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The nonextensive statistical ensembles are revisited for the complex systemswith long-range interactions and long-range correlations. An approximation, thevalue of nonextensive parameter (1-q) is assumed to be very tiny, is adoptedfor 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 numberfluctuation, and thus the power-law forms of the generalized Gibbs distributionand grand canonical distribution can be derived. These new distributionfunctions can be applied to derive the free energy and grand thermodynamicpotential in nonextensive thermodynamics. In order to establish appropriatenonextensive thermodynamic formalism, the dual thermodynamic interpretationsare necessary for thermodynamic relations and thermodynamic quantities. Byusing a new technique of parameter transformation, the single-particledistribution can be deduced from the power-law Gibbs distribution. Thistechnique produces a link between the statistical ensemble and thequasi-independent system with two kinds of nonextensive parameter having quitedifferent physical explanations. Furthermore, the technique is used toconstruct nonextensive quantum statistics and effectively to avoid thefactorization difficulty in the power-law grand canonical distribution.

The nonextensive statistical ensembles with dual thermodynamic interpretations