  Based on the Tsallis entropy, the nonextensive thermodynamic properties are
studied as a q-deformation of classical statistical results using only
probabilistic methods and straightforward calculations. It is shown that the
constant in the Tsallis entropy depends on the deformation parameter and must
be redefined to recover the usual thermodynamic relations. The notions of
variance and covariance are generalised. A partial derivative formula of the
entropy is established. It verifies important relations from which most of the
nonextensive thermodynamics relations can be recovered. This leads to a new
proof of a highly nontrivial conclusion that thermodynamic relations and
Maxwell relations in non extensive thermodynamic have the same forms as those
in ordinary nonextensive statistical mechanics. Theoretical results are applied
to ideal gas. The case of fermions and bosons systems with fractal
distributions is also considered.
