Microcanonical Entropy — Boltzmann State Counting

Boltzmann's formula: S = k_B ln Ω, where Ω is the number of microstates. For N two-level spins with E up-spins: Ω = C(N,E). The entropy peaks at E = N/2 (maximum disorder), tracing the foundations of statistical mechanics.

N (spins)20

Energy E (up-spins)10

Ω (microstates)

184756

S/k_B = ln Ω

12.13

T (β = ∂S/∂E)⁻¹

∞

f = E/N (filling)

0.50

Ω = C(20,10) = 184756 S = k_B ln(184756) = 12.13 k_B