The Information Geometric Structures of the Sizes of Grains

The parameter space of sizes of normal grown grain was investigated from the viewpoint of information geometry. As the diameters of grains could be described with lognormal distributions, the geometric structures of the manifold with lognormal distributions were considered to analyze the diameters of grains. The obtained manifold is a hyperbolic space with a negative constant curvature. Meanwhile, the geodesic equations were obtained and solved. And then the relations between the Kullback divergence and the arc length were analyzed. At last, the divergence of the Jacobi fields was considered.