= Macrostate
{wiki}
A <macrostate> groups <microstates> sharing prescribed coarse data. The grouping depends on what is retained: here retain only total <energy>. For the two distinguishable <two-level systems>, the <macrostates> $E=0,\Delta,2\Delta$ have <state degeneracies> $1,2,1$. Although two units are not macroscopic in size, they make this loss of information visible: knowing $E=\Delta$ does not tell us which unit is excited.
At <inverse temperature> $\beta\Delta=\ln3$, each unit's excited-state <probability> is $p=1/4$. Their <independence> gives <microstate> <probabilities> $9/16,3/16,3/16,1/16$ in the order above. Adding within each <macrostate> gives
$$
P(E=0)=\frac9{16},\qquad P(E=\Delta)=\frac6{16},\qquad P(E=2\Delta)=\frac1{16}.
$$
The total <expected value> of <energy> is therefore $\Delta(6/16)+2\Delta(1/16)=\Delta/2$, matching two units each with <expected value> of <energy> $\Delta/4$. Equal <state degeneracies> would give the wrong answer: <macrostate> <probabilities> combine counting with the <Boltzmann factor>. At very high positive <temperature>, all four <microstates> become equally likely, but these three <macrostate> <probabilities> approach $1/4,1/2,1/4$.
\Image[/microstate-energy-degeneracy.png]
{title=Four microstates grouped into three energy macrostates, with their thermal probabilities}
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