the grain

Move the grain with the sim paused — the entropy number changes while the microstate stays exactly the same. Coarser cells see the mixing as a big rise; the finest grain barely moves. The gas didn't change; your description did.

Let it run, then press reverse: it flips every particle's velocity, and the gas un-mixes back into the corner. The microscopic law has no arrow. Reverse every velocity and the gas un-mixes. The arrow came from the starting condition, not the law.

// two points, made physical

entropy is read off a grain, not stored in the gas. the arrow of time is in the low-entropy start, not in the reversible law.

Same identical microstate, two different entropies, because entropy counts how many microstates share your coarse description — coarsen the description and the count rises. And the dynamics that mix the gas will just as exactly un-mix it: nothing in the law prefers forward.

// what this is, and isn't

This is a toy showing two established points. First: coarse-grained (Gibbs–Boltzmann) entropy depends on the grain you measure with — the microscopic Gibbs entropy is conserved under the exact dynamics (Liouville's theorem); any rise you see lives in the coarse-graining (standard Gibbs–Tolman reading), and which number you read is description-relative (Jaynes 1965). Second: the reversible law has no built-in arrow — flip the velocities and the gas retraces (Loschmidt 1876). The arrow of the everyday world comes from a special low-entropy starting condition, the Past Hypothesis.

Real systems cannot be reversed in practice. You would need to flip about 10²³ velocities perfectly, and the tiniest error explodes under chaotic amplification, which is why reversals are never seen in practice. The arrow itself comes from the special low-entropy start, the Past Hypothesis. This page reverses only because it is small, integer-exact, and non-interacting. It is a simulation, not a measurement, and it does not challenge the second law: the second law is robust, and coarse-grained entropy really does rise for any grain you would actually use. The claim here is narrow, and it is a proposal — that entropy's status is a description-relative relation, not a fundamental substance in the gas. Cites: Jaynes 1965; Loschmidt 1876; Liouville's theorem.

// kin

does entropy exist — the ontological question, written up · love & entropy — that paper reads the thermodynamic, coarse-grained entropy, which is real and rises; this page asks the separate question of whether entropy is fundamental, and the two are compatible