Physics

Entropy

Lift the wall and a gas fills its box. Every collision could run backwards, and flipping every velocity really does send the gas home. Entropy counts the arrangements that look alike, S = k ln Ω, and chaos is why that count wins.

gas Agas Bwhere they started (after a flip)

Entropy only sees the cells: Ω counts the ways to place the particles so every cell holds the same number. Finer cells see more detail, so S and its ceiling both grow.

Entropy S/k—

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Chapter

Start

Particles N 240

Cells 32

coarsefine

Speed 10 steps / frame

Shade the cells by count

Numbers

Ways to arrange Ω—

Entropy S/k = ln Ω—

Largest possible S/k—

Steps since the wall lifted—

Return to the start—

What to observe

Lift the wall and the gas fills the box. Every collision could run backwards, yet S climbs to its ceiling and stays there.
Flip every velocity and the film runs in reverse: the gas crowds back into its rings and S falls to where it began. The laws have no arrow of time.
Flip again, but first move one particle by a billionth of the box. Collisions blow that error up and the gas never finds its way home.
With 6 particles and two cells, all of them land in one half every few seconds. The second law is a law of large numbers.

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