  Starting with the average particle distribution function for bosons and
fermions for non-extensive thermodynamics , as proposed in \cite{CMP}, we
obtain the corresponding density matrix operators and hamiltonians. In
particular, for the bosonic case the corresponding operators satisfy a deformed
bosonic algebra and the hamiltonian involves interacting terms in powers of
$a^{\dagger}_ja_j$ standard creation and annihilation operators. For the
unnormalized density matrix we obtain a nonlinear equation that leads to a
two-parameter solution relevant to anomalous diffusion phenomena.
