Whether we’re considering global trade, new technologies or scientific research, everything seems to become more complex over time. In the past, polymaths became well versed in various disciplines and made important contributions to each. But today it is far more difficult for a person to excel in multiple domains, in part because of greater specialization.
This is not an “everything used to be better” rant. It’s just a sober observation: the world used to be simpler. And that is true from a mathematical point of view. Physicist and philosopher Ludwig Boltzmann recognized it back in 1872.
Boltzmann studied the behavior of gases and liquids, among other things. Just decades earlier, it had been proposed that everything in the world was made up of tiny building blocks—specifically, atoms and molecules.
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If you wanted to describe how liquids and gases behave, you would have to track every single particle. Boltzmann knew that this was not feasible. He therefore developed an equation to describe the movement of particles on average. But this led to a mysterious insight: the individual atoms and molecules seem to follow completely different laws than their collective body. How can this be?
The Direction of Time
Imagine I show you a short video of balls colliding several times and rolling along a magical, friction-less billiard table. (For the purposes of this thought experiment, imagine there are no holes on the table.)
Now I ask you, “Did I play the video forward or backward?” It is actually impossible to answer my question.
Newton’s laws of motion, which describe the elastic impacts of billiard balls, do not depend on the direction of time. They give the same result both forward and backward. In the 19th century, experts assumed that the particles that make up gases or liquids would also move like small balls through empty space, bumping into one another from time to time and following Newton’s laws of motion, just like the aforementioned billiard balls.
Today we know that the truth is more complicated. Atoms and molecules obey quantum mechanics, which is much more complex. But interestingly, quantum mechanics is also often assumed to be invariant under time reversal. That means that, like our hypothetical billiard balls, at the atomic and subatomic scale, such as that of molecules of gas, it makes no difference whether you look at the process forward or backward.
At the macro level, however, things look quite different. If you zoom out of this microscopic representation, the direction of time plays a decisive role. Imagine pouring milk into coffee. The different substances mix over time to form a homogeneous liquid. That process cannot be reversed. You cannot remove the milk from the coffee.
This fundamental difference between the micro and macro worlds was Boltzmann’s main concern. How can it be that countless equations that describe the movements of individual particles and are invariant under time reversal cause irreversible behavior? If every collision between individual particles can be theoretically reversed, why can’t the milk be separated from the coffee?
The phenomenon can be explained intuitively. The particles within a gas or liquid collide again and again. This slows down fast particles, while slow particles are accelerated. If you wait long enough, an equilibrium is reached at some point, and the particles all move at the same speed on average. There may be individual outliers, but on average, the particles have roughly the same speed and are evenly distributed in space.
What Does “Complex” Actually Mean?
Boltzmann was able to express parts of this behavior with an equation that is now named after him. This so-called Boltzmann equation shows how the speed and distribution of particles in space changes depending on time and place. He also introduced a “collision operator,” which, depending on conditions such as density or temperature, takes into account the effects of elastic collisions on the particles.
The Boltzmann equation is a differential equation. This means that it contains derivatives, among other things. Such equations are usually difficult to solve. Nevertheless, Boltzmann managed to use this equation to prove that our world is becoming increasingly complex.
To do this, he first had to define what was meant by the term “complex.” A homogeneous mixture of milk and coffee does not seem particularly complex at first. From a mathematical and physical point of view, however, it is. There are countless different ways in which the microscopic particles in the mixture could move and behave, and each of them would produce the same macroscopic result. This means that even if the temperature, density, volume and mixing ratio of the mixture are known precisely, it is not possible to deduce exactly how the molecules are arranged or what their respective speeds are. Many different microscopic states lead to the same end result.
Boltzmann referred to this complexity as “entropy.” The higher the entropy of a system is, the more possibilities there are for its microscopic components to cause the same macroscopic phenomenon. If milk and coffee are separate, their system’s complexity—and therefore the entropy—is low. This is because the molecules of the milk are still separate from those of the coffee. If the liquids are poured together, they gradually mix more and more. Their system’s complexity and entropy therefore increase steadily over time. As soon as a state of equilibrium is reached, the entropy remains constant.
Boltzmann both described this qualitatively and derived it mathematically with his equation. He thus proved beyond doubt that the complexity of a system increases on the way to equilibrium.
This principle applies to our universe as a whole. Viewed holistically, nothing in the cosmos is in equilibrium, not even the universe itself, which continues to expand. This means that entropy—and therefore complexity—is constantly and inexorably increasing over time.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.
