Information Geometry of q-Gaussian Densities and Behaviors of Solutions to Related Diffusion Equations

This paper presents new geometric aspects of the behaviors of solutions to the porous medium equation (PME) and its associated equation. First we discuss thermostatistical structure with information geometry on a manifold of generalized exponential densities. A dualistic relation between the two existing formalisms by Naudts and Eguchi is elucidated. Next by equipping the manifold of what is called q-Gaussian densities with such a structure, we derive several physically and geometrically interesting properties of the solutions. Since the manifold of the q-Gaussian densities is proved invariant for the equations, it plays a central role in our analysis. We characterize the moment-conserving projection of a solution to the manifold as a geodesic curve. Further, the evolutional velocities of the second moments and the convergence rate to the manifold are evaluated in terms of the Bregman divergence. Finally we show the self-similar solution is geometrically special in the sense that it is simultaneously a geodesic curve for the dually flat connections.