Boltzmann's Stone: Counting W

Plate 12 speaks for itself: “Three boxes, three quanta, ten ways.”

The Physics

Entropy counts microstates. Three identical quanta shared among three distinguishable boxes can be arranged in C(5,2) = 10 ways.

S = k_B ln W

• S: entropy, J K⁻¹
• k_B: Boltzmann constant, 1.380649×10⁻²³ J K⁻¹ (exact)
• W: number of microstates, dimensionless

q = 3: 3+0+0 (3 ways), 2+1+0 (6 ways), 1+1+1 (1 way) = 10.

Honest caveat: The formula is engraved on Boltzmann's tombstone in Vienna; the k_B notation is Planck's.

Sources

• L. Boltzmann, Wien. Ber. 76, 373 (1877).
• SI Brochure, 9th ed. (2019): exact k_B.

On the poster

Plate 012, “Boltzmann's Stone: Counting W”: the pixel-art plate with its slogan “Three boxes, three quanta, ten ways.”, the key equation and the facts panel shown above. Pixelated Physics. Physics. Personified.