Complexity Miscellany for Beginners: Part I

Introduction

At this moment the light from the screen illuminates your face, photons jump from the luminous screen to your retinas, your brain begins to decode a bunch of symbols that you call letters, the combinations of those letters generate words, then your cerebral cortex begins to interpret the union of those words and couples them into sentences that have an abstract meaning.

It takes a few nanoseconds for the information you are reading to be converted into electricity, to be synthesized and deciphered by your brain, that organ in your head. It is so complex that it is not fully understood nowadays; the number of enigmas that exist regarding the functioning of the brain is far greater than the number of mysteries within theoretical physics.

Let’s put our central nervous system into practice and do some neurobics. The fundamental component of this organ are neurons, small cells that receive and transmit information through a process called synapse. Although the structure and functioning of neurons is relatively simple, the interaction between millions of them allows human beings to develop consciousness, language and other characteristics that almost distinguish them from other living beings.

The brain shows a kind of emergence independently of the individual neurons that make it up, its behavior goes beyond the sum of its parts, and a structure that meets the above characteristic is called a complex system. Today there is no general theory that predicts and explains the behavior of complex systems as a whole, however, there is a set of tools that allow us to study the general properties that describe this group of phenomena.

Feedback, entropy, autopoiesis, criticality… There are so many ways to characterize complex systems. Some are more ambiguous or abstract than others. The present essay—divided in three volumes—has as its main objective to give you a general—but not ambiguous—notion of those mysterious concepts. The first volume of this issue is devoted to some of the physical aspects that define complex systems. The second installment, on the other hand, will focus more on the biological aspects that shape the notions we understand today as complexity. Finally, the third and last part of the series will seek to unravel the concept of emergence with emphasis on phenomena related to cognition. As such, there should be no problem if the reader wishes to approach any of these volumes in random order. However, I suggest that you follow the ascending numerical order, as this will give you a much clearer perspective of what complex systems are.

But how to start? It is estimated that only 200,000 years ago the first homo sapiens sapiens appeared. It is captivating to know that for the Romans the pyramids of Giza were older than they are for us, but it is perhaps more surprising to realize that the history of humanity is only 0.0004% of the history of the observable universe, which is approximately 13.7 billion years. As astronomer and science communicator Carl Sagan rightly says in the headline of the present article, in order to understand and manufacture a brain—or any other object—we must first create the universe.

Feedback

The number of neurons in the human brain is of the order of $10^{11}$; thus, we can say that the brain is a very large place in a very small space, and although this number may seem immeasurable, is only 0.00000000001% of the total number of stars in the universe, whose order is around $10^{24}$.

It is during galaxy formation that we first observe the feedback phenomenon; there have already been authors—such as theoretical physicist Lee Smolin—who argue that galaxies should be considered as living systems. But where does such a claim come from? To answer this question, let’s start with a simpler one: why do most galaxies have a spiral shape? This fact is exclusively due to a feedback mechanism.

All galaxies come from immense gas clouds that can contain up to a million times the mass of the Sun: gravity collapses small clusters of all this gas and thus stars form. When the stars begin to shine, the energy of this light forms a bubble within the gas cloud and this bubble tends to slow down star formation. However, once the stars grow and explode into supernovae, the shock wave puts pressure on the nearby interstellar clouds and causes them to begin to contract.

Waves from different supernovae mix with each other, sweeping through the interstellar material and forming new gas clouds that produce more stars and thus more supernovae. This is how star formation maintains itself: using energy absorption through a feedback process. Importantly, the density in the clouds is ideal for this self-sustaining process to continue.

If the density is too high, gravity will cause the inner part to contract very fast, forming a few stars with short life cycles that will burst the cloud into pieces and form a thinner cloud. If the cloud is not very dense, many stars will be born, which sooner or later will become supernovae whose multiple shock waves will make the interstellar gas denser. The spiral shape is the result of the rotation of the galaxy and the force subjected by dark matter. It is analogous to when we put milk into the coffee and we mix it: we not only rotate the mixture, but also the milk is subjected to the caffeine.

The diversity of composition, temperature and density in the interstellar medium is far from being uniform, it is not in equilibrium and it also self-organizes by means of the described feedback. By virtue of the above, galaxies are zones where entropy is reduced and consequently they are very similar to life. It is worth taking a short pause to dismember the concept of entropy; we will do it in the most pleasant possible way.

Entropy

Go to the kitchen and take out a plate, then go to the nearest refrigerator and take an ice; now put the ice on the plate and wait. Surely, even without psychic abilities, you already know what is going to happen. The ice will melt to leave a small puddle of water on the plate. But understanding entropy and its secrets goes beyond that, for we must think of the molecular version of the experiment. At the beginning, the water molecules are embedded in the ice, all tightly bound and arranged in a cubic lattice, where each edge is a water molecule.

As time progresses, the water molecules, agitated by the temperature, will leave the delicate structure that forms the ice and spread throughout the dish. Now, why don’t we consider the possibility that the water molecules remain in the cube? This is a matter of probability. When comparing the number of possibilities in which the molecules can stay forming the ice structure and the number of ways in which they can be dancing around the dish, it is clear that the difference is enormous.

We are playing in a roulette where almost all the squares are of the same color, we already know in advance what the winning bet is! That is why on every occasion, in every game played, we see the ice cube melting. So, what the second principle of thermodynamics essentially tells us is that systems tend to disordered states and entropy is nothing more than a number that says how disordered the configuration is.

Let’s put the pieces of the puzzle on the table: for self-organization to emerge, feedback processes are necessary and the way in which systems are kept out of thermodynamic equilibrium is by reducing entropy, which is synonymous with order, but the thaumaturgy does not end here. For now, let us continue studying the evolution of the universe.

Abiogenesis

Once the stars formed, they began to use gravity to form structures that hovered around them; these remnants of matter clumped together spherically and became planets, giving rise to the era of solar systems. When we look at a starry sky, in front of us we contemplate a bunch of twinkling dots that capture the eye and elevate our imagination.

If an observer points his telescope at our solar system from the Andromeda galaxy, he will surely find that what we call the Sun is just another sparkling speck in the immense celestial dome; each one of those sparkling points that we perceive is the Sun of another solar system! Is there anything that distinguishes our solar system from the others? There are gaseous and rocky planets in all of them, some of them have extremely hot or extremely cold temperatures.

However, for some stars we know that there is a zone of habitability in which the flux of incident radiation allows the presence of water in a liquid state on the surface of any rocky planet (or satellite); it is as if the Sun of certain solar systems functioned as an incubator for life. Let us proceed with mental gymnastics: the diameter of the observable universe is about 90 billion light years with at least 100 billion galaxies, each with between 100 billion and one trillion stars.

The above mentioned gives us the guideline to propose the existence of trillions of habitable planets, which shows that the probability that life has developed and exists is very high. Shouldn’t the universe be full of spaceships and extraterrestrial species wanting to communicate with us? This unknown is called Fermi’s paradox. Even if there are alien civilizations in other galaxies it is impossible for us to get to know them due to the expansion of the universe, so it is worth focusing only on our own galaxy.

The Milky Way has about 20 billion stars similar to the Sun and it is estimated that one fifth of them have a planet the size of the Earth in the habitable zone. If only 0.1% of those planets had life, there would be one million planets in the Milky Way with such a characteristic. In fact, the first candidate for a habitable planet was Mars, and in 1975 the Viking space probes were launched with the aim of searching for organisms beyond our planet.

Some have compared NASA’s attempt to search for life to searching for fish in the Sahara Desert. Most of the instruments used to analyze the rocky and atmospheric material on Mars are based purely on our image and likeness. But it is clear that living things on other planets need not necessarily be like those we find on our own; there are hypothetical biochemistries such as silicon, boron, and arsenic that make carbon-based organisms one of many possibilities.

Nucleic acid analogues have also been investigated that could form the genetic structure of other living beings on other planets, which makes it seem that those non-corporeal life forms presented in the Star Trek saga are not so impossible at all. During the time of 1965 James Lovelock proposed a series of parameters that would allow the detection of any type of living being on some distant planet. According to him, the best way to look for entropy reduction processes on a planet is by studying the chemistry of its atmosphere.

If there were no life, the gases in the atmosphere would be in a state of thermodynamic and chemical equilibrium, dominated by stable components such as carbon dioxide. If there were life, then the waste products of life processes would be dumped into the planet’s atmosphere, contributing reagents such as methane and oxygen to its composition, which would lower the entropy of the atmosphere. By analyzing the atmosphere we can also detect the sounds emitted by the atmosphere, since the chimes emitted by living beings contain information in the form of pink noise.

Scaling

To understand pink noise we must imagine a rare event of large magnitude, e.g., think of a major earthquake, a climatic cataclysm or a mass extinction. Since these events rarely happen, consequently we can say that the frequency of the event is equal to 1 divided by some power of its magnitude. Conversely, we can say that the magnitude of an event is proportional to 1 divided by some power of its frequency. This is what characterizes pink noise, also called 1/f noise.

In this way, whether it is the Martian equivalent of a bird’s song, the chirping of crickets on Kepler-186f or the music of an alien Mozart living on Andromeda, all would be characterized by pink noise: for life is also considered an extremely rare event in the universe. To conclude this chapter, we will talk about power laws, which happen to be synonymous with pink noise.

Power laws have a significant omnipotence in all complex phenomena, but even something as simple as the volume of an object also follows a power law. If we have a cube such that the length of its edges is $l$, the volume is proportional to $l^3$, regardless of what the value of $l$ is. If we have a sphere of radius $r$, the volume is proportional to $r^3$, regardless of what the value of $r$ is.

The volume follows a power law of exponent 3 and the diversity of phenomena presenting power laws is enormous. Not only do we find them in physics in the form of the law of universal gravitation or Coulomb’s law, they also represent probability distributions, learning curves and self-organized criticality, which for many is the signature of complexity. We will discuss the latter in the second part of this trilogy.

All phenomena characterized by a power law show scale invariance, so the phenomenon described by them can happen at any time regardless of the brevity of what happened before. In the mid-1980s, researchers studying metabolic rates in animals of different sizes were intrigued to discover that, although the mass of the animal increases according to a power law of exponent 3, the metabolic rate grows following a power law of exponent 2.25.

In this sense, animals behave as if their size does not conform to a three-dimensional volume, but as something intermediate between a volume and a two-dimensional surface. A mathematician would immediately interpret this power law as suggesting that living things are fractal surfaces within finite volumes. Our nature is fractal. Little by little we have glimpsed the framework of complexity, and although you now know some of the most important conceptualizations for its understanding, we must warn you that there is still a long way to go.

In order to understand brain dynamics, for example, it is necessary to be familiar with the concept of criticality, an abstraction that is omnipresent in various biological and social phenomena. As we previously mentioned, it is also a cornerstone term for understanding complexity. This idea and others will be explored in the next volume, in which we will examine a central topic in our discussion: Life.