Size Matters
I don’t think we have to tell you that the Universe is a very complex and huge place. But in case we actually do, here it is: the universe is bigger and more complex than the human mind can fathom. Think about our galaxy, with its millions of stars, which have their own solar systems with some planets and hundreds of asteroids and general debris. All of them attract each other gravitationally and modify the path that each other has,literally all the time! How can one even start to try to predict how the Universe works with such staggering number of bare elements?
And if you feel like that the astronomical example is a bit far off, take a very deep zoom in now, to a city like New York, Berlin or Shanghai. In these cities, there are millions of people moving in public transport, by car,biking, or walking. Imagine every single one of them moving, with a little pin on a map. Now you want to track them and, let’s say, predict points at which traffic jams are produced, or even more complicated issues.
The complexity in any of these systems is, plainly put, overwhelming. As we said before, how can one even start to work on this wildly tangled network of…. stuff? The answer is a mental process that we actually all do intuitively in our everyday lives. We physicists, however, have given this idea a fancy name, separation of scales.
What is this? Well, the idea is always simple, the realization, however, sometimes not so much. At the heart of the problem lies the discrimination between the strength of effects. Mostly these are distances (or sizes), energies (and momenta/velocities) and the strength of interactions. We will see everything is connected in the end. To be able to compare them, we physicists resort to the mystical art of the dimensionless ratio. And what is that? Well, a fraction where the numerator and denominator have the same units, and therefore, we get something without them.
Why?
Well, because we human beings have the limitation that we can only measure ratios without any units. “How about kilometers and kilograms and other weird volty stuff you can actuallymeasure?” You retort, rightfully so. Well, you don’t measure kilograms, or kilometres or the volty stuff. You measure the ratio between the quantity and a reference. It is still a ratio, between the quantity and the ruleryou decide to pair it against.
We then take these dimensionless ratios and compare how much the system changes if you take/don’t take into account that quantity. There is no strict definition, but the rule of thumb is normally that below a couple of percentage points in what you want to measure, you can forget about the effect. And that’s it. That is the whole trick. I know it sounds pretty dumb but it is so useful that wephysicists use it every single day in most of our arguments and discussions. And when used well, it works wonders.
I will give you some more examples for you to convince yourself that size matters depending on the size of your lens. Let’s zoom progressively. An atom is neutral, right? Well, yes, by definition (ions are charged). But it has a nucleus with a pretty high charge, and a bunch of electrons flying around. And yes, as you imagined, this depends on how big you are and where you are. If you are inside the electronic cloud, electrons will fly past you, and you WILLsee charge, outside you won’t. So that question clearly depends on whether you are equally tiny, but are quite far away from the atom, the distance of their orbits will be too small for you to see.
Zoom in more, and the same happens for the nucleus. Before, we saw a blob, with a definite positive charge, now we see a bunch of protons and neutrons, which bounce around. Different pieces of the nucleus may have more charge than others some times. But from outside you can only see the net charge, as with the atom.
Zoom in more, and we enter the insides of the proton. We will see little particles called quarks zooming around at neck-breaking speeds. They have their own pieces of charge, which add up to the net charge of the proton, once again. Now you can ask yourself what can be the effect of a quark on the overall effects of the atom, if already the proton isone hundred thousand times smaller…
A quick example
Now let’s use a concrete example to explain this. My favourite one, the one I use to teach is in fact the solar system. Apart from the Sun, it has eight planets (I know this is a sensitive topic for some of you) and a whole bunch of planetoids and moons. We have one, Mars has two, but Jupiter has 79 moons! All of these objects interact gravitationally with each other by means of Newton’s Law of Gravity, meaning that the force increases with the two bodies masses, and decreases with the square of the distance between them. This is a blurry mess, how do we even predict orbits and eclipses, and other close astronomical phenomena?
We take ratios of the forces, naturally. For example, let’s say you want to calculate the orbit of the Earth around the sun. Well, you take the biggest input, which is the sun, and compare the other effects to it. Which would intuitively be the next one? The Moon. Then, let’s compare it. Remember, we have to calculate the ratio of the forces that the Earth will feel from both the moon and the Sun. As you can see in my nicely scribbled across mess, it boils down to two ratios, one of the distances and one of the masses. The Sun is roughly 100 million timesheavierthan the Moon, while it is around 1000 timesfarther away. Using those numbers, we can calculate quickly that the force the Moon exerts on our home planet is around1%of the force the Sun exerts. So, we can just forget about it and solve the puzzle without even thinking of the Moon.
Effective descriptions (or why you should care about this)
There is much more to this than a simple mechanical example, of course. Using this technique, we can radically change the way we tackle the system. Is it quantum? is it astronomical? Does relativity represent an important aspect? Is it complex, or simple? Is this a fluid? Is the system in equilibrium and does equilibrium even pose a meaningful concept in the system? There are, of course, many other questions you can ask, but in principle, this little trick allows you to quickly surf the inmense sea of possibilities that is the Universe.
But because this is Many Body Physics, I will focus now on the question:Is it complex, or simple? And this question is sometimes quite hard to answer, to be honest, but it relies on a very interesting concept of emergent properties. To explain them, I want you hold in your imagination that youhave a unit object, something, doesn’t really matter what it is. However, you can describe this object quite well. You know the basic rules of engagement of two of these basic, fundamental thingsthat you have. Think of two billiard balls, or gravitational attraction of two bodies. Then you add a third one. Ok, the system becomes more complicated, but those unit objects still behave like individual thingies, right? So you include more, and more, and more, and there is a subtle, but powerful transition. Your collection of gizmos now host extra things, wave motions, weird behaviour. Maybe, you zoom out and they look like a fluid? What just happened is that they exhibit emergent properties. Basically we can say that
The whole is different than the strict sum of its parts.
Two examples I absolutely love, actually happen in society. The first example is very interesting and non-orthodox, let’s say. It is crowds and moshpits in metal concerts. You have a designated concert area, right? Put three or four metalheads in there to headbang, and they will barely come across each other. But now put there a huge concert crowd, and see the magic unfold. In moshpits, the crowd exhibits flow dynamics, which means, basically that it acts as a fluid (actual scientific paper about this here), in which the collective behaviour arises -or emerges-from a very simple set of rules. This is fascinating, that such a complex system can be explained from such simple rules, while the collectivity is created on the fly!
Now, let’s go to traffic.
It is such a wonderful nightmare for basically everyone, but we can learn so much from it. Our unit object here is the car, of course. So the question here is basically, what are jams, and how do they happen? Well, there are many kinds, but basically, what happens here is that when you add enough cars, they behave as medium, like a fluid, namely. This means they can host some kind of waves. In this case, they move parallel to the “velocity of the fluid” (along the street), so they are actually analogous to sound. To create a jam, you just need a seed for the waves to propagate, like throwing a stone into a pond. And voilá, your favourite phenomenon ever just happened (you can see a video of an experiment here).
But how does that relate to what I talked about before? Well, the classification is due, of course, and the relevant parameter is density. With our trick we can separate two cases: if the mean distance between cars is way bigger than the cars themselves, the cars are individual particles, and they actually don’t see each other. This system is analogous to a dilute perfect gas. If the density is way bigger, which means that the mean distance is very similar to the sizes of the cars, well, my friend, we are in a jammed region. Now, this sounds very trivial, and borderline dumb, but now remember the video I just gave you. This scenario doesn’t mean that we are in a jam, it means that basically anything can trigger a jam. Now our cars behave actually like a compressible fluid.
Once again, so what with all of that?
I hope it is not like it, but all of this may sound a bit like word salad. Like the ramblings of a physics major which got too much coffee and started comparinng the real world with physics. It is not at all like that. More and more, these comparisons are being made to describe new exciting, collective phenomena, with non-trivial emergent properties. You have a complex system, maybe it will be described by something physics has already. Crowds, traffic? What about gases, fluids? Stock markets? Let’s try quantum mechanics. It is an ever rising field in science to start thinking about theses questions. And it all starts with one question. Is it simple? Is it complex? What can we simplify from it?
For that, you may not know the answer yet, but you now know the method.
