Scott Aaronson, Sean M. Carroll, Lauren Ouellette
5 min
In contrast to entropy, which increases monotonically, the "complexity" or "interestingness" of closed systems seems intuitively to increase at first and then decrease as equilibrium is approached. For example, our universe lacked complex structures at the Big Bang and will also lack them after black holes evaporate and particles are dispersed. This paper makes an initial attempt to quantify this pattern. As a model system, we use a simple, two-dimensional cellular automaton that simulates the mixing of two liquids ("coffee" and "cream"). A plausible complexity measure is then the Kolmogorov complexity of a coarse-grained approximation of the automaton's state, which we dub the "apparent complexity." We study this complexity measure, and show analytically that it never becomes large when the liquid particles are non-interacting. By contrast, when the particles do interact, we give numerical evidence that the complexity reaches a maximum comparable to the "coffee cup's" horizontal dimension. We raise the problem of proving this behavior analytically.
This paper investigates the intuitive observation that closed systems, such as a cup of coffee mixing with cream or the evolution of the universe, exhibit a specific pattern of complexity. While entropy increases monotonically toward equilibrium, complexity—the amount of information required to describe interesting structures—seems to rise from a low-complexity initial state, peak at an intermediate state, and eventually decline as the system reaches a high-entropy, featureless equilibrium. The authors aim to formalize this pattern using a two-dimensional cellular automaton that simulates the mixing of two liquids.
To quantify this, the authors utilize "apparent complexity," defined as the Kolmogorov complexity of a coarse-grained approximation of the system's state. This measure acts as a proxy for the information content of the system after "incidental" or random noise is smoothed out. By comparing non-interacting particle systems to interacting ones, the researchers test whether this complexity measure captures the expected rise-and-fall behavior.
Numerical evidence indicates that when particles interact, the apparent complexity reaches a maximum proportional to the horizontal dimension of the system before declining. Conversely, in non-interacting systems, the complexity never reaches significant levels. This work provides a quantitative foundation for understanding why complex structures emerge and persist in closed systems before eventually succumbing to the second law of thermodynamics. It highlights the distinction between entropy, which measures disorder, and complexity, which measures the presence of non-random, interesting structures.
Alex: Oh, I see. It's like the difference between a high-resolution photograph and a blurry thumbnail. The thumbnail can't show you individual grains of sand, but it can still clearly show you a sandcastle. The random noise disappears, but the meaningful shapes remain.
Sam: That's a good way to put it. When you compress that blurred image, the resulting file size gives you what the researchers call "apparent complexity." A simple, uniform pattern compresses to almost nothing. A structured, patterned image takes more space to store. And a purely random image—after blurring—also compresses well, because the blur has averaged out the noise into something featureless. So apparent complexity peaks in the middle, where genuine structure exists, and stays low at both extremes.
Alex: So you end up with a curve that rises and then falls. The universe starts simple, gets interesting, and then winds down into simplicity again—and this measure actually captures that arc.
Sam: That's the claim, yes. And it matters because other proposed measures of complexity have real limitations. One alternative, called "sophistication," tries to separate the meaningful structure in a pattern from the random noise mathematically. It's a theoretically appealing idea, but it turns out to be extraordinarily difficult to compute in practice—you can define it precisely but rarely calculate it for real systems.
Alex: So apparent complexity is the practical version. Less perfect in theory, but actually usable.
Sam: That's the trade-off. The researchers argue that apparent complexity is grounded in what we can observe and compute. It gives you a number you can actually work with, which means you can track how a real system—or a simulated one—moves through that arc from simple, to structured, to simple again.
Alex: And if that arc is genuinely universal—if it shows up consistently across different types of systems—then this could be a meaningful tool for understanding how structure emerges and dissolves, not just in a coffee cup, but potentially in physical systems at much larger scales.
Sam: That's where the paper points. The researchers are careful not to overclaim—this is a framework demonstrated in simulations, and extending it to, say, cosmological scales would require a great deal more work. But the underlying logic is sound: if you want to measure the "interestingness" of a system over time, apparent complexity gives you a principled, computable way to do it.
Alex: It's a modest but meaningful step. Instead of just saying "the universe gets complex and then doesn't," we now have a way to draw that curve and ask where on it any given system sits. Thanks for listening to ResearchPod.