The power low in information geometry: Attempt from the viscoelastic relaxation of rock

It is known that power law exists in the background of various natural phenomena. One example is the viscoelastic behavior of rocks. In the flow laws of high temperature of rocks, strain rate is in proportion to the power of stress. It can be replaced by the one that relaxation modulus (the ratio of stress to strain) is in proportion to the power of time with fractal dimension as power exponent. From Laplace transform for the relaxation modulus, the distribution of relaxation time (relaxation spectrum) with the power of relaxation time is derived. It indicates the existence of fractal distribution of different relaxation times in material elements in rocks. On the other hand, these strain-relaxation modulus-stress relations can be recaptured as the input-response-output relation in an ideal complex system with the power law of component. When input and output are stochastic with probability functions, the response corresponds to the change in differential geometric structure on a statistical manifold with a point as a probability function. Although previous studies suggested the correspondence between the power exponent (fractal dimension) and the constant (alpha) characterizing invariant geometric structure (alpha-connection), its details have not been discussed yet. In this presentation, we would reveal the correspondence between the power exponent (fractal dimension) and the constant (alpha) based on q-exponential family in information geometry, which is a more general exponential family.