Microscopic foundation of nonextensive statistics

A combination of the Lie-Poisson equation with the $q$-averaged energy $U_q=〈H{〉}_q$ leads to a microscopic framework for nonextensive $q$ thermodynamics. The resulting von Neumann equation is nonlinear: $i\dot{\rho }=[H,{\rho }^q].$ In spite of its nonlinearity the dynamics is consistent with linear quantum mechanics of pure states. The free energy $F_q{=U}_q-{\mathrm{TS}}_q$ is a stability function for the dynamics. This implies that $q$-equilibrium states are dynamically stable. The (microscopic) evolution of ρ is reversible for any $q,$ but for $q\ne 1$ the corresponding macroscopic dynamics is irreversible.