subject to the normalization constraint that ∑ ∑ p i = 1 and the constraint that ∑ ∑ p i ε ε i equals a particular mean energy value, except for two special cases. (These special cases occur when the mean value is either the minimum or maximum of the energies ε i. In these cases, the entropy maximizing distribution is a limit of Boltzmann distributions where T approaches zero from above or below, respectively.)