From Boltzmann-Gibbs ensemble to generalized ensembles

We reconsider the Boltzmann-Gibbs statistical ensemble in thermodynamics using the multinomial coefficient approach. We show that an ensemble is defined by the determination of four statistical quantities, the element probabilities $p_i$, the configuration probabilities $P_j$, the entropy $S$ and the extremum constraints (EC). This distinction is of central importance for the understanding of the conditions under which a microcanonical, canonical and macrocanonical ensemble is defined. These three ensembles are characterized by the conservation of their sizes. A variation of the ensemble size creates difficulties in the definitions of the quadruplet $\{p_i, P_j, S, \mt{EC}\}$, giving rise for a generalization of the Boltzmann-Gibbs formalism, such one as introduced by Tsallis. We demonstrate that generalized thermodynamics represent a transformation of ordinary thermodynamics in such a way that the energy of the system remains conserved. From our results it becomes evident that Tsallis's formalism is a very specific generalization, however, not the only one. We also revisit the Jaynes's Maximum Entropy Principle, showing that in general it can lead to incorrect results and consider the appropriate corrections.