A basic problem in the correlations between statistics and thermodynamics

By using a fonctionelle of probability distributions, several different statistical physics including extensive and nonextensive statistics are unified in a general method. The essential equivalence between the MaxEnt process of the most probable probility distribution in these statistics and the famous thermodynamical relation dU=TdS is strictly proved without any additional assumption. Moreover, it is expounded that all the conclusions of these different statistics can be directly derived from the equivalent relation.