Multiplicity (Ω)

The multiplicity Ω of a macrostate is the count of microstates compatible with it. For N two-state objects (coins showing heads or tails, molecules in the left or right half of a box), the multiplicity of the macrostate 'k in the first state' is the binomial coefficient Ω = C(N, k) = N! / [k!(N−k)!]. It measures how many distinct microscopic arrangements look identical from the macroscopic outside.

Multiplicities are astronomically large and grow with ferocious speed: for 100 coins the central macrostate has C(100, 50) ≈ 10²⁹ microstates, while 'all heads' has exactly one. This disparity is why systems settle into their most probable macrostate and effectively never leave it — the equilibrium macrostate outnumbers all alternatives by factors like 10 raised to 10²³.

Boltzmann's formula S = k_B ln Ω turns the multiplicity into the thermodynamic entropy. Because Ω multiplies when independent systems combine while entropy adds, the logarithm is the unique bridge between the microscopic count and the macroscopic state function.

Multiplicity counting underlies Boltzmann's 1877 statistical definition of entropy and the de Moivre–Laplace theorem (1730s–1810s) that gives its Gaussian large-N limit.