Memory of fracture in information geometry

In this study, the memory effect of the fracture phenomenon in information geometry is discussed. The input–output relation in a complex system as an application of fractional calculus generates the power law for the time and response time distribution, which determines the memory effect. The exponent of the response time distribution is related to the one of the various power laws for fracture phenomena, including earthquakes. The one of them is the shape parameter of the Weibull distribution, which indicates uniformity in the material. The exponent of the response time distribution is also linked to the magnitude of the change rate in the information density and the non-extensivity of the information in the statistical manifold for the response time distribution. From the discussion of the properties of their exponents, the memory effect of a fracture depends on the response time distribution with the uniformity of the material and reflects the information density for parameters related to the fracture and the non-extensivity of the information in the statistical manifold for the response time distribution. Moreover, we propose a method to understand fracture phenomena using information geometry for the response time distribution.