Evo-SETI: Life Evolution Statistics on Earth and Exoplanets

✍ Scribed by Claudio Maccone

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 878
• Category: Library

✦ Synopsis

This book offers a vision of how evolutionary life processes can be modelled. It presents a mathematical description that can be used not only for the full evolution of life on Earth from RNA to modern human societies, but also the possible evolution of life on exoplanets, thus leading to SETI, the current Search for ExtraTerrestrial Intelligence.

The main premise underlying this mathematical theory is that the Geometric Brownian Motion (GBM) can be applied as a key stochastic process to model the evolution of life. In the resulting Evo-SETI Theory, the life of any living thing (a cell, an animal, a human, a civilization of humans, or even an ET civilization) is represented by a b-lognormal, i.e., a lognormal probability density function starting at a precise instant (b, birth) then increasing up to a peak time, then decreasing to senility time and then continuing as a straight line down to the time of death.

Using this theory, Claudio Maccone arrives at remarkable hypotheses on the development of life and civilizations, the possibility of extraterrestrial life, and when computers will take over the reins from us humans (Singularity). The book develops the mathematical Evo-SETI Theory by integrating a set of articles that the author has published in various journals on Astrobiology and Astronautical Research.

✦ Table of Contents

Preface
Contents
OVERCOME Theorem, that is PEAK-LOCUS Theorem, Proven by Maxima
1 OVERCOME Theorem, that is PEAK-LOCUS Theorem
2 Evo-Entropy(p): Measuring “How Much Evolution” Occurred
3 Perfectly LINEAR Evo-Entropy When the Mean Value Is Perfectly Exponential (A GBM): This Is just the Molecular Clock
4 Introducing EE, the Evo-SETI Unit: An Amount of Information Equal to the Evo-Entropy Reached by Life Evolution on Earth Nowadays
5 Conclusions About Evo-Entropy
Appendix
References
Evo-SETI Mathematics: Part 1: Entropy of Information. Part 2: Energy of Living Forms. Part 3: The Singularity
1 Introduction
2 Part 1: Entropy of Information as the Measure of Evolution, Peak-Locus Theorem, and Scale of Biological Evolution (Evo-SETI Scale)
2.1 Purpose of This Chapter
2.2 A Simple Proof of the b-Lognormal Probability Density Function (PDF)
2.3 Biological Evolution as the Exponential Increase of the Number of Living Species
2.4 Biological Evolution on Earth Was just a Particular Realization of Geometric Brownian Motion in the Number of Living Species
2.5 During the Last 3.5 Billion Years Life Forms Increased like a Lognormal Stochastic Process
2.6 Mean Value of the Lognormal Process L(t)
2.7 L( t ) Initial Conditions at ts
2.8 L( t ) Final Conditions at te > ts
2.9 Important Special Cases of mL ( t )
2.10 Boundary Conditions When mL ( t ) Is a First, Second or Third Degree Polynomial in the Time (t - ts)
2.11 Peak-Locus Theorem
2.12 Evo-Entropy(p): Measuring “How Much Evolution” Occurred
2.13 Perfectly Linear Evo-Entropy When the Mean Value Is Perfectly Exponential (a GBM): This Is just the Molecular Clock
2.14 Introducing EE, the Evo-SETI Unit: An Amount of Information Equal to the Evo-Entropy Reached by Life Evolution on Earth Nowadays
2.15 Markov-Korotayev Alternative to Exponential: A Cubic Growth
2.16 Evo-Entropy of the Markov-Korotayev Cubic Growth
2.17 Comparing the Evo-Entropy of the Markov-Korotayev Cubic Growth to a Hypothetical (1) Linear and (2) Parabolic Growth
2.18 Conclusions About Evo-Entropy
2.19 Life as a Finite b-Lognormal as Assumed by This Author Prior to 2017
2.20 b-Lognormal History Formulae and Their Applications to Past History
3 Part 2: Energy of Living Forms by “Logpar” Power Curves
3.1 Introduction to Logpar Power Curves
3.2 Finding the Parabola Equation of the Right Part of the Logpar
3.3 Finding the b-Lognormal Equation of the Left Part of the Logpar
3.4 Area Under the Parabola on the Right Part of the Logpar Between Peak and Death
3.5 Area Under the Full Logpar Curve Between Birth and Death
3.6 The Area Under the Logpar Curve Depends on Sigma Only, and Here Is the Area Derivative W.R.T. σ
3.7 Exact “History Equations” for Each Logpar Curve
3.8 Considerations on the Logpar History Equations
3.9 Logpar Peak Coordinates Expressed in Terms of ( b,p,d ) Only
3.10 History of Ancient Rome as an Example of How to Use the Logpar History Formulae
3.11 Area Under Rome’s Logpar and Its Meaning as “Overall Energy” of the Roman Civilization
3.12 The Energy Function of d Regarded as a Function of the Death Instant d, Hereafter Renamed D
3.13 Discovering an Oblique Asymptote of the Energy Function, Energy(D), While the Death Instant D Is Increasing Indefinitely
3.14 The Oblique Asymptote for the “History of Rome” Case
3.15 What if Hadn’t Rome Fallen? Discovering the Straight Line Parallel to the Asymptote but Starting at the Rome Power Peak
3.16 Energy Output of the Sun as a G2 Star Over the About 10 Billion Years of Its Lifetime
3.17 Energy Output of an M Star Over 45 Billion Years of Lifetime
3.18 Mean Power in a Lifetime
3.19 Lifetime Mean Value
3.20 Logpar Power Curves Versus b-Lognormal Probability Densities
3.21 Conclusions About Logpars
4 Part 3: Before and After the Singularity According to Evo-SETI Theory
4.1 Every Exponential in Time Has just a Single Knee: The Instant at Which Its Curvature Is Highest
4.2 GBM Exponential as Mean Value of the Increasing Number of Species Since the Origin of Life on Earth
4.3 Deriving the Knee Time for GBMs
4.4 Knee-Centered Form of the GBM Exponential
4.5 Finding WHEN the GBM Knee Will Occur According to the Author’s Conventional Values for ts and B
4.6 Ray Kurzweil’s 2006 Book “the Singularity Is Near”
4.7 Kurzweil’s Singularity Is the Same as Our GBM’s Knee in Our Evo-SETI Theory
4.8 Measuring the Pace of Evolution B by the Average Number m0 of Species Living on Earth NOW
4.9 An Unexpected Discovery: The “Origin-to-Now” (“OTN”) Equation Relating the Time of the Origin of Life on Earth (ts) to m0 (the Average Number of Species Living on Earth Right Now)
4.10 Solving the “Origin-to-Now” Equation NUMERICALLY for the Two Cases of -3.5 and -3.8 Billion Years of Life on Earth
4.11 But… Biologists Are UNABLE to Measure m0 Experimentally!
4.12 Lognormal pdf of the GBM
4.13 Finding the GBM Parameter σ
4.14 Numerical Standard Deviation Nowadays for the Two Cases of Life Starting 3.5 and 3.8 Billion Years Ago
4.15 Numerical σ for the Two Cases of Life Starting 3.5 and 3.8 Billion Years Ago
4.16 Conclusions
References
SETI, Evolution and Human History Merged into a Mathematical Model
1 SETI and Darwinian Evolution Merged Mathematically
1.1 Introduction: The Drake Equation (1961) as the Foundation of SETI
1.2 Statistical Drake Equation (2008)
1.3 The Statistical Distribution of N Is Lognormal
1.4 Darwinian Evolution as Exponential Increase of the Number of Living Species
1.5 Introducing the ‘Darwin’ (D) Unit, Measuring the Amount of Evolution that a Given Species Reached
1.6 Darwinian Evolution Is just a Particular Realization of Geometric Brownian Motion in the Number of Living Species
2 GBM as the Key to Stochastic Evolution of All Kinds
2.1 The N(t) GBM as Stochastic Evolution
2.2 Our Statistical Drake Equation Is the Static Special Case of N(t)
2.3 GBM as the Key to Mathematics of Finance
3 Darwinian Evolution Re-defined as a GBM in the Number of Living Species
3.1 A Concise Introduction to Cladistics and Cladograms
3.2 Cladistics: Namely the GBM Mean Exponential as the Locus of the Peaks of b-Lognormals Representing Each a Different Species Started by Evolution at  Time t = b
3.3 Cladogram Branches Are Increasing, Decreasing or Stable (Horizontal) Exponential Arches as Functions of Time
3.4 KLT-Filtering in Hilbert Space and Darwinian Selection Are “the Same Thing” in Our Theory…
3.5 Conclusions About Our Statistical Model for Evolution and Cladistics
4 Lifespans of Living Beings as b-Lognormals
4.1 Further Extending b-Lognormals as Our Model for All Lifespans
4.2 Infinite b-Lognormals
4.3 From Infinite to Finite b-Lognormals: Defining the Death Time, d, as the Time Axis Intercept of the b-Lognormal Tangent Line at Senility s
4.4 Terminology About Various Time Instants Related to a Lifetime
4.5 Terminology About Various Time Spans Related to a Lifetime
4.6 Normalizing to One All the Finite b-Lognormals
4.7 Finding the b-Lognormals Given b and Two Out of the Four a, p, s, d
5 Golden Ratios and Golden b-Lognormals
5.1 Is σ Always Smaller Than 1?
5.2 Golden Ratios and Golden b-Lognormals
6 Mathematical History of Civilizations
6.1 Civilizations Unfolding in Time as b-Lognormals
6.2 Eight Examples of Western Historic Civilizations as Finite b-Lognormals
6.3 Plotting All b-Lognormals Together and Finding the Trends
6.4 b-Lognormals of Alien Civilizations
7 Extrapolating History into the Past: Aztecs
7.1 Aztecs–Spaniards as an Example of Two Suddenly Clashing Civilizations with Large Technology Gap
7.2 ‘Virtual Aztecs’ Method to Find the ‘True Aztecs’ b-Lognormal
8 b-Lognormal Entropy as ‘Civilization Amount’
8.1 Introduction: Invoking Entropy and Information Theory
8.2 Exponential Curve in Time Determined by Two Points Only
8.3 Assuming that the Exponential Curve in Time Is the GBM Mean Value Curve
8.4 The ‘No-Evolution’ Stationary Stochastic Process
8.5 Entropy of the ‘Running b-Lognormal’ Peaked at the GBM Exponential Mean
8.6 Decreasing Entropy for an Exponentially Increasing Evolution: Progress!
8.7 Six Examples: Entropy Changes in Darwinian Evolution, Human History Between Ancient Greece and Now, and Aztecs and Incas Versus Spaniards
8.8 b-Lognormals of Alien Civilizations
9 Conclusion: Summary of Technical Concepts Described
References
Evolution and Mass Extinctions as Lognormal Stochastic Processes
1 Introduction: Mathematics and Science
2 A Summary of the ‘Evo-SETI’ Model of Evolution and SETI
3 Important Special Cases of mL(t)
4 Introducing b-lognormals
5 Peak-Locus Theorem
6 Entropy as the Evolution Measure
7 Evo-SETI
8 Mass Extinctions of Darwinian Evolution Described by a Decreasing GBM
8.1 GBMs to Understand Mass Extinctions of the Past
8.2 Example: The K–Pg Mass Extinction Extending Ten Centuries After Impact
9 Mass Extinctions Described by an Adjusted Parabola Branch
9.1 Adjusting the Parabola to the Mass Extinctions of the Past
9.2 Example: The Parabola of the K–Pg Mass Extinction Extending Ten Centuries After Impact
10 Cubic as the Mean Value of a Lognormal Stochastic Process
10.1 Finding the Cubic When Its Maximum and Minimum Times Are Given, in Addition to the Five Conditions to Find the Parabola
11 Markov–Korotayev Biodiversity Regarded as a Lognormal Stochastic Process Having a Cubic Mean Value
11.1 Markov–Korotayev’s Work on Evolution
12 Conclusions
Supplementary Material
References
New Evo-SETI Results About Civilizations and Molecular Clock
1 Part I: New Results About Civilizations in Evo-SETI Theory
1.1 Introduction
1.2 A Simple Proof of the b-Lognormal’s pdf
1.3 Defining ‘Life’ in the Evo-SETI Theory
1.4 History Formulae
1.5 Death Formula
1.6 Birth–Peak–Death (BPD) Theorem
1.7 Mathematical History of Nine Key Civilizations Since 3100 bc
1.8 b-Scalene (Triangular) Probability Density
1.9 Uniform Distribution Between Birth and Death
1.10 Entropy Difference Between Uniform and b-Scalene Distributions
1.11 ‘Equivalence’ Between Uniform and b-Lognormal Distributions
1.12 b-Lognormal of a Civilization’s History as CLT of the Lives of Its Citizens
1.13 The Very Important Special Case of Ci Uniform Random Variables: E-Pluribus-Unum Theorem
2 Part 2: New Results About Molecular Clock in Evo-SETI Theory
2.1 Darwinian Evolution as a Geometric Brownian Motion (GBM)
2.2 A Leap Forward: For Any Assigned Mean Value mL(t) We Construct Its Lognormal Stochastic Process
2.3 Completing [3]: Letting ML(t) There Be Replaced Everywhere by mL(t), the Assigned Trend
2.4 Peak-Locus Theorem
2.5 tsGBM and GBM Sub-cases of the Peak-Locus Theorem
2.6 Shannon Entropy of the Running b-Lognormal
2.7 Introducing Our… Evo-Entropy(p) Measuring How Much a Life Form Has Evolved
2.8 The Evo-Entropy(p) of tsGBM Increases Exactly Linearly in Time
3 Conclusions
Supplementary Material
References
Life Expectancy and Life Energy According to Evo-SETI Theory
1 Part 1: Logpar Curves and Their History Equations
1.1 Introduction to Logpar “Finite Lifetime” Curves
1.2 Finding the Parabola Equation of the Right Part of the Logpar
1.3 Finding the b-Lognormal Equation of the Left Part of the Logpar
1.4 Area Under the Parabola on the Right Part of the Logpar Between Peak and Death
1.5 Area Under the Full Logpar Curve Between Birth and Death
1.6 The Area Under the Logpar Curve Depends on Sigma Only, and Here Is the Area Derivative w.r.t. Sigma
1.7 Exact “History Equations” for Each Logpar Curve
1.8 Considerations on the Logpar Least-Energy History Formulae
1.9 Logpar Peak Coordinates Expressed in Terms of ( b,p,d ) Only
2 Part 2: Energy as the Area Under All Logpar Power Curves
2.1 The Area Under a Logpar and Its Meaning as “Lifetime Energy”
3 Part 3: Mean Energy in a Lifetime and Lifetime Mean Value
3.1 Mean Energy in a Lifetime
3.2 Lifetime Mean Value
4 Part 4: Adolescence Formulae (Or Puberty Formulae)
4.1 Logpar’s Increasing Inflexion Time as Adolescence Time (Or Puberty Time for Living Beings)
5 Part 5: Life Expectancy and Fertility in Logpars
5.1 Reconsidering the Death Time d as a Living Being’s Life Expectancy
5.2 Introducing the Living Being’s End-Of-Fertility (EOF) Time
5.3 Life Expectancy of Living Beings
5.4 Fertility Span of Living Beings
5.5 A Numerical Example About the Most Important Case: Man
5.6 Checking Numerically the (Small) Difference Between History Formulae and Adolescence Formulae for Man
6 Conclusions
References
Evo-SETI: A Mathematical Tool for Cladistics, Evolution, and SETI
1 Purpose of This Chapter
2 During the Last 3.5 Billion Years, Life Forms Increased as in a (Lognormal) Stochastic Process
3 Mean Value of the Lognormal Process L(t)
4 L(t) Initial Conditions at ts
5 L(t) Final Conditions at te > ts
6 Important Special Cases of m(t)
7 Boundary Conditions When m(t) Is a First, Second, or Third Degree Polynomial in the Time (t - ts)
8 Peak-Locus Theorem
9 EvoEntropy(p) as a Measure of Evolution
10 Perfectly Linear EvoEntropy When the Mean Value Is Perfectly Exponential (GBM): This Is just the Molecular Clock
11 Markov-Korotayev Alternative to Exponential: A Cubic Growth
12 EvoEntropy of the Markov-Korotayev Cubic Growth
13 Comparing the EvoEntropy of the Markov-Korotayev Cubic Growth, to the Hypothetical (1) Linear and (2) Parabolic Growth
14 Conclusions
Proof of the CUBIC MEAN VALUE Equation (31)
References
The Statistical Drake Equation
1 Introduction
2 Step 1: Letting Each Factor Become a Random Variable
2.1 Step 2: Introducing Logs to Change the Product into a Sum
2.2 Step 3: The Transformation Law of Random Variables
3 Step 4: Assuming the Easiest Input Distribution for Each Di: The Uniform Distribution
3.1 Step 5: A Numerical Example of the Statistical Drake Equation with Uniform Distributions for the Drake Random Variables Di
3.2 Step 6: Computing the Logs of the Seven Uniformly Distributed Drake Random Variables Di
3.3 Step 7: Finding the Probability Density Function of N, but Only Numerically, Not Analytically
4 The Central Limit Theorem (CLT) of Statistics
5 The Lognormal Distribution Is the Distribution of the Number N of Extraterrestrial Civilizations in the Galaxy
6 Comparing the CLT Results with the Non-CLT Results
7 Distance of the Nearest Extraterrestrial Civilization as a Probability Distribution
7.1 Classical, Non-probabilistic Derivation of the Distance of the Nearest ET Civilization
7.2 Probabilistic Derivation of the Probability Density Function for ETDistance
7.3 Statistical Properties of This Distribution
7.4 Numerical Example of the ETDistance Distribution
8 The “DATA ENRICHMENT PRINCIPLE” as the Best CLT Consequence upon the Statistical Drake Equation (Any Number of Factors Allowed)
9 Conclusions
References
SETI and SEH (Statistical Equation for Habitables)
1 Introduction to SETI
2 The Key Question: How Far Are They?
3 Computing N by Virtue of the Drake Equation (1961)
4 The Drake Equation Is Over-Simplified
5 The Statistical Drake Equation by Maccone (2008)
6 Solving the Statistical Drake Equation by Virtue of the Central Limit Theorem (CLT) of Statistics
7 An Example Explaining the Statistical Drake Equation
8 Finding the Probability Distribution of the ET-Distance by Virtue of the Statistical Drake Equation
9 The “Data Enrichment Principle” as the Best CLT Consequence upon the Statistical Drake Equation (Any Number of Factors Allowed)
10 Habitable Planets for Man
11 The Statistical Dole Equation
12 The Number of Habitable Planets for Man in the Galaxy Follows the Lognormal Distribution
13 The Distance Between Any Two Nearby Habitable Planets Follows the Maccone Distribution
14 A Numerical Example: A Some Hundred Million Habitable Planets Exist in the Galaxy!
15 Distance (Maccone) Distribution of the Nearest Habitable Planet to Us According to the Previous Numerical Inputs
16 Comparing the Statistical Dole and Drake Equations: Number of Habitable Planets Versus Number of ET Civilizations in This Galaxy
17 SEH, the “Statistical Equation for Habitables” Is just the Statistical Dole Equation
18 Conclusions
References
Societal Statistics by Virtue of the Statistical Drake Equation
1 Introducing the Drake Equation
2 The Drake Equation is Over-Simplified
3 The Statistical Drake Equation by Maccone [3]
4 Solving the Statistical Drake Equation by Virtue of the Central Limit Theorem (CLT) of Statistics
5 An Example Explaining the Statistical Drake Equation
6 The “Data Enrichment Principle” as the Best CLT Consequence upon the Statistical Drake Equation (Any Number of Factors Allowed)
7 Habitable Planets for Man
8 The Statistical Dole Equation
9 The Number of Habitable Planets for Man in the Galaxy Follows the Lognormal Distribution
10 An Example Explaining the Statistical Dole Equation: Some Hundred Million Habitable Planets Exist in the Galaxy!
11 Comparing the Statistical Dole and Drake Equations: Number of Habitable Planets Versus Number of ET Civilizations in This Galaxy
12 The Probability Distribution of the Ratio of Two Lognormally Distributed Random Variables
13 Breaking the Drake Equation up into the Dole Equation Times the Drake Equation’s Societal Part
14 Conclusions
References
Evolution and History in a New “Mathematical SETI” Model
1 Introduction: Interstellar Flight and SETI Are the Two Sides of the Same Coin
1.1 Astronautics and Interstellar Flight Since the Moon Landings
1.2 The “FOCAL” Space Missions Enabling Us to Use the Sun as a Gravitational Lens (1992)
1.3 A “RADIO BRIDGE” Among Two Stars to Enable the Radio Link Among Their Civilizations (2011)
1.4 A Galactic Internet Might Be in Use Already, but by Aliens, Not by Humans yet (2021)
2 SETI and Darwinian Evolution Merged Mathematically
2.1 Introduction: The Drake Equation (1961) as the Foundation of SETI
2.2 Statistical Drake Equation (2008)
2.3 The Statistical Distribution of N Is Lognormal
2.4 Darwinian Evolution as (Overall) Exponential Increase in the Number of Living Species
2.5 Darwinian Evolution Is Just a Particular Realization of Geometric Brownian Motion in the Number of Living Species
3 Geometric Brownian Motion (GBM) Is the Key to Stochastic Evolution of All Kinds
3.1 The N(t) GBM as Stochastic Evolution
3.2 Our Statistical Drake Equation Is the Static Special Case of N(t)
3.3 The N(t) Stochastic Process Is a Geometric Brownian Motion
4 Darwinian Evolution Re-defined as a GBM in the Number of Living Species
4.1 Introducing the “DARWIN” (D) Unit, Measuring the Amount of Evolution that a Given Species Reached
4.2 Cladistics, Namely the GBM Mean Exponential as the Locus of the Peaks of b-Lognormals Representing Each a Different Species Started by Evolution at the Time t = b
4.3 Cladogram Branches Are Made up by Increasing, Decreasing or Stable (Horizontal) Exponential Arches
4.4 Conclusions About Our Statistical Model for Evolution and Cladistics
5 Life-Spans of Living Beings as b-Lognormals
5.1 Further Extending b-Lognormals as Our Model for All Life-Spans
5.2 Infinite b-Lognormals
5.3 From Infinite to Finite b-Lognormals: Defining the Death Time, d, as the Time Axis Intercept of the Tangent at Senility
5.4 Terminology of Various Time Instants Related to a Lifetime
5.5 Terminology of Various Time Spans Related to a Lifetime
5.6 Normalizing to One All Finite b-Lognormals
5.7 Finding the b-Lognormals Given b and Two Out of the Four a, p, s, d
6 Golden Ratios and Golden b-Lognormals
6.1 Is σ Always Smaller Than 1?
6.2 Golden Ratios and Golden b-Lognormals
7 Mathematical History of Human Civilizations
7.1 Civilizations Unfolding in Time as b-Lognormals
7.2 Eight Examples of Western Civilizations as Finite b-Lognormals
7.3 Plotting All b-Lognormals Together and Finding Exponentials
8 Extrapolating History into the Past: Aztecs
8.1 Aztec–Spaniards as an Example of Two Suddenly Clashing Civilizations
8.2 “Virtual Aztecs” Method to Find the “True Aztecs” B-Lognormal
9 b-Lognormal Entropy as “Civilization Amount”
9.1 Introduction: Invoking Entropy and Information Theory
9.2 Exponential Curve in Time Determined by Two Points Only
9.3 Assuming that the Exponential Curve in Time Is the GBM Mean Value Curve
9.4 The “No-Evolution” Stationary Stochastic Process
9.5 Entropy of the “Running B-Lognormal” Peaked at the GBM Exponential Mean
9.6 Decreasing Entropy for an Exponentially Increasing Evolution: Progress!
9.7 Six Examples: Entropy Changes in Darwinian Evolution, Human History Between Ancient Greece and Now, and Aztecs and Incas Versus Spaniards
9.8 b-Lognormals of Alien Civilizations
10 Aliens: How Much More Advanced Than US?
10.1 Extrapolating the Human Past 8000 Years into the Future
10.2 Extrapolating 100,000 Years into the Future
10.3 Extrapolating a Million Years into the Future
10.4 Fermi Paradox: Extrapolating Ten Million Years into the Future
11 Spaceflight, SETI and the Future of Humankind
11.1 Spaceflight as of 2013
11.2 Big History: One More Step Ahead
11.3 Conclusion: The Grand Vision of Universal Evolution
References
SETI as a Part of Big History
1 Introduction: New Statistical Mechanisms
2 Merging SETI and Darwinian Evolution Statistically
2.1 The Drake Equation (1961) as the Foundation of SETI
2.2 Statistical Drake Equation (2008)
2.3 The Statistical Distribution of N is Lognormal
2.4 Darwinian Evolution as the Exponential Increase of the Number of Living Species
2.5 Introducing the “Darwin” (d) Unit, Measuring the Amount of Evolution that a Given Species Reached
2.6 Darwinian Evolution Is Just a Particular Realization of Geometric Brownian Motion in the Number of Living Species
3 Geometric Brownian Motion (GBM) as the Key to Stochastic Evolution of All Kinds
3.1 The N(t) GBM as Stochastic Evolution
3.2 Statistical Drake Equation as the Static Special Case of N(t)
3.3 GBM as the Key to the Mathematics of Finance
3.4 Adjusting the GBM: Letting It Take the Value of One at Its Start (Time t = tSTART) and Deriving Its Current Mean Value and Standard Deviation
3.5 Example: Darwinian Evolution as a GBM Taking the Value of One at Its Start (Time t = tSTART) with Known Current Mean Value and Standard Deviation
4 Big History as the Statistical Drake Equation Extended by Adding the “Missing Initial Part”
4.1 Big Bang to Current Stars: The “Missing Initial Part” of the Drake Equation
5 Mass Extinctions in the Course of Darwinian Evolution Understood by Virtue of a Decreasing GBM
5.1 A Brand-New Discovery: GBMs to Understand Mass Extinctions of the Past
5.2 Example: The K–Pg Mass Extinction Extending Ten Centuries After Impact
6 Conclusion
Appendix: Cyclic Phenomena as Lognormal Stochastic Processes
References
Lognormals for SETI, Evolution and Mass Extinctions
1 Introduction: New Statistical Mechanisms
2 Mass Extinctions of Darwinian Evolution Described by a Decreasing Geometric Brownian Motion
2.1 GBMs to Understand Mass Extinctions of the Past
2.2 Example: The K–Pg Mass Extinction Extending Ten Centuries After Impact
3 Mass Extinctions Described by an Adjusted Parabola Branch
3.1 Adjusting the Parabola to the Mass Extinctions of the Past
3.2 Example: The Parabola of the K–Pg Mass Extinction Extending Ten Centuries After Impact
4 Conclusions
References
Statistical Drake–Seager Equation for Exoplanet and SETI Searches
1 Introduction
2 The Classical Drake Equation (1961)
3 Transition from the Classical to the Statistical Drake Equation
4 Step 1: Letting Each Factor Become a Random Variable
5 Step 2: Introducing Logs to Change the Product into a Sum
6 Step 3: The Transformation Law of Random Variables
7 Step 4: Assuming the Easiest Input Distribution for Each Di: The Uniform Distribution
8 Step 5: Computing the Logs of the 7 Uniformly Distributed Drake Random Variables Di
9 The Central Limit Theorem (CLT) of Statistics
10 The Lognormal Distribution is the Distribution of the Number N of Extraterrestrial Civilizations in the Galaxy
11 Data Enrichment Principle
12 The Statistical Seager Equation
13 The Extremely Important Particular Case When the Input Random Variables Are Uniform
14 Conclusion
Appendix
References
Evo-SETI Entropy Identifies with Molecular Clock
1 Purpose of This Chapter
2 During the Last 3.5 Billion Years Life Forms Increased Like a (Lognormal) Stochastic Process
3 Important Special Cases of m(t)
4 Peak-Locus Theorem
5 Entropy as Measure of Evolution
6 Conclusions
References
Evo-SETI SCALE to Measure Life on Exoplanets
1 Purpose of This Chapter
2 During the Last 3.5 Billion Years Life Forms Increased Like a (Lognormal) Stochastic Process
3 Mean Value of the Lognormal Process L(t)
4 L(t) Initial Conditions at ts
5 L(t) Final Conditions at te > ts
6 Important Special Cases of mL(t)
7 Boundary Conditions When mL(t) is a First, Second or Third Degree Polynomial in the Time (t–ts)
8 Peak-Locus Theorem
9 Entropy as Measure of Evolution
10 Kullback–Leibler Divergence (or, Better, “Distance”) Among Any Two Living Species
11 Conclusions
References
Evo-SETI Theory and Information Gap Among Civilizations
1 Introduction
2 A Simple Proof of the b-Lognormal’s Pdf
3 History Formulae
4 Death Formula
5 Birth-Peak-Death (BPD) Theorem
6 Information Entropy (SHANNON ENTROPY) as the Measure of a Civilization’s Advancement
7 Information Gaps, Namely Entropy Differences Among the Nine Historic Western Civlizations
8 Conclusion
References
Kurzweil’s Singularity as a Part of Evo-SETI Theory
1 Geometric Brownian Motion (GBM) Is Key to Exponential Stochastic Evolution
1.1 Darwinian Evolution as the Exponential Increase of the Number of Living Species
1.2 Darwinian Evolution Was Just a Particular Realization of Geometric Brownian Motion in the Number of Living Species
2 Knee of Any Exponential
2.1 Every Exponential in Time Has Just a Single Knee: The Instant at Which Its Curvature Is Highest
2.2 (GBM) Exponential as Mean Value of the Increasing Number of Species Since the Origin of Life
2.3 Deriving the Knee Time for GBMs
2.4 Knee-Centered Form of the GBM Exponential
2.5 Finding When the GBM Knee Will Occur According to the Author’s Conventional Values for ts and B
3 Kurzweil’s “the Singularity Is Near” (Is Nowadays)
3.1 Ray Kurzweil’s 2006 Book “the Singularity Is Near”
3.2 Kurzweil’s Singularity Is the GBM’s Knee in Our Evo-SETI Theory
3.3 Measuring the Pace of Evolution B by Measuring the Average Number m0 of Species Living on Earth Right Now
3.4 An Unexpected Discovery: The “Origin-to-Now” (“OTN”) Equation Relating the Time of the Origin of Life on Earth ts to m0 the Average Number of Species Living on Earth Right Now
3.5 Solving the “Origin-to-Now” Equation Numerically for the Two Cases of -3.5 and -3.8 Billion Years of Life Development
3.6 No Way for the Biologists to Measure m0 Experimentally!
4 Upper and Lower Standard Deviation Curves
4.1 Lognormal PDF of the GBM
4.2 Finding the GBM Parameter σ
4.3 Numerical Standard Deviation Nowadays for the Two Cases of Life Starting 3.5 and 3.8 Billion Years Ago
4.4 Numerical σ for the Two Cases of Life Starting 3.5 and 3.8 Billion Years Ago
5 Evo-entropy of the b-Lognormals Having Their Peaks on the GBM Exponential
5.1 Peak-Locus Theorem
5.2 Entropy as Measure of Evolution
6 The Korotayev-Markov Alternative Evolution Theory with a Cubic-like Mean Value in Time
6.1 Peak-Locus Theorem When the Mean Value Is a Polynomial in the Time
6.2 Finding the Cubic When Its Maximum and Minimum Times Are Given, in Addition to the Five Conditions to Find the Parabola
6.3 Markov-Korotayev Biodiversity Regarded as a Lognormal Stochastic Process Having a Cubic Mean Value
7 Conclusions
APPENDIX, i.e. our: Evo-SETI SINGULARITY THEOREM. A NEW RESULT connecting three quite different facts like: (1) The time ts of the origin of life on Earth (RNA);  (2) The number m0 of Species living NOW 140 million; (3) The SINGULARITY is just (a few years from) NOW !
References
Energy of Extra-Terrestrial Civilizations According to Evo-SETI Theory
1 Part 1
1.1 Logpar Curves and Their History Equations
2 Part 2
2.1 Energy as the Area Under Logpar Curves
3 Part 3
3.1 Mean Energy in a Lifetime and Lifetime Mean Value
4 Part 4
4.1 Conclusions: The Advantages of Logpar Power Curves
Appendix A: Supplementary Data
References
The Evo-SETI Unit of Evolution is EE = Earth Evolution = 25.575 bit
1 Purpose of This Chapter
2 During the Last 3.5 Billion Years Life Forms Increased Like a (Lognormal) Stochastic Process
3 Mean Value of the Lognormal Process L(t)
4 L(t) Initial Conditions at ts
5 L(t) Final Conditions at te > ts
6 Important Special Cases of m(t)
7 Boundary Conditions When m(t) Is a First, Second or Third Degree Polynomial in the Time (t - ts)
8 Peak-Locus Theorem
9 EvoEntropy(p) as Measure of Evolution
10 Perfectly Linear EvoEntropy When the Mean Value Is Perfectly Exponential (GBM): This Is Just the Molecular Clock!
11 Introducing the EE Evo-SETI Unit: Information Equal to the EvoEntropy Reached by the Evolution of Life on Earth Nowadays
12 Conclusions
13 Appendix A: Supplementary Data
References
Evo-SETI Quartics Yielding ET Civilizations’ Energy
1 Introduction to Evo-SETI Theory
2 Quartic Curve in the Time t Representing the Power (Measured in Watts) of a Civilization Between Its Inception (b = Birth) and Its End (d = Death)
3 Conditions Imposed on Our Quartic Power Curve
4 Separating Our Quartic Curve into the Product of the Two Above Boundary Conditions Times a Quadratic Polynomial in the Time
5 Expressing the Separated Quartic in Terms of b, d and the Derivatives Db and Dd of this Quartic at the Initial and End Times, Respectively
6 The General Quartic and Its Five Coefficients Expressed in Terms of b, d and the Derivatives Db and Dd of the Quartic at the Initial and End Times
7 Energy of the General Quartic, i.e. Area Under the General Quartic and the Time Axis, i.e. Integral of the Quartic Between b and d
8 Defining the Smooth-Start Quartic, That Is the Quartic with Zero Derivative at b
9 Energy of the Smooth-Start Quartic
10 Defining the Symmetric Quartic, i.e. When the Derivatives Db and Dd of the Quartic at the Initial and End Times are Equal to Each Other
11 Energy of the Symmetric Quartic, i.e. Area Under the Symmetric Quartic, i.e. Integral of the Symmetric Quartic Between b and d
12 The Special Case of the Symmetric Quartic with Zero Derivatives at Both b and d
13 Energy of the Symmetric Quartic with Zero Derivatives at Both b and d
14 Determining the Last Unknown, A, If One Knows the Energy that a Civilization or a Living Being Used (or Produced) During Its Own Lifetime
15 Conclusions
Appendix
Reference
KLT for an Expanding Universe with SETI Applications
1 Introduction
2 Claudio Maccone’s 1981 Exact Analytical Solution Yielding the KLT for All Time-Rescaled Gaussian Stochastic Processes (that is Gaussian Noises)
3 The Machinery of Maccone’s 1981 Exact KLT for Relativistic Frames in Arbitrary Motion
4 The 1981 Exact KLT Machinery in Case of Relativistic Frames in Uniform Motion
5 Friedman-Lemaître-Robertson-Walker (FLRW) Metric and the Friedman Equations with Λ
6 KLT Time-Rescaling Function f( t ) for Motion of Spaceships, Particles and Light Inside Any Expanding Universe Given Its a( t ) Function
7 Motion of a “Star Trek Spaceship” with “Flight Law” Equal to ν( t ) Inside an Expanding Universe Having the Expansion Law a( t )
8 Motion of a Constant-Speed νconst Beam of Particles Inside an Expanding Universe Having the Expansion Law a( t )
9 Motion of c-Speed Electromagnetic Waves, Neutrinos and Gravitational Waves in an Expanding Universe Having the Expansion Law a( t )
10 Applications of the KLT to SETI
11 Conclusions
References
SETI Space Missions
1 Introduction
2 A SETI Space Mission that Never Was: Quasat
3 This Author’s Activity About Exploiting for SETI the Moon Farside Radio Quietness in the Fifteen Years 1995–2010
4 This Author’s First Presentation Ever at the United Nations COPUOS About Legally Protecting the Central Part of the Moon Farside Against Man-Made Radio Pollution
5 Defining PAC, the “Protected Antipode Circle”
6 Need for RFI-Free Radio Astronomy, as Pointed Since 1974 by Both ITU and Jean Heidmann (1923–2000)
7 A Short Review About the Five Lagrangian Points of the Earth-Moon System
8 The Quiet Cone Above the Moon Farside Depends on the Orbits of Telecommunication Satellites Orbiting the Earth
9 Selecting Crater Daedalus Near the Farside Center, i.e. the Near the Earth Antipode
10 Our 2010 Vision of the Moon Farside for RFI-free Science, Likely not Valid After 2018
11 The Further Two Lagrangian Points L1 and L2 of the Sun-Earth System: Their “Polluting” Action on the Farside of the Moon
12 Attenuation of Man-Made RFI on the Moon Farside
13 A New Mathematical Contribution of Ours: The Blocking Equation for Electromagnetic (em) Waves Emitted at Height H Above the Earth and Reaching the Earth-Moon Axis at a Distance X Above Moon Farside
14 The Years 2018–2019
15 The Farside Spectrum Still Is not Polluted (in August 2019) Except in the S, X and UHF Bands
16 Queqiao and Chang’e 4 Communications Bands Above the Moon Farside
17 COSMOLOGY: Need for Ultra-Low Frequency Radio Astronomy in Space Within the Quiet Cone Above the Moon Farside, I.E. just at the Lagrangian Point L2, Where Queqiao Is
18 SETI (or “Technosignatures”, According to NASA’s 2018 “New Jargon”)
19 Summary About This Author’s Work to Legally Protecting Radio Astronomy on the Farside and Within the Quiet Cone in the Space Above the Farside
20 CONCLUSIONS: The Moon Village Should Be Located Outside the PAC and Along the 180 Degrees Meridian, Possibly Close to the South Pole
References
Power and Energy of Civilizations by Logpar and Logell Power Curves (with Ancient Rome Example)
1 Part 1: Logell Curves and Their History Equations
1.1 Introduction to Logell “Finite Lifetime” Curves
1.2 Finding the Ellipse Equation of the Right Part of the Logell
1.3 Finding the b-Lognormal Equation of the Left Part of the Logell
1.4 Area Under the Ellipse on the Right Part of the Logell Between Peak and Death
1.5 Area Under the Full Logell Curve Between Birth and Death
1.6 The Area Under the Logell Curve Depends on Sigma Only, and Here Is the Area Derivative w.r.t. Sigma
1.7 Exact “History Equations” for Each Logell Curve
1.8 Considerations on the Logell History Equations
1.9 Logell Peak Coordinates Expressed in Terms of ( b,p,d ) Only
1.10 History of Rome as an Example of How to Use the Logell History Formulae
2 Part 2: Energy as the Area Under Logell Power Curves
2.1 Area Under Any Logell Power Curve and Its Meaning as “Lifetime Energy” of that Living Being
2.2 Discovering an Oblique Asymptote of the Energy Function Energy (D) While the Death Instant D Is Increasing Indefinitely
3 Part 3: Mean Power in a Logell Lifetime
3.1 Mean Power in a Logell Lifetime
4 Part 4: Logell Lifetime Mean Value
4.1 Lifetime Mean Value
5 Part 5: Conclusions: Which One Is Better? Logell or LOGPAR?
5.1 Conclusions About Rome’s Civilization
5.2 Conclusions About Evo-Seti Theory as of 2018
References
Logpars and Energy of Nine Historically Important Civilizations
1 Summary of Chapter “Energy of Extra-Terrestrial Civilizations According to Evo-SETI Theory” Results that Will Be Used in This Chapter
2 Finding the Logpars of Nine Civilizations that Made the History of the World
3 Peak Height of Each of the Nine Civilizations
4 Total Energy of Each Civilization
5 After-Peak Energy and Longterm History, that Is, Letting D Approach Infinity
6 SURPRISE: The MINIMUM AFTER-ENERGY Value Is THE SAME for All Nine Civilizations!
7 AFTER-PEAK-ENERGY Curves for All Nine Civilizations
8 Central Star, Longterm Energy and the Evolution of a Civilization There
9 LONGTERM POWER for All Nine Civilizations
10 MAXIMA CODE for All Calculations Described in This Chapter
Appendix
LOGPAR (=LOGnormal + PARabola) Power Curves of Nine Western Civilizations (3100 bc–2100 ad)
Finding the Expression of the Analytic LOGPAR
Numeric TRIPLET Input for Each Civilization
Numeric LOGPARs of Each Civilization
Plotting the LOGPARs
P = Power Peak of Every Civilization
Total Energy of Every Civilization
Longterm History (that is letting d -> infinity). Longterm Energy and Power for all Civilizations
MOLECULAR CLOCK as a Stochastic Process: Evo-Entropy (Shannon Entropy of Evolution) of a Geometric Brownian Motion (GBM) with a LINEAR MEAN VALUE
1 Introduction
2 Summary of the Mathematical Appendix Found in Chapter “OVERCOME Theorem, that is PEAK-LOCUS Theorem, Proven by Maxima” of This Book
3 A Strong Upper Bound upon the Standard Deviation in the Number of Species Living Today: Delta Ne << Ne2
4 MOLECULAR CLOCK: From a Straight-Line to a Lognormal Process having a straight-line Mean Value!
WORNOUT TIME (wt) at the End of One’s Lifetime: When the Lower Standard Deviation Curve (LSDC) of the Lognormal Process Finally Intercepts the Time Axis
1 Introduction
2 Time of Exctintion Risk (Wornout Time) for any Living Being
3 Consequences of the WORNOUT TIME Coming for any Living Being
4 INVERTING the WORNOUT TIME Equation
5 Conclusion