Untangling Complex Systems

  Untangling Complex Systems

  Untangling Complex Systems

  A Grand Challenge for Science

  Pier Luigi Gentili

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  Dedication

  To God, my family, all my past, present, and next students, and

  whoever will read this book with curiosity and open-mindedness.

  Contents

  Preface..............................................................................................................................................xv

  Acknowledgments ..........................................................................................................................xvii

  About the Author ............................................................................................................................xix

  Chapter 1 Introduction ..................................................................................................................1

  1.1

  The Never-Ending Journey to Discovering the Secrets of Nature ....................1

  1.1.1 The “Practical Period” .........................................................................4

  1.1.2 The “Philosophical Period” ..................................................................5

  1.1.3 The “Experimental Period” ..................................................................8

  1.1.4 The “Computational Period” .............................................................. 15

  1.2

  What Is Science, Today? .................................................................................. 16

  1.3

  Purpose and Contents of This Book ................................................................ 18

  1.4

  Key Questions ..................................................................................................20

  1.5

  Key Words .......................................................................................................20

  1.6

  Hints for Further Reading ...............................................................................20

  Chapter 2 Reversibility or Irreversibility? That Is the Question! ............................................... 21

  2.1 Introduction ..................................................................................................... 21

  2.2

  The Thermodynamic Approach ......................................................................22

  2.2.1 The Classical Definition of Entropy ...................................................23

  2.2.2 The Statistical Definition of Entropy .................................................24

  2.2.3 The Logical Definition of Entropy .....................................................27

  2.3

  An Exhausting Fight against Entropy .............................................................30

  2.3.1 The Maxwell’s Demon .......................................................................30

  2.3.2 A First Mechanical Attempt ...............................................................30

  2.3.3 Another Mechanical Attempt ............................................................. 31

  2.3.4 The Involvement of Artificial Intelligence: A “Thought

  Experiment” ....................................................................................... 32

  2.3.5 The Embodiment of Maxwell’s Demon: A “Real Experiment” ........34

  2.3.6 The Surprising Behavior of Small Systems ....................................... 35

  2.3.7 There Is Still an Open Question ......................................................... 38

  2.4

  Key Questions .................................................................................................. 38

  2.5

  Key Words ....................................................................................................... 38

  2.6

  Hints for Further Reading ............................................................................... 39

  2.7 Exercises .......................................................................................................... 39

  2.8

  Solutions to the Exercises ................................................................................ 39

  Chapter 3 Out-of-Equilibrium Thermodynamics ....................................................................... 41

  3.1 Introduction ..................................................................................................... 41

  3.2

  Definition of the Entropy Change for an Out-of-Equilibrium System ............ 41

  3.2.1 Heat Conduction ................................................................................. 43

  3.2.2 Chemical Reactions ............................................................................44

  3.2.3 Diffusion............................................................................................. 47

/>   3.2.4 Migration ............................................................................................ 47

  3.2.5 Generalization ....................................................................................50

  vii

  viii

  Contents

  3.3

  Non-equilibrium Thermodynamics in Linear Regime .................................50

  3.3.1

  Fourier’s Law: The Law of Heat Conduction ..................................50

  3.3.2

  Ohm’s Law: The Law of Electrical Conduction .............................. 51

  3.3.3

  Poiseuille’s Law: The Law of Laminar Flow of Fluids ................... 52

  3.3.4

  Fick’s Law: The Law of Diffusion ................................................... 53

  3.3.5

  Generalization: Symmetry Principle and Onsager Reciprocal

  Relations ...........................................................................................54

  3.3.6

  An Experimental Proof of the Reciprocal Relations ....................... 57

  3.3.7 Cross-Diffusion ................................................................................ 59

  3.3.8

  Thermal Diffusion ........................................................................... 61

  3.4

  Evolution of Out-of-Equilibrium Systems in Linear Regime ....................... 63

  3.4.1

  The Case of Heat Conduction ..........................................................64

  3.4.2

  The Case of Diffusion ......................................................................65

  3.5

  The Theorem of Minimum Entropy Production in Linear Regime .............. 67

  3.5.1

  A Single Force and Flow ................................................................. 67

  3.5.2

  The Case of More Than One Force and One Flow ..........................68

  3.6

  Evolution of Out-of-Equilibrium Systems in Nonlinear Regime ..................69

  3.6.1

  Chemical Reactions .........................................................................69

  3.6.2

  The Glansdorff-Prigogine Stability Criterion .................................. 71

  3.7

  The Chemical Transformations and the Linear Regime ............................... 72

  3.7.1

  Onsager’s Reciprocal Relations for Chemical Reactions................. 73

  3.7.2

  A Particular Case ............................................................................. 73

  3.8

  The Evolution of Chemical Reactions in Open Systems ............................... 74

  3.8.1

  The Mono-Dimensional Case .......................................................... 75

  3.8.2

  The Bi-Dimensional Case ................................................................ 76

  3.8.3

  The Multi-Dimensional Case ........................................................... 81

  3.9

  Key Questions ............................................................................................... 81

  3.10

  Key Words ..................................................................................................... 82

  3.11

  Hints for Further Reading ............................................................................. 82

  3.12 Exercises ........................................................................................................ 82

  3.13

  Solutions to the Exercises ..............................................................................86

  Chapter 4 An Amazing Scientific Voyage: From Equilibrium up to Self-Organization

  through Bifurcations ..................................................................................................97

  4.1 Introduction ...................................................................................................97

  4.2 Bifurcations ................................................................................................. 101

  4.2.1

  Saddle-Node Bifurcation ............................................................... 101

  4.2.2

  Trans-Critical Bifurcation .............................................................. 102

  4.2.2.1 From a Lamp to a Laser: An Example of

  Trans-Critical Bifurcation .............................................. 103

  4.2.3

  Pitchfork Bifurcation ..................................................................... 105

  4.2.3.1 Chiral Symmetry Breaking ............................................ 105

  4.2.4

  Hopf Bifurcations ........................................................................... 110

  4.3

  Key Questions ............................................................................................. 111

  4.4

  Key Words ................................................................................................... 111

  4.5

  Hint for Further Reading ............................................................................. 111

  4.6 Exercises ...................................................................................................... 112

  4.7

  Solutions to the Exercises ............................................................................ 112

  Contents

  ix

  Chapter 5 The Emergence of Temporal Order in the Ecosystems ............................................ 117

  5.1 Introduction ................................................................................................. 117

  5.2

  Predator-Prey Relationship: The Lotka-Volterra Model ............................. 117

  5.3

  Entropy Production in the Lotka-Volterra Model ....................................... 121

  5.4

  More about Predator-Prey Relationships ..................................................... 123

  5.5

  Other Relationships within an Ecosystem ................................................... 129

  5.6

  Mathematical Modeling of Symbiotic Relationships .................................. 130

  5.6.1 Antagonism .................................................................................... 130

  5.6.2 Mutualism ...................................................................................... 133

  5.7

  Key Questions ............................................................................................. 135

  5.8

  Key Words ................................................................................................... 135

  5.9

  Hints for Further Reading ........................................................................... 135

  5.10 Exercises ...................................................................................................... 136

  5.11

  Solutions to the Exercises ............................................................................ 137

  Chapter 6 The Emergence of Temporal Order in the Economy ............................................... 147

  6.1 Introduction ................................................................................................. 147

  6.2

  The Economic Problem ............................................................................... 148

  6.3

  Linear and Circular Economy ........................................
............................. 149

  6.4

  The Law of Supply and Demand ................................................................. 151

  6.5

  The Business Cycles .................................................................................... 152

  6.5.1

  Goodwin’s Predator-Prey Model.................................................... 153

  6.5.2

  The Multiplier and Accelerator Model .......................................... 156

  6.5.3

  Other Models ................................................................................. 158

  6.5.4

  The Real Business Cycles .............................................................. 159

  6.6

  Key Questions ............................................................................................. 160

  6.7

  Key Words ................................................................................................... 160

  6.8

  Hints for Further Reading ........................................................................... 160

  6.9 Exercises ...................................................................................................... 160

  6.10

  Solutions to the Exercises ............................................................................ 161

  Chapter 7 The Emergence of Temporal Order within a Living Being ..................................... 167

  7.1 Introduction ................................................................................................. 167

  7.2

  Metabolic Events ......................................................................................... 167

  7.2.1

  Michaelis-Menten Kinetics ............................................................ 168

  7.2.2

  Hill Kinetics ................................................................................... 169

  7.2.3

  The Nonlinearity of Allosteric Enzymes ....................................... 170

  7.2.4 Glycolysis ....................................................................................... 174

  7.3

  Cellular Signaling Processes ....................................................................... 176

  7.3.1

  The Simplest Signal Transduction System..................................... 176