Deriving partition functions and entropic functionals from thermodynamics

Given an arbitrary probability distribution and a nondegenerate energy spectrum, so that a mean energy U can be computed, we derive the partition function Z and the entropic functional S that satisfy the basic relation d U = T d S . The procedure is illustrated by considering examples of typical distributions found currently in nature. In particular, the power-law spectrum is shown to correspond to a critical state, associated with Tsallis’ entropy.