Entropy only climbs. Complexity rises, then falls — and the gap is the whole story.
The second law of thermodynamics says the entropy — the disorder — of a closed system only ever goes up. Yet the universe was structureless at the Big Bang and will be structureless again at heat death, with all the stars and galaxies appearing only in between. So whatever we mean by “interestingness” plainly does not behave like entropy: it rises and then falls. This paper takes that vague intuition and pins a number on it.
The bet is one definition: measure complexity not on the raw state, but on a coarse-grained — blurred — snapshot of it, and take its description length. On the blurred view, both perfectly separated and fully mixed coffee are simple; only a half-swirled cup is hard to describe. The number this gives bumps up and back down. And the authors show that bump appears only when particles interact — structure is never a free gift of mixing.
Blur the picture before you measure it, and “the cream is half-swirled” finally registers as more complex than either pure layers or uniform brown.
The puzzle
If entropy only rises, what is it that rises and then falls?
Step 1 · The running example
Pour cream into coffee, and watch it go interesting, then boring.
Here is the whole subject in one picture — and it is the example we will ride through every step. Pour cream onto coffee and leave the cup alone. It passes through three states:
Notice the asymmetry. The start (a clean layer of cream over coffee) and the end (uniform brown) are both trivial to describe in a sentence. The middle — feathered tendrils, half-formed swirls — is the one state where a short description fails. That, informally, is what we mean by complexity: the amount of information needed to describe everything interesting about the system.
Meanwhile a second quantity, the entropy — the disorder, how much structure has been lost — climbs the whole way and never comes back down. The puzzle of the paper is the difference between these two curves: one humped, one monotone.
A good complexity measure must be low for the simple ends and high only in the middle. Entropy can't do this — it only rises. So complexity has to be measured differently. That difference is the engine of the whole page.
From the introduction: “Imagine a cup of coffee into which cream has just been poured… it is easy to describe the state of the cup [at the start and end]; however, when the cup is in an intermediate state… it seems more difficult to describe what the contents of the cup look like.”
Step 2 · Two curves, told apart
Entropy is a staircase that only goes up. Complexity is a hill.
You've watched the cup go from easy, to hard, to easy. Now plot what each of our two quantities does over that same time. The point of the whole paper lives in the difference between these two shapes.
Read both curves left to right as the cup mixes. They start together at zero — a fresh, perfectly layered cup has neither disorder nor interesting structure — but they part ways immediately:
Entropy is the easy one — the second law guarantees it. As the sharp cream/coffee boundary dissolves, structure is irreversibly lost, so disorder climbs and then flattens at a maximum. It never dips.
Complexity is the interesting one, and it is exactly what has no general theory. It must be near zero at both ends and large only in between. The paper's job is to (1) write down a measure that actually behaves like this teal hill, and (2) explain why the hill exists at all.
This is the same story as the universe: structureless plasma at the start, heat death at the end, planets and galaxies only in between. There is no general principle explaining that intermediate peak — this paper is the first quantitative attempt at one.
From the paper: “There is no general principle that quantifies and explains the existence of high-complexity states at intermediate times in closed systems… as far as we know this is the first quantitative exploration of the phenomenon.”
The machine
A toy coffee cup · measuring entropy · the blur trick · and apparent complexity
Step 3 · The coffee automaton
A real coffee cup is too hard. So model it as a grid of bits.
You have the two curves you want to reproduce. But to measure anything you need an exact, computable system — not a real fluid. So the authors built the simplest cup that still mixes: a two-dimensional cellular automaton (a grid that updates by a fixed local rule).
The setup is deliberately spare. Take a grid of cells — the paper's main runs use 100 × 100. Each cell holds one bit: 1 = cream, 0 = coffee. The cup starts perfectly layered: the top half all cream, the bottom half all coffee.
That swap rule is the interacting model: only one particle per cell, so a cream particle can move only into a coffee cell, never on top of another cream particle. Particles get in each other's way — and, as Step 8 will show, that mutual blocking is exactly what makes structure possible.
From this grid we will read off our two numbers at every moment in time: an entropy (Step 4) and an apparent complexity (Steps 5–6). Both are computed by the same humble tool — a file compressor.
The model is fully closed and deterministic-in-spirit: nothing enters or leaves the cup, and the only thing that happens is conservative swapping. So any rise in complexity has to come from the dynamics alone, not from outside structure being injected.
From the paper: “The automaton begins in a state in which the top half of the cells are filled with ones, and the bottom half is filled with zeros. At each time step, one pair of horizontally or vertically adjacent, differing particles is selected, and the particles' positions are swapped.”
Step 4 · Measuring entropy
Entropy = how big the snapshot is after you zip it.
You have a grid that mixes. The first number to read off it is the easy one — the entropy, the disorder. The authors needed a definition that is concrete and computable, and they found one in an unexpected place: a file compressor.
The connection runs through Kolmogorov complexity K(x) — the length of the shortest program that outputs x. A highly patterned grid (all-cream-on-top) can be printed by a tiny program, so its K is small. A random-looking grid has no shortcuts; the shortest program basically hard-codes it, so its K is large. K(x) can't be computed exactly, but a good stand-in is the gzipped file size: zip compresses the patterned grid to almost nothing and the random one barely at all.
Run this on the automaton every so often and you get the entropy curve from Step 2: it climbs as the layers dissolve and saturates once the grid is effectively random. There is the catch, though, and it is the heart of the paper: a fully mixed grid is just as random as it gets, so by this measure the boring uniform end scores the highest, not the most interesting middle.
Entropy, Shannon entropy, and Kolmogorov complexity are all maximized by random states. None of them can ever peak in the middle. To capture “interesting,” we need to break that — which is the next step's whole trick.
Every standard disorder measure is largest for random objects. But a complexity measure must be small for random objects too (uniform brown is boring) and large only for the in-between. So complexity cannot be any flavor of entropy applied to the raw grid.
From the paper: “Kolmogorov complexity… is well-known to be uncomputable… one can often estimate K(x) reasonably well by the compressed file size, when x is fed to a standard compression program such as gzip.”
Step 5 · The blur trick (coarse-graining)
Squint at the cup first. Then measure. That one move does all the work.
Entropy failed because it measures the raw grid, where mixed and random look identical. The fix is the conceptual pivot of the whole paper, and it is almost embarrassingly simple: don't measure the raw grid — measure a blurred version of it first.
Coarse-graining means replacing each cell by the average of the small block around it — the cup as a human would see it from a few feet away, not pixel by pixel. The paper averages over g × g blocks (grain size g), then snaps each average into a few buckets: near 0 = mostly coffee, near 1 = mostly cream, near 0.5 = mixed.
Here is the paper's own worked example, a grain size of 3. Click any cell to compute the average of its 3 × 3 neighborhood — exactly the number that lands in the blurred grid:
Click a cell — the 3×3 window around it lights up; read its average below
Every blurred value lands in [0, 1]. Layered regions average to 0 or 1 (“pure”); a feathered border averages near 0.5 (“mixed”). The blurred grid is what we will compress — never the raw bits. Values here are exact, from the paper's Figure 1.
Why does blurring rescue us? Because it changes which states look simple. A fully mixed grid, blurred, is uniform 0.5 everywhere — trivially compressible, low complexity. Separated layers, blurred, are two flat blocks — also trivial. Only the half-mixed middle, with its tendrils, survives blurring as something genuinely irregular and hard to compress.
Raw random and raw mixed both look maximally complex. Blur first, and uniform brown collapses to a flat sheet while the swirling middle keeps its structure. Coarse-graining is the small modeling choice that does most of the work.
From the paper: “for an automaton state represented by a string x, its coarse-grained version is analogous to a typical set S which contains x… we then threshold its floating-point values into… buckets.”
Step 6 · Apparent complexity, defined
Apparent complexity = the zipped size of the blurred snapshot.
You can blur a grid (Step 5) and you can zip a grid to measure disorder (Step 4). The paper's definition is simply those two moves composed, in that order — and that ordering is the entire idea.
They call it apparent complexity: take the blurring function f, apply it to the state x, then measure the description length of the result. In symbols it is H(f(x)) — an entropy H of the de-noised picture f(x), not of x itself. In practice: coarse-grain the grid, then gzip the blurred array. That compressed size is the number.
apparent complexity=H(f(x))
f = coarse-grain (blur) · H = compressed size (≈ Kolmogorov complexity) · measured on the blurred grid f(x), never the raw grid x
Now contrast the two numbers on the three states of our cup. Entropy reads the raw grid; apparent complexity reads the blurred grid. The ordering of f and H is everything:
Apparent complexity is just “how hard is the picture once you squint at it.” Pure layers squint flat; uniform brown squints flat; only the tendril stage stays detailed when squinted. Blur, then compress — that is the whole measure.
From the paper: “By the apparent complexity of an object x, we mean H(f(x))… where f is some ‘denoising’ or ‘smoothing’ function… For example, if x is a bitmap image, then f(x) might simply be a blurred version of x.”
The verdict
Run the cup · why interaction is the whole story · and what it left open
Step 7 · Run the cup
Now watch both numbers as the cup mixes — and catch the hump live.
You have everything: a grid that mixes (Step 3), an entropy (Step 4), and an apparent complexity (Steps 5–6). Now turn the crank and watch all three move at once. This is the experiment the paper actually ran.
Drag the time slider. The left pane is the raw cup (its zipped size is the entropy); the right pane is the same cup blurred (its zipped size is the apparent complexity). Watch the two dots ride their curves:
Drag time — the raw cup keeps getting messier, but the blurred cup peaks in the middle
raw cup → entropy
Sharp boundary — almost all order intact.
blurred cup → complexity
Two flat blocks — trivially simple.
The two curves disagree by design. Slide to either end and the blurred cup is a flat sheet — complexity near zero — while entropy sits low (start) or high (end). Only in the middle does the blurred cup keep its tendrils, and complexity peaks. The pictures are schematic; the rise-and-fall shape is the paper's measured result.
Both models reproduce it: entropy rises and saturates, while apparent complexity rises, peaks, and collapses. The intuition about coffee — interesting in the middle, dull at both ends — is now a curve you measured, not a feeling.
From the paper: “Both the interacting and non-interacting models show the predicted increasing, then decreasing pattern of complexity… the fine-grained representation continues to grow more complicated with time, while the coarse-grained representation first becomes more and then less complicated.”
Step 8 · Why interaction is the whole story
Turn off the bumping, and the hump disappears.
You watched the hump appear. Now the deepest question: where does it come from? To find out, the authors built a second cup that mixes the same way but with one rule removed — and the contrast is the paper's sharpest result.
In the non-interacting model, cream particles ignore each other: any number can pile into one cell, each just taking an independent random walk. It still mixes, entropy still rises. But the swirls never form. Toggle between the two cups and compare their complexity curves:
Switch the model — only the interacting cup grows a real complexity hump
Interacting cup — the complexity hump rises to roughly the cup's width, then collapses.
Same mixing, same rising entropy — but only the interacting cup grows structure. When particles can't share a cell, they jam at the border and form tendrils; when they can pass through each other, the front just spreads smoothly and stays simple. Curves are schematic; the contrast is the paper's finding.
For the non-interacting cup, the authors prove analytically that the
blurred grid stays compressible — its complexity can be at most about
log₂(n) + log₂(t), i.e. it never gets large. Structure is not
a free byproduct of mixing; it requires the particles to interact.
From the paper: shown analytically that the non-interacting model “never becomes large,” while numerically the interacting model's complexity “reaches a maximum comparable to the coffee cup's horizontal dimension.” A later artifact-corrected metric (7 thresholds + majority smoothing) flattens the non-interacting curve to a low value, matching the theory.
Step 9 · The scorecard & what's left open
The measured landscape — and one honest gap.
You've built the cup, defined both numbers, watched the hump, and seen that interaction causes it. Final question: how solid is this, what scales how, and have you actually got it?
Report card · the measured behavior
Entropy vs. grid size · ∝ particle count
Peak complexity vs. size · ∝ side length
Time to the peak · ∝ particle count
non-interacting cup · true complexity
Now you can explain it. Five questions — answer each out loud before opening it. If all five come easily, you've genuinely got this paper.
Why can't entropy capture “interestingness”?
Entropy (and Shannon / Kolmogorov complexity) is maximized by random states, so it just keeps rising — the boring, fully-mixed cup scores highest. A complexity measure must be low at both ends and high only in the middle, which entropy can never do.
What does coarse-graining actually do?
It blurs the grid — replaces each cell by the average of its g×g neighborhood, then buckets into mostly-coffee / mostly-cream / mixed. A uniform or layered cup blurs to a flat sheet (simple); only the tendril stage survives blurring as something irregular.
Define apparent complexity in one line.
H(f(x)): blur the state with f, then take the description length H of the result — in practice, the gzipped size of the coarse-grained grid. Blur first, then compress.
Why does only the interacting cup grow structure?
With one particle per cell, particles jam at the cream/coffee border and form tendrils. With no interaction, each cream particle random-walks independently, the front spreads smoothly, and it's provably compressible — at most ≈ log₂(n)+log₂(t).
How big is the peak, and what's unproven?
Numerically the peak scales linearly with the cup's side length (the time to reach it scales with particle count, ∝ n²). What's left open is a proof that the interacting model genuinely peaks — it's conjectured, not yet a theorem.
The honest gap — the rise-and-fall is reproduced numerically and proven only on the easy (non-interacting) side. Proving the interacting cup truly peaks is stated as an open problem, not papered over. That candor is part of why the toy model endures.
“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.” — Aaronson, Carroll & Ouellette, 2014
A measuring stick for “interestingness” — and a model that won't go away.
Complexity-rises-then-falls had been folklore for decades. This paper gave it a concrete, computable definition and a toy system you can actually run — and in doing so handed three different fields the same clean question to argue about.
Why the universe is interesting now
The same curve describes the cosmos: structureless plasma at the Big Bang, heat death at the end, galaxies only in between. The cup is a tabletop model of why complexity is a transient.
Complexity ≠ entropy
It sharpened a distinction the field had argued for years: a good complexity measure is low for both order and randomness, and high only for the structured middle — captured here as “blur, then compress.”
A theorem still wanted
Proving the interacting cup truly peaks remains unsolved. The honest open problem keeps the coffee automaton a live benchmark for anyone defining complexity rigorously.
A pour of cream became the cleanest known toy model of why closed systems get interesting in the middle — and a definition you can compute with gzip.