Thus the canonical partition function Z becomes Z ≡ ≡ ∑ ∑ i e − − β β E i , where β β ≡ ≡ 1 / ( k B T ) is defined as the thermodynamic beta. Finally, the probability distribution ρ ρ i and entropy S are respectively ρ ρ i = 1 Z e − − β β E i , S = U T + k B ln ⁡ ⁡ Z .