Deriving partition functions and entropic functionals from thermodynamics

•The inverse problem for obtaining the entropic functional is presented.•We do not make use of the MaxEnt principle.•An arbitrary probability distribution and an energy spectrum are necessary.•A mean energy UU can be computed and we derive also the partition function ZZ.•The entropic functional SS and mean energy UU satisfy the basic relation dU=TdS.

Given an arbitrary probability distribution and a nondegenerate energy spectrum, so that a mean energy UU can be computed, we derive the partition function ZZ and the entropic functional SS that satisfy the basic relation dU=TdS. 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.