Boltzmann distribution

Remarks

The factor $e^{-\beta \varepsilon_i}$ is the Boltzmann factor, and $Z$ is the partition function. For two states, $p_i / p_j = e^{-\beta (\varepsilon_i - \varepsilon_j)}$. Example: the levels $\varepsilon_n = n \hbar \omega$, $n \ge 0$, of a quantum harmonic oscillator give $Z = 1 / (1 - e^{-\beta \hbar \omega})$. So $p_n = (1 - e^{-\beta \hbar \omega}) e^{-n \beta \hbar \omega}$, a geometric distribution. Non-example: the levels $\varepsilon_n = -1/n^2$, $n \ge 1$, accumulate at $0$, so $\sum_n e^{-\beta \varepsilon_n} = \infty$ and no Boltzmann distribution exists. For a continuous set of states one replaces the sum by an integral against a reference measure.