晶粒直径的信息几何结构

The Information Geometric Structures of the Sizes of Grains

从信息几何的角度研究了晶粒直径所呈现的几何特性.由于某物质的晶粒直径服从对数正态分布,通过分析对数正态分布全体所构成的统计流形的几何结构,达到对晶粒直径研究的目的,得到了该统计流形是具有负常曲率的双曲空间.同时,通过求解测地线方程,得到流形的测地线.并且讨论了Kullback散度和弧长之间的关系.最后,讨论了Jacobi场的敛散性.

The parameter space of sizes of normal grown grain was investigated from theviewpoint 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.