How chaos drives the arrow of time

This post is a continuation of our last post on chaos. We therefore recommend you to read that one first, but it’s by no means necessary.

Time only moves forward – this is what we experience in our everyday lives. Often, people connect this to the fact that there are clearly processes in nature which cannot be undone: when you boil an egg you can not make the egg-white transparent again, your coffee is very unlikely to go back into the coffee powder, which will never become a full bean again itself.

In physics, this irreversibility of all macroscopic processes is known as the second law of thermodynamics. It more specifically says that entropy can only grow or stay constant – but never be reduced. Entropy is a notion which is very hard to explain without becoming too technical, but very loosely speaking, one could say that a system has very low entropy, if it is ordered and high entropy if it isn’t.

On the left, many balls are assembled in one corner of the box – a state with very low disorder, i.e. low entropy. On the right, the opposite is the case. The second law of thermodynamics states that configurations like the left one will evolve to one like the right one and never the other way around – this is what some people call the arrow of time.

Put differently, the second law says that it becomes increasingly unlikely to find a configuration of the particles where they are all in one corner of the box in above picture – most often you will find them randomly spread over the whole space (the configuration with the lowest degree of order).

The unipresent experience in our everyday life of ever-growing entropy is however in stark contrast to all the fundamental laws of physics that we know. Because in contrast to the second law, which states that all processes are irreversible, these equations state that microscopic dynamics are reversible! Newton’s famous second law (not to confuse with the second law of thermodynamics!) is one example for those, it states that


Why is this equation in disagreement with the second law? Well, it states that if you start out with some initial configuration that you know exactly, you will know at every future point in time where the system has evolved to. And, more crucially, you know where it came from! In a thought experiment, you can just use these equations to go forwards in time, and then let time move backwards and end up at the state that you were starting from again. This is the definition of reversible dynamics!

Reversible dynamics und Newton’s equations: A car is made to move along a road because of a force F acting on it. If we let it drive for some time t it will end up at the end of the road. But equally as well, we could then make the car move back to its initial place again by acting on it with the opposite force, or, put differently, let it evolve for the same amount of time, but “backwards”.

While what I stated above is not exactly true for the successor of Newton’s theory of classical mechanics, quantum mechanics, the basic message stays the same: Also the equations governing those more fundamental physical theories are time reversible! The second law of thermodynamics is therefore not derivable from microscopic theories, it is a phenomenological observation we have about the world, not something which is mathematically proven.


But why is the second law such a universal principle then? This is a question already the old giants of thermodynamics such as Boltzmann, the inventor of the notion of entropy, were wondering about. He derived something like the second law from Newton’s equations under great mathematical difficulty, but he had to make some simplifications which essentially meant that he put in the irreversibility “by hand”. He (and everyone trying after him) was therefore not able to derive the second law from more fundamental physical principles.

He was however able to give some notion for what you need in order to motivate the second law. You need to make some assumption about how much the system “remembers” where it came from. For example, the assembly of balls in a box which we drew above behave according to the second law if they effectively forget after a while that they once were crumpled up in one corner. It was however not clear to Boltzmann where this should come from.

It took another discovery to enable full understanding why at least classical many body systems effectively forget their initial state: Chaos! We have discussed Chaos before in this post, but the most important thing you have to know about it here is summarized in this picture:

Very slight changes in the position where the ball is dropped onto the slope mean enormous changes in its final position when it hits the ground. (adapted from our earlier post)

Basically, chaos means that you would have to know the initial configuration of a system to infinite precision in order for you to be able to calculate where it came from, even when using the time reversible and fully deterministic Newton’s equation. As this is in practice never possible it means that in our everyday experience, reversibility never actually happens: You cannot track the position and velocity of every single atom in your panfried eggwhite in order to find out how you could make it transparent again. Hence, the many body system effectively forgets its initial state due to its chaotic dynamics.

This also makes our initial example of the little balls spreading out in the whole of space understandable: The configuration of them sticking around in one corner is an extremely special one and after a short amount of time, the balls will have spread around in the whole of space. Them behaving chaotically then means that they quickly forget that they once were assembled in that corner. Moreover, when observing these chaotic dynamics over time, it is far more likely to find them in a configuration where all of them are spread in the whole of the box, the configuration with the highest entropy. This is because there are just many more possibilities to distribute the balls over the whole box than to stick them in one corner. Ultimately, the second law of thermodynamics or the “arrow of time” emerges from the loss of memory of the balls about their initial state due to their chaotic dynamics.

It leaves to say that while this picture of the second law emerging from chaos is a widely accepted picture in classical systems, things are much less clear in the more fundamental theory of quantum mechanics. In fact, a great number of people try to generalize the notions of chaos to quantum systems and this is also one of the topics that I deal with in my PhD work. I hope to be able to write about this exciting quest in the future, so stay tuned !

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