Generalized ensemble theory with non-extensive statistics

The non-extensive canonical ensemble theory is reconsidered with the method of Lagrange multipliers by maximizing Tsallis entropy, with the constraint that the normalized term of Tsallis' q−average of physical quantities, the sum ∑pjq, is independent of the probability pi for Tsallis parameter q. The self-referential problem in the deduced probability and thermal quantities in non-extensive statistics is thus avoided, and thermodynamical relationships are obtained in a consistent and natural way. We also extend the study to the non-extensive grand canonical ensemble theory and obtain the q-deformed Bose-Einstein distribution as well as the q-deformed Fermi-Dirac distribution. The theory is further applied to the generalized Planck law to demonstrate the distinct behaviors of the various generalized q-distribution functions discussed in literature.