  We study the nonextensive thermodynamics for open systems. On the basis of
the maximum entropy principle, the dual power-law q-distribution functions are
re-deduced by using the dual particle number definitions and assuming that the
chemical potential is constant in the two sets of parallel formalisms, where
the fundamental thermodynamic equations with dual interpretations of
thermodynamic quantities are derived for the open systems. By introducing
parallel structures of Legendre transformations, other thermodynamic equations
with dual interpretations of quantities are also deduced in the open systems,
and then several dual thermodynamic relations are inferred. One can easily find
that there are correlations between the dual relations, from which an
equivalent rule is found that the Tsallis factor is invariable in calculations
of partial derivative with constant volume or constant entropy. Using this
rule, more correlations can be found. And the statistical expressions of the
Lagrange internal energy and pressure are easily obtained.
