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Numerical Analysis of Statistical Manifolds Associated with Nonequilibrium Processes

与非平衡过程相关的统计流形的数值分析

基本信息

批准号：
07680320
负责人：
OBATA Tsunehiro
金额：
$ 1.34万
依托单位：
Gunma National College of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1996
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07680320/
关键词：
information geometry correlated walk random walk stability nonequilibrium visco-elastic complex system quantum gase 粘弾性物質

项目摘要

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

利用信息几何的方法，在由跳跃概率构成的参数空间上引入了度量和连接。这些几何对象通常依赖于阶跃时间。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。讨论了线性格上的两个相关行走模型。在模型中，步行者不停留，参数空间是二维的。在另一种模型中，行走者有时会停留，对称行走时参数空间是三维的。通过一个停留参数将三维空间划分为二维空间，并研究了每个叶片的黎曼标量曲率R的时间演化。结果表明，从过程的稳定性、连续两步之间的相关性以及阶梯的活动性等方面可以很好地理解两种模型的R的动态特性。随着时间的推移，参数空间趋近于均匀空间。无限时间内的R反映了系统的稳定性和路径的规律性。曲率也进行了研究。=1的曲率在任何情况下都收敛于零。这一性质被认为是包括热力学平衡系统在内的广泛系统的普遍性质。时间演化参数空间的数学结构与牛顿-卡坦引力理论非常相似。此外，还提出了新的随机过程模型。通过考虑动物的行为，得到了内部状态随时间变化的物体运动的随机方程。引入了另一种模型，该模型使用了由无限种所谓粘弹性材料构成的假设复粘弹性材料的响应函数。利用无穷多个复杂粘弹性材料，可以构造出更复杂的粘弹性材料。发现高度复杂的材料受超慢动力学的影响。新模型参数空间的几何性质尚不清楚。少