热物理中的诱导Hilbert空间、Markov链、扩散映射和Fock空间
Induced Hilbert Space, Markov Chain, Diffusion Map and Fock Space in Thermophysics
摘要：
在这篇文章中，我们将继续探索我们在上一篇文章中提出的概率符号（PBN）。利用Dirac向量括号表示法（VBN）和PBN，定义了诱导Hilbert空间和诱导样本空间，并证明了Hilbert空间与由相同基观测量构造的样本空间之间存在等价关系.然后，我们研究马尔可夫转移矩阵及其特征向量，使扩散映射两个例子：一个简单的图论的例子，作为一个原型的双向转移算子;一个著名的文本文件的例子，在IR文献中，作为一个教程的扩散映射在文本文件空间。证明了马氏链的样本空间与转移矩阵的特征向量所张成的Hilbert空间是不等价的。最后，我们将我们的PBN和等价方案应用到热物理中，将样本（相）空间与单粒子的Hilbert空间和多粒子系统的Fock空间联系起来。
英文摘要：
In this article, we continue to explore Probability Bracket Notation (PBN), proposed in our previous article. Using both Dirac vector bracket notation (VBN) and PBN, we define induced Hilbert space and induced sample space, and propose that there exists an equivalence relation between a Hilbert space and a sample space constructed from the same base observable(s). Then we investigate Markov transition matrices and their eigenvectors to make diffusion maps with two examples: a simple graph theory example, to serve as a prototype of bidirectional transition operator; a famous text document example in IR literature, to serve as a tutorial of diffusion map in text document space. We show that the sample space of the Markov chain and the Hilbert space spanned by the eigenvectors of the transition matrix are not equivalent. At the end, we apply our PBN and equivalence proposal to Thermophysics by associating sample (phase) space with the Hilbert space of a single particle and the Fock space of many-particle systems.
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