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.