= Boltzmann distribution ratios and energy reference
{c}
{title2=$p_i/p_j=e^{-\beta(E_i-E_j)}$}
For two individual <microstates>, the <partition function> cancels:
$$
\frac{p_i}{p_j}=e^{-\beta(E_i-E_j)}.
$$
At positive <temperature>, a higher-<energy> individual <microstate> is less probable, and equal-<energy> individual <microstates> are equally probable. For example, a gap $E_i-E_j=k_BT\ln2$, using the <natural logarithm>, gives $p_i/p_j=1/2$. This comparison concerns individual <microstates>; a whole <macrostate> can contain several of them.
Changing the reference for <energy> leaves every <probability> unchanged. If all <energies> become $E_i'=E_i+C$ for the same constant $C$, then
$$
Z'=e^{-\beta C}Z,\qquad
p_i'=\frac{e^{-\beta(E_i+C)}}{e^{-\beta C}Z}=p_i.
$$
The <partition function> itself changes, but its common factor cancels in the <Boltzmann distribution>. Thus choosing a <ground state> <energy> of zero simplifies calculations without changing populations. The <expected value> of <energy> shifts by $C$, as it must when its reference changes.
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