  The nonextensive statistical ensembles are revisited for the complex systems
with long-range interactions and long-range correlations. An approximation, the
value of nonextensive parameter (1-q) is assumed to be very tiny, is adopted
for the limit of large particle number for most normal systems. In this case,
Tsallis entropy can be expanded as a function of energy and particle number
fluctuation, and thus the power-law forms of the generalized Gibbs distribution
and grand canonical distribution can be derived. These new distribution
functions can be applied to derive the free energy and grand thermodynamic
potential in nonextensive thermodynamics. In order to establish appropriate
nonextensive thermodynamic formalism, the dual thermodynamic interpretations
are necessary for thermodynamic relations and thermodynamic quantities. By
using a new technique of parameter transformation, the single-particle
distribution can be deduced from the power-law Gibbs distribution. This
technique produces a link between the statistical ensemble and the
quasi-independent system with two kinds of nonextensive parameter having quite
different physical explanations. Furthermore, the technique is used to
construct nonextensive quantum statistics and effectively to avoid the
factorization difficulty in the power-law grand canonical distribution.
