Understanding the mathematics of scaling towards a sustainable future for humanity.

Complexity, physics

There are some truly pressing problems in our times. These concern the survival of humanity. They can range from environmental to socio-economic such as climate change, world poverty, migration and gender inequality. These are problems which are prevalent in every city around the world. 

Today, the world is urbanizing at an alarming rate. The population is reaching unsustainable levels, especially since resources are finite. Growing population brings with it growing rates of crime, inequality and pollution. Thus, it is now more urgent than ever that we model and understand our cities better. 

While cities can be the problem, they can also point us towards a potential solution. They can be facilitators of social interaction leading to innovation. Due to their multi-variate and highly non-linear nature, cities have evaded a quantitative understanding until now. However, recent insights in the field of complex adaptive systems could be the key to solving these important problems.

A key feature of complex adaptive systems is the scaling properties they follow. These scaling properties are underpinned by predictive mathematical frameworks and can be written as power law equations. Such an equation looks like:

Y = A * X^k, 

The value of the exponent k determines the type of growth in the system. Linear growth — where if X doubles, Y doubles, and so on — results if k = 1.

‘Sub-linear scaling’ is the case where k < 1, in which changes in X result in less-than-proportionate changes in Y: if X doubles, Y will increase by less than double.

In physics, such scaling is observed in things like Kepler’s Laws, which relate the time period of a planet orbiting the sun and its radius. (In this case, k = 2/3.)

Biological systems, on the other hand, are continuously evolving, which makes them difficult to predict. But nonetheless, despite the extraordinary complexity and diversity of life, many of its most fundamental metrics follow similar scaling laws. This can be true of systems as small as cells and as large as ecosystems.

For example, comparing metabolic rates and body mass across species gives k = 3/4. Nature exhibits an inbuilt economy of scaling.

The problem arises when we have k > 1, which is known as ‘super-linear’ scaling. This implies unbounded growth and is unsustainable in the presence of finite resources. Unfortunately, as we will see, cities are systems which follow this kind of growth.

According to studies by complexity expert, Geoffrey West, the origin of such power laws lies in the dynamics of the underlying networks which constitute such complex systems. Every system  needs energy to survive. Networks help deliver energy in a system efficiently. In this way, they enable interaction between seemingly unrelated parts of a system leading to an emergent large scale behaviour. This is the field of complex adaptive systems. 

In the biological world, such a network is the circulatory system. The total (finite) amount of input energy is allocated between maintenance and the growth of a system. As the system grows, a larger volume requires higher allocation of energy towards maintenance which implies less energy for growth. This is what leads to bounded growth and eventually death.

West and his collaborators observed that there is a similarity between growth in the biological world and growth in our cities. Cities, like living systems, are continuously growing and evolving. More importantly, cities, just like biological systems require energy to survive and grow. However, unlike biological systems, cities seem to avoid senescence!

Studying the growth of infrastructural measures in a city, such as numbers of gas stations and lengths of roads and electrical cables, with population size, we see that these follow the same sub linear scaling (k < 1) as in the biological world. However, cities are much more than just infrastructure! In fact, the growth of socioeconomic quantities involving human interaction, such as wages, patents, AIDS cases, and violent crime with respect to increase in population size, follows a super-linear scaling (k >1)! This is a new and surprising class of phenomena separate from anything observed in the biological world. It also implies unbounded growth which explains why cities never stop growing! 

What is even more interesting is that despite their unique histories, cities all over the world exhibit universality by obeying the same scaling laws. These regularities have led to the beginnings of a quantitative framework of cities in terms of their underlying networks. In this case, these are social networks or transportation networks which are universal features of every city. Unfortunately, on earth, resources are finite. And thus, such an unbounded growth is unsustainable and predicts a collapse of the system once resources are depleted. This is known as a finite time singularity. 

Interestingly, human beings have avoided such finite time singularities by making a drastic paradigm shift after regular finite time intervals. Or in other words, we have had to innovate continuously in order to use our resources more efficiently. Examples of such paradigm shifts have been the discovery of coal, the industrial revolution, the discovery of clean energy sources, the IT revolution, digitalization of technology and discovery of automobiles. However, there is a catch. Each subsequent interval is smaller and smaller which implies that in order to survive, humanity needs to innovate faster and faster. Thus, it is important to understand what drives innovation in cities. The answer, perhaps unsurprisingly, lies in diversity. The more diverse a city, the more adaptable and resilient is it. Again, this has parallels in the biological world.

Thus, the need of the hour is this: Can we come up with a recipe for a sustainable future? The answer might lie in combining insights of complexity theory into concrete and smart implementable policies. 

Tricksters in the Sky.


Following my last post on black holes, I want to write a small post focusing on what other kinds of objects could be in the sky. A more specific question is: What other objects does General Relativity (GR) predict?

Also, physicists have long known that GR as it stands currently, cannot account for all the matter in the universe. In fact, 85% of the universe must be what we call “Dark Matter“. Now, whether Dark Matter could be made up of many possible things. And it could very well be made up of “dark” bosonic particles.

Turns out there are many possible “compact” objects in the sky. Schwarzschild metric is the metric which describes the space-time around a spherical symmetric static object and turns out it can be applicable to different kinds of stars such as normal stars (Luminous stars made up of hydrogen and helium), which could turn into red and white dwarfs, neutron stars and some other hypothetical stars known as Boson Stars. Mind you, here we are assuming that the Standard Model of Particle Physics is the only valid model for these stars.

My personal favourites are called Boson Stars, especially because they can mimic black holes. So what are these stars and how are they different from the usual stars as we know them?

Boson Stars (BS) are stars which are made up of elementary particles called “Bosons”. They are solutions to Einsteins Equations which can be found using numerical GR. One of the simplest kind of Boson Stars would have no self interaction force between the bosons other than being held together by gravity. However, in order to stabilise them, you would probably require some kind of repulsive potential between the bosons in order to stabilise them against gravitational collapse due to gravity pulling everything in.

Such a simple BS, would not have any electromagnetic radiation or light being emitted which would make them very hard to detect. In fact, they can have masses close to masses of black holes and without having any light emission, might not be indistinguishable from them! And this is why they caught my interest. They are like silent dark spies in the night sky, steadily spinning around without any one knowing about their existence. How exciting is that!

In a recent paper, me and my collaborators modelled a very simple Boson Star with a repulsive potential which could be made up of “dark” Bosons. Since these do not have any electromagnetic coupling, when two BS which are in a binary system, rotating around each other, finally collapse, they will radiate energy in the form of gravitational waves, just like black holes. We calculated the amount of gravitational wave radiation would be emitted from mergers of binary Boson star systems in the universe such as this one and we discussed whether detectors such as LISA and EPTA would be able to detect them or not. Turns out in certain ranges LISA might just be lucky!

Here a few favourite things about Bosons Stars that people have said on the internet and some lists it has found a place on:

All I can say is, if you exist, you go girl!! I am rooting for your existence!

Black Holes!


As a kid, I used to look up to the sky and try to count the number of stars. But I wasn’t as intrigued by what happened inside of them till I heard the story of black holes. For the first time, I was super intrigued by these seemingly super mysterious points in our universe. What are they? How are they made? What happens inside of them?

While the question of what happens inside of them might be the subject of another upcoming post, if scientists manage to crack that code, I did manage to learn about how they are theoretically predicted as well as experimentally formed. Lo and behold, stars had a lot to do with it!

In Einstein’s General Relativity, we can write down the Einstein’s equations and calculate the solutions to these equations. A particular class of solutions which are called as Vacuum Solutions, exist, among which the “Schwarzschild Solution” is a special one. This is a metric which describes the space-time around a spherically symmetrical mass in the universe and it looks like this:

If we calculate the poles of this equation, we find that the equation predicts an event horizon as well as a singularity! If a star many times the mass of the sun goes through the whole chain of nuclear fusion events and ends up quite heavy, it would at some point collapse under its own weight and would form a blackhole if it is trapped inside it’s own event horizon! At this point, the mass and gravitational force of this star becomes so high that not even light can escape the confines of the event horizon! And thus, we do not have a direct probe for the inside of this hole due to which it looks black to us!

And thus, it is a black hole!! And to top this post off, here is a picture of a blackhole swallowing a star!

Taken from here.