概率括号表示、Wick-Matsubara关系、密度算符与微观概率建模
Probability Bracket Notation, Wick-Matsubara Relation, Density Operators, and Microscopic Probability Modeling
AI总结：
本文基于概率括号表示（PBN），利用Wick-Matsubara关系将量子系统转化为概率空间中的微观概率过程，恢复了量子统计量并再现理想气体内能，为微观概率建模提供了新途径。
摘要：
继Dirac矢量括号表示（VBN）之后，我们在前一篇论文中提出了概率括号表示（PBN）。我们曾提到，在特殊的Wick旋转（虚时间）下，Hilbert空间中的定态Schrodinger方程会转化为概率空间中微观概率过程（MPP）的主方程。在本文中，我们首先研究单粒子系统的MPP，表明MPP的能量期望最终会趋近于其初始条件中的最低能级，且其von Neumann熵最终会消失。接着，我们探讨Fock空间中全同粒子量子系统的MPP，通过将时间与温度联系起来（即Wick-Matsubara关系），恢复了量子统计中预期的粒子占据数和巨配分函数。我们还利用该关系再现了热力学中理想气体的内能。为了解决熵的问题并将PBN与文献中的统计学研究课题联系起来，我们利用PBN在概率空间中表示密度算符和von Neumann熵。Wick-Matsubara关系加上PBN可能为微观概率建模提供一种新途径。
英文摘要：
Following the Dirac vector bracket notation (VBN), we proposed the probability bracket notation (PBN) in our previous paper. We mentioned that under the special Wick rotation (imaginary time), a stationary Schrodinger equation in the Hilbert space transforms into the master equation of a microscopic probabilistic process (MPP) in the probability space. In this article, we first study the MPP of the system of a single particle, we show that the energy expectation of the MPP eventually approaches the lowest energy level in its initial condition and its von Neumann entropy finally vanishes. Then we explore the MPP for the quantum system of identical particles in the Fock space, we recover the expected occupation number of particles and the grand partition function in quantum statistics by connecting time with temperature (the Wick-Matsubara relation). We also reproduce the internal energy of an ideal gas in thermodynamics by using the relation. To address the entropy issue and relate the PBN with research topics of statistics in the literature, we express the density operators and the von Neumann entropy in the probability space by using the PBN. The Wick-Matsubara relation plus the PBN might provide a new way of microscopic probability modeling.
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