Classical and quantum distributions

We began the previous chapter by pointing out that the energy of a system of non-interacting classical particles could either be written as the sum of the energies of each single particle, $E = \sum_i \epsilon_{r_i}$, or as the sum of the single particles energies, each one multiplied by the number of particles with that particular value of energy, $E = \sum_r n_r \epsilon_r$. We now want to discuss the probabilities of the individual occupation numbers $n_r$, and the resulting expressions for their averages $\bar{n}_r$. We will do this both for classical and for quantum systems.