According to statistical mechanics, the equilibrium probability distribution over the energy levels of a particular mode is given by: P r = e − − β β E ( r ) Z ( β β ) . where we use the reciprocal temperature β β = d e f 1 k B T . The denominator Z (β), is the partition function of a single mode. It makes P r properly normalized, and can be evaluated as Z ( β β ) = ∑ ∑ r = 0 ∞ ∞ e − − β β E ( r ) = e − − β β ε ε / 2 1 − − e − − β β ε ε , with