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Numerical Analysis of Generalized Statistical Manifolds Associated with Stochastic Processes

与随机过程相关的广义统计流形的数值分析

基本信息

批准号：
10680322
负责人：
OBATA Tuhiriro
金额：
$ 0.64万
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：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

项目摘要

1.A generalized statistical manifold is constructed for (time-) discrete stochastic processes characterized by n parameters 6=(θ_1・・・θ_n). According to information geometry, a metric tensor and an α-connection are introduced on an n-parameter family S_N={p(X, θ, N) θ∈Ω} of probability density functions p(X, θ, N) for a random variable X(N) at a discrete time N. As the discrete time goesby, a time series of statistical manifolds・・・, S_1, S_2, ・・・, S_N・・・ is generated, andgathering them leads to a stratified statistical manifold, which is the direct product space S×Z of a statistical manifold S and a set Z of integers. On the generalized statistical manifold an extended connection connecting two layers is introduced after the Newton-Cartan theory of Newton gravity and a theory in elementary particle physics in which Higgs fields are formulated as extended gauge fields. And then extended curvature tensors are constructed from the extended connection and the α-connection. Applying this new geometrical method to a random walk model, we find that its generalized statistical manifold has a duality structure.2.The physical role of the Riemann scalar curvature R of thermodynamic equilibrium systems is studied. Janyszek and Mrugala (JM) interpreted the tfas a measure of the instability of the systems.3.To examine wider roles of curvatures in physics, we also construct new models of thermodynamic systems or stochastic processes. Three germs are obtained, but geometrical characteristics in these models are left as future works.

1.构造了具有n个参数6=(θ_1···θ_n)的(时间)离散随机过程的广义统计流形。根据信息几何理论，在随机变量X(N)在离散时间N的概率密度函数p(X，α，N)的n参数族S_N={p(X，θ，N)θ∈Ω}上引入度量张量和θ-联络。随着离散时间的推移，产生统计流形的时间序列···，S_1，S_2，···，S_N···，将它们聚集在一起，得到一个分层的统计流形，它是统计流形S和一组整数的直积空间S×Z。在广义统计流形上，继牛顿-卡坦引力理论和基本粒子物理学中希格斯场被表示为扩展规范场之后，引入了连接两层的扩展连接。然后由扩展连接和α连接构造扩展曲率张量。将这种新的几何方法应用于一个随机游动模型，我们发现它的广义统计流形具有对偶结构。2.研究了热力学平衡系统的黎曼标量曲率R的物理作用。Janyszek和Mrugala(JM)将TFA解释为系统不稳定性的度量。3.为了考察曲率在物理学中的更广泛作用，我们还构建了热力学系统或随机过程的新模型。得到了三个胚芽，但这些模型中的几何特征有待于下一步的工作。