What is Microstates and macrostates? — Physics

Definition 16.1 University Physics — Year 3 · Chapter 16 — The Microcanonical Ensemble

A microstate is a complete microscopic specification of a system — every quantum number of every particle (or, classically, every position and momentum, counted in phase-space cells of $h^3$ per particle, Proposition 7.5). A macrostate is what thermodynamics can see: energy $E$ , volume $V$ , particle number $N$ , magnetisation… The multiplicity $\Omega(E, V, N)$ is the number of microstates wearing the same macrostate.

The multiplicity of the spin system against its up-fraction, for growing N: the peak sharpens as 1/√ N. At N 1020 the “distribution” is, for every practical purpose, a single value — macroscopic definiteness out of microscopic democracy. — The multiplicity of the spin system against its up-fraction, for growing $N$ : the peak sharpens as $1/\sqrt N$ . At $N \sim 10^{20}$ the “distribution” is, for every practical purpose, a single value — macroscopic definiteness out of microscopic democracy.

Examples

Example 16.3 (A magnet of $N$ coins)

$N$ independent spins, each up or down: $2^N$ microstates. The macrostate “ $n$ up” has multiplicity $\Omega(n) = \binom Nn$ , overwhelmingly peaked at $n = N/2$ with relative width $\sim1/\sqrt N$ . For $N = 100$ : the all-up state is one; the balanced macrostate holds $1 \times 10^{29}$ . For $N = 10^{20}$ , deviations of even $10^{-8}$ in the up-fraction are suppressed by factors like $\eu^{-10^4}$ : the system is at its peak, and what we call equilibrium is the macrostate with the most microstates.

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