multiplicity

How many microscopic arrangements produce the same macroscopic appearance? That count is the multiplicity. Toss a hundred coins: only one arrangement gives all heads, but roughly 10^29 arrangements give fifty heads and fifty tails. The fifty-fifty macrostate has vastly higher multiplicity, which is exactly why you always see about half heads and never all of them.

The multiplicity W (or Omega) of a macrostate is the number of microstates consistent with it. For an isolated system it is a function W(E, V, N), and entropy is S = k_B ln W. Multiplicity is generally an astronomically large and extremely sharply peaked function: the equilibrium macrostate has a multiplicity so overwhelmingly greater than any other that fluctuations away from it are unmeasurable, the relative width of the peak scaling like 1/sqrt(N).

The second law of thermodynamics is, at bottom, just this: an isolated system evolves toward the macrostate of maximum multiplicity, because that is where nearly all the microstates are. An honest caveat: counting W requires a rule for what counts as a distinct microstate, whether quantum discreteness (states within a shell) or classical phase-space cells of size h^(3N), divided by N! for identical particles.