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 is elucidated. Next by equipping the manifold of q-Gaussian densities with such a structure, we derive several physically and geometrically interesting properties of the solutions. The manifold is proved invariant and attracting for the evolving solutions, which play crucial roles in our analysis. We demonstrate that the moment-conserving projection of a solution coincides with a geodesic curve on the manifold. 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 that the self-similar solution is geometrically special in the sense that it is simultaneously geodesic with respect to the mutually dual two affine connections.

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