Numerical Analysis of Statistical Manifolds Associated with Nonequilibrium Processes

利用信息几何的方法，在由指定相关步长的跳跃概率构成的参数空间上引入度量和联系。这些几何对象通常取决于步长时间。研究了曲率张量的时间演化与过程稳定性等物理性质之间的关系。讨论了线性格子上的两个关联行走模型。在模型中，步行器不会停留，参数空间就是二维的。在另一种模型中，步行者有时会停留，在对称行走的情况下，参数空间是三维的。通过滞留参数将三维空间分成两维空间，并研究了每个叶的黎曼标量曲率R的时间演化。结果表明，这两个模型的R‘s的动态性质可以从过程的稳定性、连续两步之间的相关性以及STEPPI…的活性来很好地理解再来一次。随着时间的推移，参数空间趋于一致空间。证明了无穷时间内的R反映了系统的稳定性和路径的正则性。文中还研究了α-曲率。在任何情况下，α=1的曲率都收敛到零。这一性质被认为是包括热力学平衡系统在内的一大类系统的普遍性质。指出时间演化参数空间的数学结构与牛顿-卡坦引力理论的数学结构非常相似。此外，还提出了新的随机过程模型。通过考虑动物的行为，得到了描述内部状态随时间变化的对象的运动的随机方程。介绍了另一种模型，该模型使用由无限多个所谓的粘弹性材料构成的假设的复合粘弹性材料的响应函数。利用无限多的复杂粘弹性材料，可以构造出更复杂的粘弹性材料。这种高度复杂的材料被发现受到超慢动力学的影响。新模型的参数空间的几何性质仍然未知。较少

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