Boltzmann distribution ratios and energy reference ($p_i/p_j=e^{-\beta (E_i-E_j)}$)

For two individual microstates, the partition function cancels:At positive temperature, a higher-energy individual microstate is less probable, and equal-energy individual microstates are equally probable. For example, a gap $E_i-E_j=k_{B}T\ln 2$, 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 $E_i^{\prime }=E_i+C$ for the same constant $C$, thenThe 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.