Numerical Analysis of Statistical Manifolds Associated with Nonequilibrium Processes

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Metrics and connections are introduced on parameter spaces constituted from the jump probabilities specifying correlated walks, by the method of information geometry. These geometrical objects depend on a step time in general. Relations between the time evolution of curvature tensors and physical properties such as the stability of processes are investigated. Two correlated-walk models on a linear lattice are treated. In a model, the walkers do not stay, and the parameter space is then two-dimensional. In another model, the walkers sometimes stay, and the parameter space is three-dimensional in case of symmetric walking. The three-space is foliated into two-dimensional spaces by a stay parameter, and the time evolution of the Riemann scalar curvature R of each leaf is investigated. It is shown that the dynamic properties of the R's for both models can be well understood in terms of the stability of processes, the correlation between successive two steps, and also the activity of steppi … More ng. As time goes by, the parameter spaces approach uniform spaces. The R's in the infinite time are shown to reflect the stability of systems and also the regularity of paths. The alpha-curvature is also investigated. The alpha=1 curvature converges to zero for any case. This property is suggested to be an universal property in a broad class of systems including thermodynamic equilibrium systems. The mathematical structure of time evolbing parameter spaces is pointed out to be very analogous to that of the Newton-Cartan theory of gravity. Moreover, new models of stochastic processes are proposed. Stochastic equations prescribing the motion of objects whose inner states change in time are obtained through considering the behavior of animals. Another model is introduced, using reponse functions of hypothetical complex visco-elastic materials made from an infinitude of so-called visco-elastic materials. Using an infinitude of complex visco-elastic materials, one can construct more complicated visco-elastic materials. The highly complicated materials are found to be subject to super-slow dynamics. The geometrical properties of the parameter spaces for the new models are still unknown. Less