Tuesday, 4 July 2017


UNDERSTANDING NATURAL PHENOMENA: Self-Organization and Emergence in Complex Systems

Vinod Wadhawan

Book details
  • Paperback: 586 pages
  • Publisher: CreateSpace Independent Publishing Platform; 1 edition (July 4, 2017)
  • Language: English
  • ISBN-10: 1548527939
  • ISBN-13: 978-1548527938
  • Product Dimensions: 6.7 x 1.3 x 9.6 inches
  • Shipping Weight: 2.5 pounds

Can be ordered directly from the CreateSpace eStore also:

Legend for the front cover
A flower is a work of art, but there is no artist involved. The flower evolved from lesser things which, in turn, evolved from still lesser things, and so on. For example, the symmetry of a flower is the end result of a long succession of spontaneous processes and events, as also of some simple ‘local rules’ in operation, all constrained by the infallible second law of thermodynamics for ‘open’ systems. In fact, the second law is the mother of all organizing principles, leading to the enormous amounts of cumulative self-organization, structure, symmetry, and ‘emergence’ we see in Nature.

About the book
Science is all about trying to understand natural phenomena under the strict discipline imposed by the celebrated scientific method. Practically all the systems we encounter in Nature are dynamical systems, meaning that they evolve with time. Among them there are those that are either 'simple' or 'simplifiable' and can be handled by traditional, reductionistic science; and then there are those that are 'complex', meaning that nonreductionistic approaches must be attempted for understanding their evolution. In this book the author makes a case that the best way to understand a large number of natural phenomena, both simple and complex, is to focus on their self-organization and emergence aspects. Self-organization and emergence are rampant in Nature and, given enough time, their cumulative effects can be so mind-boggling that many people have great difficulty believing that there is no designer involved in the emergence of all the structure and order we see around us. But it is really quite simple to understand how and why we get so much 'order for free'. It all happens because, as ordained by the infallible second law of thermodynamics, the 'thermodynamically open' systems in our ever-expanding and cooling (and therefore gradient-creating) universe constantly tend to move towards equilibrium and stability, often ending up in ordered configurations if that helps minimize the free energy. In other words, order emerges because Nature tends to find efficient ways of annulling gradients of all types.

This book will help you acquire a good understanding of the essential features of natural phenomena, via the complex-science route. It has four parts: (1) Complexity Basics; (2) Pre-Human Evolution of Complexity; (3) Humans and the Evolution of Complexity; and (4) Appendices. The author gives centre-stage to the second law of thermodynamics for 'open' systems, which he describes as 'the mother of all organizing principles'. He also highlights a somewhat unconventional statement of this law: 'Nature abhors gradients'.

The book is written at two levels, one of which hardly uses any mathematical equations; the mathematical treatment of some relevant topics has been pushed to the last part of the book, in the form of ten appendices. Therefore the book should be accessible to a large readership. It is a general-science book written in a reader-friendly language, but without any dumbing down of the narrative.

I am a scientist and I take pride in the fact that we humans have invented and perfected the all-important scientific method for investigating natural phenomena. Wanting to understand natural phenomena is an instinctive urge in all of us. In this book I make a case that taking the complexity-science route for satisfying this urge can be a richly rewarding experience. Complexity science enables us (fully or partially) to find answers to even the most fundamental questions we may ask about ourselves and about our universe. We can call such questions the Big Questions: How did our universe emerge out of ‘nothing’ at a certain point in time; or is it that it has been there always? Why and how has structure arisen in our universe: galaxies, stars, planets, life forms? How did life emerge out of nonlife? How does intelligence emerge out of nonintelligence? These are difficult questions. But, as Mark Twain remarked, ‘There is something fascinating about science. One gets such wholesale of conjecture out of such a trifling investment of fact’. As you will see in this book, the Big Questions, as also many others, can be answered with a good amount of credibility by using just the following ‘trifling investment of facts’:

1.          Gradients tend to be obliterated spontaneously. Concentration gradients, temperature gradients, pressure gradients, etc. all tend to decrease spontaneously, till a state of equilibrium is reached, in which the gradients cannot fall any further. This is actually nothing but the infallible second law of thermodynamics, stated here in its most basic and general form. [Why do gradients arise at all? The original cause of all gradients in the cosmos is the continual expansion and cooling of our universe. At the local (terrestrial) level, the energy impinging on our ecosphere from the Sun is the main cause of gradients.]

2.          It requires energy to prevent a gradient from annulling itself, or to create a new gradient. A refrigerator works on this principle, as also so many other devices.

3.          Left to themselves, things go from a state of less disorder to a state of more disorder, spontaneously. This is the more familiar version of the second law of thermodynamics. Examples abound. Molecules in a gas occupy a larger volume spontaneously if the larger volume is made available; but there is practically no way they would occupy the smaller volume again, on their own.

4.          If a system is not left to itself, i.e. if it is not an isolated system and can therefore exchange energy and/or matter with its surroundings, then a state of lower disorder can sometimes arise locally. Growth of a crystal from a fluid is an example. A crystal has a remarkably high degree of order and design, even though there is no designer involved. To borrow a phrase from Stuart Kauffman, this is ‘order for free’.

5.          If a sustained input of energy drives a system far away from equilibrium, the system may develop a structure or tendencies which enable it to dissipate energy more and more efficiently. This is called dissipation-driven adaptive organization. England (2013) has shown that all dynamical evolution is more likely to lead to structures and systems which get better and better (i.e., more and more efficient) at absorbing and dissipating energy from the environment.

6.          The total energy of the universe is conserved. Since energy and mass are interconvertible, the term ‘energy’ used in this statement really means ‘mass plus energy’.

7.          Natural phenomena are governed by the laws of quantum mechanics. Classical mechanics, though adequate for understanding many day-to-day or ‘macroscopic’ phenomena, is only a special, limiting, case of quantum mechanics.

8.          There is an uncertainty principle in quantum mechanics, one version of which says that the energy-conservation principle can be violated, though only for a very small, well-specified duration. The larger the violation of energy conservation, the smaller this duration is.

9.          It can be understood in terms of the second law of thermodynamics that in a system of interacting entities, entirely new (unexpected) behaviour or properties can arise sometimes if the interactions are appropriate and strong enough. ‘More is different’ (Anderson 1972). The technical term for this occurrence is emergence. Complexity science is mostly about self-organization and emergence, and we shall encounter many examples of them in this book. To mention a couple of them here: the emergence of life out of nonlife; and the emergence of human intelligence in a system of nonintelligent entities, namely the neurons. Interestingly, the second law of thermodynamics is itself an emergent law. The motion of a molecule in a gas is governed by classical or ‘Newtonian’ mechanics, which has time-inversion symmetry, meaning that if you could somehow reverse the direction of time, the Newtonian equations of motion would still hold. And yet, when you put a large number of these molecules together, there are interactions among them and there emerges a direction of time: Time increases in the direction in which entropy increases. For similar reasons, even the causality principle is an emergent principle.

10.        The most adaptable are the most likely to survive and propagate. Any species, if it is not to become extinct, must be able to survive and propagate, even in an environment in which there is intra-species and/or inter-species competition because different individuals may all have to fight for the same limited resources like food or space. The fittest individuals or groups for this task (i.e., the most adaptable ones) stand a greater chance of winning this game and, as a result, the population gets better and better (more adapted) at survival and propagation in the prevailing conditions: the more adaptable or ‘fitter’ ones not only survive but also stand a greater chance to pass on their genes to the next generation.

It is remarkable that an enormous number and variety of natural phenomena can be understood in terms of these ten ‘commonsense’ facts, by adopting the complexity-science approach. Complexity science helps us understand, to a small or large extent, even those natural phenomena which fall outside the scope of conventional reductionistic science.

By definition, a complex system is one which comprises of a large number of ‘members’, ‘elements’ or ‘agents’, which interact substantially with one another and with the environment, and which have the potential to generate qualitatively new collective behaviour. That is, there can be an emergence of new (unexpected) spatial, temporal, or functional structures or patterns. Different complex systems have different ‘degrees of complexity’, and the amount of information needed to describe the structure and function of a system is one of the measures of that degree of complexity (Wadhawan 2010).

Complexity can be defined as something we associate with a complex system (described above). It is a technical term, and is not the same thing as ‘complicatedness’.

The idea of writing this book took shape when I was working on my book Smart Structures: Blurring the Distinction between the Living and the Nonliving (Wadhawan 2007). Naturally, there was extensive exposure to concepts from the field of complexity. Like the subject of smart structures, complexity science also cuts across various disciplines, and highlights the basic unity of all science. The uneasy feeling grew in me that, in spite of the fact that complexity is so pervasive and important, it is not introduced as a well-defined subject even to science students. They are all taught, say, thermodynamics and quantum mechanics routinely, but not complexity science. Even among research workers, although a large number are working on one complex system or another (and not just in physics or chemistry, but also in biology, brain science, computational science, economics, etc.), not many have learnt about the basics of complexity science in a coherent manner at an early stage of their career. I have tried to write a book on complexity that takes this subject to the classroom at a fairly introductory but comprehensive level. There is no dumbing down of facts, even at the cost of appearing ‘too technical’ at times.

Here are some examples of complex systems: beehives; ant colonies; self-organized supramolecular assemblies; ecosystems; spin-glasses and other complex materials; stock markets; economies of nations; the world economy; and the global weather pattern. The origin and evolution of life on Earth was itself a series of emergent phenomena that occurred in highly complex systems. Evolution of complexity is generally a one-way traffic: The new emergent features may (in principle) be deducible from, but are not reducible to, those operating at the next lower level of complexity. Reductionism stands discounted.

As I said earlier, emergent behaviour is a hallmark of complex systems. Human intelligence is also an emergent property: Thoughts, feelings, and purpose result from the interactions among the neurons. Similarly, even memories are emergent phenomena, arising out of the interactions among the large number of ‘unmemory-like’ fragments of information stored in the brain.

What goes on in a complex system is essentially as follows: There is a large number of interacting agents, which may be viewed as forming a network. In the network-theory jargon, the agents are the ‘nodes’ of the network, and a line joining any two nodes (i.e. an ‘edge’) represents the interaction between that pair of agents. Any interaction amounts to communication or exchange of information. The action or behaviour of each agent is determined by what it ‘sees’ others doing, and its actions, in turn, determine what the other agents may do. The term game-playing is used for this mutual interaction in the case of those complex systems in which the agents are ‘thinking’ organisms (particularly humans). Therefore a partial list of topics covered in this book is: information theory; network theory; cellular automata; game theory.

Exchange of information in complex systems, controlled like other macroscopic phenomena by the second law of thermodynamics, leads to self-organization and emergence. Biological evolution is a natural and inevitable consequence of these ongoing processes because of the cumulative effects of mutations and natural selection. That is why this book has chapters on evolution of complexity of all types: cosmic, chemical, biological, artificial, cultural.

Networked or ‘webbed’ systems have the all-important nonlinearity feature. In fact, nonlinear response, in conjunction with substantial departure from equilibrium, is the crux of complex behaviour. There are many types of nonlinear systems. The most important for our purposes in this book are those in which, although the output (y) is proportional to the input (x), the proportionality factor is not independent of the input: y = m(x) x. This has far-reaching consequences for the (always networked) complex system. In particular, its future progression of events is very sensitive to conditions at any particular point of time (the so-called ‘initial conditions’). This sensitivity to initial conditions is also the hallmark of chaotic systems. In fact, there is a well-justified viewpoint that it is impossible to discuss several types of complex systems without bringing in concepts from chaos theory. And, what is more, complex systems tend to evolve to a configuration wherein they can operate near the so-called edge of chaos (neither too much order, nor too much chaos). There is a chapter on chaos which elaborates on these things.

Inanimate systems can also be complex. Whirlpools and whirlwinds are familiar examples of dynamic nonbiological complex systems. Even static physical systems like some nanocomposites may exhibit properties that cannot always be deduced from those of the constituents of the composite. A particularly fascinating class of complex materials are the so-called multiferroics. A multiferroic is actually a ferroic crystalline material (a ‘natural’ composite) which just refuses to be homogeneous over macroscopic length scales, so that the same crystal may be, say, ferroelectric in some part, and ferromagnetic in another. In a multiferroic, two or all three of the electric, magnetic and elastic interactions compete in a delicately balanced manner, and even a very minor local factor can tilt the balance in favour of one or the other. This class of materials offers great scope for basic research and for device applications, particularly in smart structures.

The current concern about ecological conservation and global warming points to the need for a good understanding of complex systems, particularly their holistic nature. Mother Earth is a single, highly complex system, now increasingly referred to as the system Earth. The poet Thich Nhat Hanh (1991) has beautifully described the unity and interdependence of all things:

If you are a poet, you will see clearly that there is a cloud floating in this sheet of paper. Without a cloud, there will be no rain; without rain, the trees cannot grow; and without trees, we cannot make paper. The cloud is essential for the paper to exist. If the cloud is not here the sheet of paper cannot be here either. So we can say that the cloud and the paper inter-are. “Interbeing” is a word that is not in the dictionary yet, but if we combine the prefix “inter” with the verb “to be,” we have a new verb, inter-be.

If we look into this sheet of paper even more deeply, we can see the sunshine in it. Without sunshine, the forest cannot grow. In fact, nothing can grow without sunshine. And so, we know that the sunshine is also in this sheet of paper. The paper and the sunshine inter-are. And if we continue to look we can see the logger who cut the tree and brought it to the mill to be transformed into paper. And we see wheat. We know that the logger cannot exist without his daily bread, and therefore the wheat that became his bread is also in the sheet of paper. The logger’s father and mother are in it too. When we look in this way, we see that without all these things, this sheet of paper cannot exist.

Looking even more deeply, we can see ourselves in this sheet of paper too. This is not difficult to see, because when we look at a sheet of paper, it is part of our perception. Your mind is in here and mine is also. So we can see that everything is in here with this sheet of paper. We cannot point out one thing that is not heretime, space, the earth, the rain, the minerals in the soil, the sunshine, the cloud, the river, the heat. Everything co-exists with this paper. That is why I think the word inter-be should be in the dictionary. “To be” is to inter-be  -- we cannot just be by ourselves alone. We have to inter-be with every other thing. This sheet of paper is, because everything else is.

Suppose we try to return one of the elements to its source. Suppose we return the sunshine to the sun. Do you think that this sheet of paper would be possible? No, without sunshine nothing can be. And if we return the logger to its mother, then we have no sheet of paper either. The fact is that this sheet of paper is made up only of “non-paper” elements. And if we return these non-paper elements to their sources, then there can be no paper at all. Without non-paper elements, like mind, logger, sunshine and so on, there will be no paper. As thin as this sheet of paper is, it contains everything in the universe in it.

A better understanding of complexity may well become a matter of life and death for the human race. And the subject of complexity science is still at the periphery of science. It has not yet become mainstream, in the sense that it is not taught routinely even at the college level. That cannot go on.

There are already a substantial number of great books on complexity science, and I have drawn on them. But I believe that this book is student-friendly and teacher-friendly, and it brings home the all-pervasive nature of the subject. Here are the salient features of the book:

1. It provides an update on the subject.

2. It can serve as introductory or supplementary reading for an undergraduate or graduate course on any branch of complexity science.

3. Practically all the mathematical treatment of the subject has been pushed to the appendices at the end of the book, so the main text can be comprehended even by those who are not too comfortable with equations. This is important because a large fraction of the educated public must get the hang of the nature of complexity, so that we can successfully meet the challenges posed to our very survival as a species.

4. Both among scientists and nonscientists there is a large proportion of people who are insufficiently trained about the explaining power of complexity science when it comes to some of the deepest puzzles of Nature and, hopefully, this book would help remedy the situation to some extent.

5. A proper understanding of what complexity science has already achieved will also help discredit many of the claims of mystics, supernaturalists, and pseudoscientists.

Bengaluru                                                                                                                                                                     Vinod Wadhawan
July, 2017



I. Complexity Basics

1.              Overview
1.1             Preamble
1.2             A whirlpool as an example of self-organization
1.3             Spontaneous pattern formation: the Bénard instability
1.4             Recent history of investigations in complexity science
1.5             Organization of the book

2.              The Philosophical and Computational Underpinnings of
                 Complexity Science
2.1             The scientific method for investigating natural phenomena
2.2             Reductionism and its inadequacy for dealing with complexity
2.3             The Laplace demon
2.4             Holism
2.5             Emergence
2.6             Scientific determinism, effective theories
2.7             Free will
2.8             Actions, reactions, interactions, causality
2.9             The nature of reality

3.              The Second Law of Thermodynamics
3.1             The second law for isolated systems
3.2             Entropy
3.3             The second law for open systems
3.4             Nucleation and growth of a crystal
3.5             The second law is an emergent law
3.6             Emergence, weak and strong
3.7             Nature abhors gradients
3.8             Systems not in equilibrium
3.9             Thermodynamics of small systems

4.              Dynamical Evolution
4.1             Dynamical systems
4.2             Phase-space trajectories
4.3             Attractors in phase space
4.4             Nonlinear dynamical systems
4.5             Equilibrium, stable and unstable
4.6             Dissipative structures and processes

4.7             Bifurcations in phase space

4.8             Self-organization and order in dissipative structures

5.              Relativity Theory and Quantum Mechanics
5.1             Special theory of relativity
5.2             General theory of relativity
5.3             Quantum mechanics
5.4             Summing over multiple histories

6.              The Nature of Information
6.1             Russell's paradox
6.2             Hilbert’s formal axiomatic approach to mathematics
6.3             Gödel’s incompleteness theorem
6.4             Turing’s halting problem
6.5             Elementary information theory
6.6             Entropy means unavailable or missing information
6.7             Algorithmic information theory
6.8             Algorithmic probability and Ockham’s razor
6.9             Algorithmic information content and effective complexity
6.10           Classification of problems in terms of computational complexity
6.11           Irreducible complexity’ deconstructed

7.              Darwinian Evolution, Complex Adaptive Systems, Sociobiology  
7.1             Darwinian evolution
7.2             Complex adaptive systems
7.3             The inevitability of emergence of life on Earth

7.4             Sociobiology, altruism, morality, group selection

8.              Symmetry is Supreme
8.1             Of socks and shoes
8.2             Connection between symmetry and conservation laws
8.3             Why so much symmetry?
8.4             Growth of a crystal as an ordering process
8.5             Broken symmetry
8.6             Symmetry aspects of phase transitions
8.7             Latent symmetry
8.8             Potential symmetry
8.9             The fundamental theorem of symmetry
8.10           The distinction between potential symmetry and latent symmetry
8.11           Emergent symmetry and complexity
8.12           Broken symmetry and complexity
8.13           Symmetry of complex networks

9.              The Standard Model of Particle Physics
9.1             The four fundamental interactions
9.2             Bosons and fermions
9.3             The standard model and the Higgs mechanism

10.            Cosmology Basics
10.1           The ultimate causes of all cosmic order and structure
10.2           The Big Bang and its aftermath
10.3           Dark matter and dark energy
10.4           Cosmic inflation
10.5           Supersymmetry, string theories, M-theory
10.6           Has modern cosmology got it all wrong?

11.            Uncertainty, Complexity, and the Arrow of Time
11.1           Irreversible processes can lead to order
11.2           The arrow of time and the early universe
11.3           When did time begin?
11.4           Uncertainty and complex adaptive systems

12.            The Cosmic Evolution of Complexity
12.1           Our cosmic history
12.2           We are star stuff

13.            Why Are the Laws of Nature What They Are?
13.1           The laws of Nature in our universe
13.2           The anthropic principle

14.            The Universe is a Quantum Computer
14.1           Quantum computation
14.2           Quantum entanglement
14.3           The universe regarded as a quantum computer

15.            Chaos, Fractals, and Complexity
15.1           Nonlinear dynamics
15.2           Extreme sensitivity to initial conditions
15.3           Chaotic rhythms of population sizes
15.4           Fractal nature of the strange attractor
15.5           Chaos and complexity

16.            Cellular Automata as Models of Complexity
16.1           Cellular automata
16.2           Conway's Game of Life
16.3           Self-reproducing automata
16.4           The four Wolfram classes of cellular automata
16.5           Universal cellular automata

17.            Wolfram's 'New Kind of Science'
17.1           Introduction
17.2           Wolfram’s principle of computational equivalence (PCE)
17.3           The PCE and the rampant occurrence of complexity
17.4           Why does the universe run the way it does?
17.5           Criticism of Wolfram’s NKS

18.            Swarm Intelligence
18.1           Emergence of swarm intelligence in a beehive
18.2           Ant logic
18.3           Positive and negative feedback in complex systems

19.            Nonadaptive Complex Systems
19.1           Composite materials
19.2           Ferroic materials
19.3           Multiferroics
19.4           Spin glasses
19.5           Relaxor ferroelectrics
19.6           Relaxor ferroelectrics as vivisystems

20.            Self-Organized Criticality
20.1           The sandpile experiment
20.2           Power-law behaviour and complexity
20.3           Robust and nonrobust criticality

21.            Characteristics of Complex Systems

II. Pre-Human Evolution of Complexity

22.            Evolution of Structure and Order in the Cosmos
22.1           The three eras in the cosmic evolution of complexity
22.2           Chaisson’s parameter for quantifying the degree of complexity
22.3           Cosmic evolution of information
22.4           Why so much terrestrial complexity?

23.            The Primary and Secondary Chemical Bonds
23.1           The primary chemical bonds
23.2           The secondary chemical interactions
23.3           The hydrogen bond and the hydrophobic interaction

24.            Cell Biology Basics

25.            Evolution of Chemical Complexity
25.1           Of locks and keys in the world of molecular self-assembly
25.2           Self-organization of matter
25.3           Emergence of autocatalytic sets of molecules
25.4           Positive feedback, pattern formation, emergent phenomena
25.5           Pattern formation: the BZ reaction

26.            What is Life?
26.1           Schrödinger and life
26.2           Koshland’s seven pillars of life

27.            Models for the Origins of Life
27.1           The early work
27.2           The RNA-world model for the origin of life
27.3           Dyson's proteins-first model for the origins of life

28.            Genetic Regulatory Networks and Cell Differentiation
28.1           Circuits in genetic networks
28.2           Kauffman's work on genetic regulatory networks

29.            Ideas on the Origins of Species: From Darwin to Margulis
29.1           Darwinism and neo-Darwinism
29.2           Biological symbiosis and evolution
29.3           What is a species
29.4           Epigenetics

30.            Coevolution of Species
30.1           Punctuated equilibrium in the coevolution of species
30.2           Evolutionarily stable strategies
30.3           Of hawks and doves in the logic of animal conflicts
30.4           Evolutionary arms races and the life-dinner principle

31.            Niele’s Energy-Staircase of Increasing Complexity
31.1           The thermophilic energy regime
31.2           The phototrophic energy regime
31.3           The aerobic energy regime

III. Humans and the Evolution of Complexity

32.            The Post-Human Epoch of Niele’s Energy Staircase of Evolution
32.1           The pyrocultural energy regime
32.2           The agrocultural energy regime
32.3           The carbocultural energy regime
32.4           The green-valley approach to System Earth
32.5           The imperial approach to System Earth
32.6           A nucleocultural energy regime?
32.7           A possible 'heliocultural' energy regime

33.            Computational Intelligence
33.1           Introduction
33.2           Fuzzy logic
33.3           Artificial neural networks
33.4           Genetic algorithms
33.5           Genetic programming: Evolution of computer programs

33.6           Artificial life

34.            Modelling of Adaptation and Learning in Complex Adaptive Systems
34.1           Holland’s model for adaptation and learning
34.2           The bucket brigade in Holland's algorithm

34.3           Langton’s work on adaptive computation

34.4           The edge-of-chaos existence of complex adaptive systems

 35.            Robots
35.1           Behaviour-based robotics
35.2           Evolutionary robotics
35.3           Evolution of computer power per unit cost

36.            Smart Structures
36.1           The three main components of a smart structure
36.2           Reconfigurable computers and machines that can evolve

37.             Machine Intelligence
37.1           Artificial distributed intelligence
37.2           Evolution of machine intelligence
37.3           The future of intelligence

38.            Evolution of Language

39.            Memes and Their Evolution

40.            Evolution of the Human Brain, and the Nature of Our Neocortex
40.1           Evolution of the brain
40.2           The human neocortex
40.3           The history of intelligence

41.            Minsky’s and Hawkins’ Models for how Our Brain Functions

41.1           Marvin Minsky’s ‘Society of Mind’

41.2           Can we make decisions without involving emotions?

41.3           Hawkins’ model for intelligence and consciousness

42.            Inside the Human Brain
42.1           Probing the human Brain

42.2           Peering into the human brain

43.            Kurzweil's Pattern-Recognition Theory of Mind

44.            The Knowledge Era and Complexity Science
44.1           The wide-ranging applications of complexity science
44.2           Econophysics
44.3           Application of complexity ideas in management science
44.4           Cultural evolution and complexity transitions
44.5           Complexity leadership theory
44.6           Complexity science in everyday life

45.            Epilogue

IV. Appendices

A1.            Equilibrium Thermodynamics and Statistical Mechanics
A1.1           Equilibrium thermodynamics
A1.2           Statistical mechanics
A1.3           The ergodicity hypothesis
A1.4           The partition function
A1.5           Thermodynamics of small systems

A2.            Probability Theory
A2.1           The notion of probability
A2.2           Multivariate probabilities
A2.3           Determinism and predictability

A3.            Information and Uncertainty
A3.1           Information theory
A3.2           Shannon’s formula for a numerical measure of information
A3.3           Shannon entropy and thermodynamic entropy
A3.4           Uncertainty
A3.5           Algorithmic information theory
A3.6           Entropic uncertainty relations

A4.            Thermodynamics and Information
A4.1           Entropy and information
A4.2           Kolmogorov-Sinai entropy
A4.3           Mutual information and redundancy of information

A5.            Systems Far from Equilibrium
A5.1           Emergence of complexity in systems far from equilibrium
A5.2           Nonequilibrium classical dynamics
A5.3           When does the Newtonian description break down?
A5.4           Generalization of Newtonian dynamics
A5.5           Pitchfork bifurcation
A5.6           Extension of Newton's laws

A6.            Quantum Theory and Particle Physics
A6.1           Introduction
A6.2           The Heisenberg uncertainty principle
A6.3           The Schrödinger equation
A6.4           The Copenhagen interpretation
A6.5           Time asymmetry
A6.6           Multiple universes
A6.7           Feynman's sum-over-histories formulation
A6.8           Quantum Darwinism
A6.9           Gell-Mann's coarse-graining interpretation
A6.10          Poincaré resonances and quantum theory
A6.11          Model-dependent realism, intelligence, existence
A6.12          Particle physics

A7.            Theory of Phase Transitions and Critical Phenomena
A7.1           A typical phase transition
A7.2           Liberal definitions of phase transitions
A7.3           Instabilities can cause phase transitions
A7.4           Order parameter of a phase transition
A7.5           The response function corresponding to the order parameter
A7.6           Phase transitions near thermodynamic equilibrium
A7.7           The Landau theory of phase transitions
A7.8           Spontaneous breaking of symmetry
A7.9           Field-induced phase transitions
A7.10          Ferroic phase transitions
A7.11          Prototype symmetry
A7.12         Critical phenomena
A7.13          Universality classes and critical exponents
A7.14          Scaling relations
A7.15          Renormalization-group theory

A8.            Chaos Theory
A8.1           The logistic equation
A8.2           Lyapunov exponents
A8.3           Divergence of neighbouring trajectories
A8.4           Chaotic attractors

A9.            Network Theory and Complexity
A9.1           Graphs
A9.2           Networks
A9.3           The travelling salesman problem
A9.4           Random networks
A9.5           Percolation transitions in random networks
A9.6           Small-world networks
A9.7           Scale-free networks
A9.8           Evolution of complex networks
A9.9           Emergence of symmetry in complex networks
A9.10          Kauffman’s work on genetic regulatory networks . .
A9.11         Chua’s cellular nonlinear networks as a paradigm for emergence and complexity

A10.           Game Theory
A10.1          Introduction
A10.2          Dual or two-player games
A10.3          Noncooperative games
A10.4          Nash equilibrium
A10.5          Cooperative games



About the Author


As science turns to complexity, one must realize that complexity demands attitudes quite different from those heretofore common in physics. Up to now, physicists looked for fundamental laws true for all times and all places. But each complex system is different: apparently there are no general laws for complexity. Instead, one must reach for ‘lessons’ that might, with insight and understanding, be learned in one system and applied to another. Maybe physics studies will become more like human experience.
                                                               Goldenfeld and Kadanoff (1999)

I think the next century will be the century of complexity.
                                                               Stephen Hawking (2000)

Successful ecosystems are complex adaptive systems, as are successful cities and societies. According to the scientist James Lovelock’s Gaia concept, the Earth as a whole is a complex adaptive system. One of its long-term adaptations that should be of concern to all of us may well be to get rid of our species to protect itself. Whether that happens or not could come down to whether we are able to understand the rules that govern its complexity, and whether we have the wisdom to adapt ourselves and conform to those rules.
                                                                                 Len Fisher (2009)