Microstates and multiplicity

A macrostate is everything you can observe about a system from the outside without tracking every particle — a gas's total pressure, volume and temperature, or for a set of coins, simply "how many show heads". A microstate is the complete, exact specification of every individual particle — every molecule's position and velocity, or every single coin's face — and enormously many different microstates can produce the identical macrostate. Flip four coins and "exactly two heads" is one macrostate; HHTT, HTHT, HTTH, THHT, THTH and TTHH are six distinct microstates that all satisfy it. The number of microstates sharing a given macrostate is its multiplicity.

Multiplicity is not evenly spread across macrostates, and the imbalance grows extreme fast as a system gets larger. "All four heads" has multiplicity 1 — only one sequence gives it. "Exactly two heads" has multiplicity 6, already six times more likely by simple counting. Scale this up from four coins to the roughly 10²³ molecules in a mole of gas, and the macrostate where molecules are spread evenly throughout a container has a multiplicity so astronomically larger than the macrostate where they happen to cluster in one corner that the clustered state is never observed in practice — not because it is forbidden, but because it is so improbable that no system to ever exist has waited long enough to land on it.

This is what Boltzmann's S = k ln W actually counts: W is the multiplicity of a macrostate, and entropy is essentially its logarithm. A system drifting toward higher entropy is doing nothing more mysterious than drifting toward whichever macrostate has more microstates able to realise it — the second law of thermodynamics, read this way, is a statement about counting, not about any special "force" pushing systems toward disorder.