S = k_B ln Ω(E,N) — counting microstates, deriving temperature
For N ideal gas particles sharing energy E quanta, the number of microstates is the stars-and-bars count:
Ω(E,N) = C(E+N−1, N−1). Boltzmann entropy S = k_B ln Ω is extensive: S(2E,2N) ≈ 2S(E,N).
Temperature emerges from the fundamental relation 1/T = ∂S/∂E.