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• Longitudinal Data Analysis 纵向数据分析
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## 物理代写|理论力学作业代写Theoretical Mechanics代考|Analogies between quantum mechanics and classical statistical mechanics

A simple comparison between classical statistical mechanics and quantum mechanics involves several analogies between these statistical theories (see in Table 2). Physical theories as classical mechanics and thermodynamics assume a simultaneous definition of complementary variables like the coordinate and the momentum $(\mathbf{q}, \mathbf{p})$ or the energy and the inverse temperature $(E, 1 / T)$. A different situation is found in those applications where the relevant constants as the quantum of action $\hbar$ or the Boltzmann’s constant $k_{B}$ are not so small. According to uncertainty relations shown in equations (3) and (92), the thermodynamic state $(E, 1 / T)$ of a small thermodynamic system is badly defined in an analogous way that a quantum system cannot support the classical notion of particle trajectory $[\mathbf{q}(t), \mathbf{p}(t)]$.

Apparently, uncertainty relations can be associated with the coexistence of variables with different relevance in a statistical theory. In one hand, we have the variables parameterizing the results of experimental measurements: space-time coordinates $(t, \mathbf{q})$ or the mechanical macroscopic observables $I=\left(I^{i}\right)$. On the other hand, we have their conjugated variables associated with the dynamical description: the energy-momentum $(E, \mathbf{p})$ or the generalized differential forces $\eta=\left(\eta_{i}\right)$. These variables control the respective deterministic dynamics: while the energy $E$ and the momentum $\mathbf{p}$ constrain the trajectory $\mathbf{q}(t)$ of a classical mechanic system, the inverse temperature differences, $\eta=1 / T^{m}-1 / T$, drives the dynamics of the system energy $E(t)$ (i.e.: the energy interchange) and its tendency towards the equilibrium. Similarly, the experimental determination of these dynamical variables demands the consideration of many repeated measurements due to their explicit statistical significance in the framework of their respective statistical theories.

According to the comparison presented in Table 2, the classical action $S(\mathbf{q}, t)$ and the thermodynamic entropy $S(I \mid \theta)$ can be regarded as two counterpart statistical functions. Interestingly, while the expression (10) describing the relation between the wave function $\Psi(\mathbf{q}, t)$ and the classical action $S(\mathbf{q}, t)$ is simply an asymptotic expression applicable in the quasi-classic limit where $S(\mathbf{q}, t) \gg \hbar$, Einstein’s postulate (84) is conventionally assumed as an exact expression in classical fluctuation theory. The underlying analogy suggests that Einstein’s postulate (84) should be interpreted as an asymptotic expression obtained in the limit $S(I \mid \theta) \gg k_{B}$ of a more general statistical mechanics theory. This requirement is always satisfied in conventional applications of classical fluctuation theory, which deal with the small fluctuating behavior of large thermodynamic systems. Accordingly, this important hypothesis of classical fluctuation theory should lost its applicability in the case of small thermodynamic systems. In the framework of such a general statistical theory, Planck’s constant $k_{B}$ could be regarded as the quantum of entropy.

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Final remarks

Classical statistical mechanics and quantum theory are two formulations with different mathematical structure and physical relevance. However, these physical theories are hallmarked by the existence of uncertainty relations between conjugated quantities. Relevant examples are the coordinate-momentum uncertainty $\Delta q \Delta p \geq \hbar / 2$ and the energy-temperature uncertainty $\Delta E \Delta\left(1 / T-1 / T^{m}\right) \geq k_{B}$. According to the arguments discussed along this chapter, complementarity has appeared as an unavoidable consequence of the statistical apparatus of a given physical theory. Remarkably, classical statistical mechanics and quantum mechanics shared many analogies with regards to their conceptual features: (1) Both statistical theories need the correspondence with a deterministic theory for their own formulation, namely, classical mechanics and thermodynamics; (2) The measuring instruments play a role in the existence of complementary quantities; (3) Finally, physical observables admit the correspondence with appropriate operators, where the existence of complementary quantities can be related to their noncommutative character.

As an open problem, it is worth remarking that the present comparison between classical statistical mechanics and quantum mechanics is still uncomplete. Although the analysis of complementarity has been focused in those systems in thermodynamic equilibrium, the operational interpretation discussed in this chapter strongly suggests the existence of a counterpart of Schrödinger equation (42) in classical statistical mechanics. In principle, this counterpart dynamics should describe the system evolutions towards the thermodynamic equilibrium, a statistical theory where Einstein’s postulate (84) appears as a correspondence principle in the thermodynamic limit $k_{B} \rightarrow 0$.

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MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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