= Boltzmann distribution temperature limits in a finite system
{c}
{title2=$x=k_BT/\Delta$}
Keep the gap $\Delta$ fixed and vary positive <temperature>. With $x=k_BT/\Delta$, the <Boltzmann factor> is $q=e^{-1/x}$. In the four-<microstate> example, as $T\to0^+$, $q\to0$: the unique <ground state> has <probability> tending to one, and the other <microstates> have <probabilities> tending to zero. If several <microstates> instead shared the lowest <energy>, their <state degeneracy> would split that limiting population equally.
As $T\to\infty$ in this fixed finite model, all four individual <Boltzmann factors> tend to one, so all four individual <microstates> have <probability> tending to $1/4$. The three aggregate <probabilities> instead approach
$$
\bigl(P(0),P(\Delta),P(2\Delta)\bigr)\longrightarrow
\left(\frac14,\frac12,\frac14\right).
$$
The equal-population limit applies to this finite set of <microstates>. An infinite collection of <microstates> cannot all have the same positive normalized <probability>, so the statement cannot be extended to an unbounded <energy> spectrum. Real systems may also acquire additional accessible <microstates> as <temperature> rises.
\Image[/boltzmann-populations-versus-temperature.png]
{title=Aggregate <Boltzmann distribution> populations versus positive <temperature> for <state degeneracies> 1, 2, 1}
The plotted crossing occurs at $x=1/\ln2$, where $P(0)=P(\Delta)=4/9$. For larger positive <temperatures>, the middle <macrostate> is more probable than the lowest-<energy> <macrostate>, although the lowest-<energy> individual <microstate> remains more probable than either individual middle <microstate>. The <expected value> of <energy> rises from zero toward $\Delta$ in this finite model.
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