Entropy in a closed system only ever goes up, yet the universe was structureless at the Big Bang and will be structureless again once everything disperses — so "interestingness" clearly behaves differently from entropy. This paper takes that vague intuition and tries to pin a number on it, asking a question most thermodynamics treatments dodge: what exactly rises and then falls as a system equilibrates?
Key Findings
- Apparent complexity, defined operationally. The authors model mixing coffee and cream as a 2D cellular automaton and measure the Kolmogorov complexity of a coarse-grained (blurred) snapshot of its state — a quantity they name "apparent complexity," distinct from the entropy that just keeps climbing.
- Interaction is the whole story. Analytically, non-interacting particles never reach high apparent complexity; the tendrils-and-swirls phase only appears when particles interact. Structure is not a free byproduct of mixing — it requires coupling.
- A complexity bump, not a plateau. Numerically, the interacting system's apparent complexity peaks at a value comparable to the width of the "coffee cup" before collapsing as equilibrium arrives, reproducing the rise-and-fall curve the paper set out to capture.
- An open challenge, stated honestly. Proving the observed peaking behavior analytically is left as an explicit open problem, not papered over.
How It Works
Coarse-graining is the conceptual pivot: at full resolution a mixed state is as random as an unmixed one, so complexity must be measured on a blurred view. That blurring is what makes "the cream is half-swirled" register as more complex than either pure separation or uniform brown — a small modeling choice that does most of the work.
Great Fit / When to Skip
Great fit if you care about defining and quantifying complexity, self-organization, or the arrow of time, and want a concrete toy model rather than hand-waving. Look elsewhere if you came for a finished theorem — the central behavior is shown numerically and posed as a conjecture, not proven — or if you expected a machine-learning method: despite living in an AI directory, this is a physics-and-information-theory paper at heart.
