  Thermodynamic properties of extensive but nonadditive systems are
investigated. The precise definitions of additivity and extensivity are
presented, and we will see that additivity derives several important properties
including the shape-independence of the thermodynamic functions, the concavity
of the entropy, and the equivalence of ensembles. In nonadditive systems, some
of the above properties can be violated. It is pointed out that the
shape-dependence of the entropy density in a nonadditive system results in the
violation of the Gibbs-Duhem equation. As an example, the dipolar gas is
numerically studied and the violation of the Gibbs-Duhem equation is confirmed.
The predicted violation of the Gibbs-Duhem equation should be observable in
cold-atom experiment with polarized dipolar gas.
