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 the Legendre structure with information geometry on the manifold of generalized exponential densities. Next by considering such a structure in particular on the q-Gaussian densities, we derive several physically and geometrically interesting properties of the solutions. They include, for example, characterization of the moment-conserving projection of a solution, evaluation of evolutional velocities of the second moments and the convergence rate to the manifold in terms of the geodesic curves, divergence and so on.