s are given by

102
MASS EXCTINCTION a GBM in the decreasing
number of SPECIES

• Finally the GBM upper and lower STANDARD
  DEVIATION CURVES are given by, respectively
• While the upper standard deviation curve has its
  maximum at the time

103
Part 11LOGNORMAL PROCESSES WITH ARBITRARY
MEANRATHER THEN GBMs
104

• New Evo-SETI Results about Civilizations and
• Molecular Clock.
• International Journal of Astrobiology, in press
  (2016), available on line on March 28, 2016.

105

• LOGNORMAL PROCESSES WITH ARBITRARY MEAN

• After having discovered the Peak-Locus Theorem
  for GBMs, i.e. for an EXPONENTIAL ENVELOPE like
  this
• This author was able to EXTEND the Peak-Locus
  Theorem to an ARBITRARY MEAN

106

• PEAK-LOCUS THEOREM WITH ARBITRARY MEAN

• This author was able to EXTEND the GBM Peak-Locus
  Theorem to the general case when the ENVELOPE of
  all b-lognormals is an ARBITRARY MEAN VALUE

107

• EVO-ENTROPY WITH ARBITRARY MEAN

• This author was also able to EXTEND the
  Evo-Entropy Theorem to the general case when the
  ENVELOPE of all b-lognormals is an ARBITRARY MEAN
  VALUE
• In general, this Evo-Entropy is a NON-LINEAR
  funtion of the peak time p of the Running
  b-lognormal.
• But in the particular case of the GBM mean value,
  the above Evo-Entropy reduces the LINEAR ONE
  seen.

108
Part 12MARKOV-KOROTAYEV CUBICas the Mean
Valueof a Lognormal PROCESS
109
MARKOV-KOROTAYEV MODEL OF EVOLUTION

• 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 logisti
  c 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.

110
MARKOV-KOROTAYEV MODEL OF EVOLUTION
111
OUR LOGNORMAL PROCESS WITH A CUBIC MEAN VALUE
112
The CUBIC EQUATION in terms of BOUNDARY
CONDITIONS CONDITIONCONDITIONSCONDITIONS
113
CONCLUSIONS
1) We developed here a new mathematical model
embracing all of Big History, including Darwinian
Evolution (RNA to Humans), Human History (Aztecs
to USA) (see ref. 16) and then we extrapolated
even that that into the future up to 10 million
years (see ref. 17), the minimum time requested
for a civilization to expand to the whole Milky
Way (Fermi paradox). 2) Our mathematical model
is based on the properties of lognormal
probability distributions. It also is fully
compatible with the Statistical Drake Equations,
i.e. the foundational equation of SETI, the
Search for Extra-Terrestrial Intelligence.
3) Merging all these apparently different topics
into the larger but single topic called Big
History is the achievement of this paper. As
such, our statistical theory would be crucial to
estimate how much more advanced than Humans the
Aliens would be when SETI scientists will succeed
in finding the first ET Civilization.
114

• Thank you very much !