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第719行：
第719行： − === 与概率论的结合 ===+ 第808行：
第808行： − === 计算热力学总能 ===+ 第839行：
第839行： − === 与热力学变量的关联 ===+ 第874行：
第874行： − === 子系统配分函数 ===+ 第898行：
第898行： − === 影响 ===+
→正则配分函数
== 与概率论的结合 ==
For simplicity, we will use the discrete form of the partition function in this section. Our results will apply equally well to the continuous form.
For simplicity, we will use the discrete form of the partition function in this section. Our results will apply equally well to the continuous form.
</math>
</math>
== 计算热力学总能 ==
为了证明配分函数的有用性，让我们计算总能量的热力学值。这仅仅是能量的期望值，或者说总体均值，它是微状态能量的总和，加上它们的概率:
为了证明配分函数的有用性，让我们计算总能量的热力学值。这仅仅是能量的期望值，或者说总体均值，它是微状态能量的总和，加上它们的概率:
这为我们提供了一种计算许多微观量的期望值的方法。我们将这个量人为地加到微态能量上(或者用量子力学的语言，加到哈密顿量上) ，计算出新的配分函数和期望值，然后在最终的表达式中将 λ 设置为零。这类似于量子场论路径积分表述中使用的源场方法。
这为我们提供了一种计算许多微观量的期望值的方法。我们将这个量人为地加到微态能量上(或者用量子力学的语言，加到哈密顿量上) ，计算出新的配分函数和期望值，然后在最终的表达式中将 λ 设置为零。这类似于量子场论路径积分表述中使用的源场方法。
== 与热力学变量的关联 ==
== 子系统配分函数 ==
Suppose a system is subdivided into N sub-systems with negligible interaction energy, that is, we can assume the particles are essentially non-interacting. If the partition functions of the sub-systems are ζ<sub>1</sub>, ζ<sub>2</sub>, ..., ζ<sub>N</sub>, then the partition function of the entire system is the product of the individual partition functions:
Suppose a system is subdivided into N sub-systems with negligible interaction energy, that is, we can assume the particles are essentially non-interacting. If the partition functions of the sub-systems are ζ<sub>1</sub>, ζ<sub>2</sub>, ..., ζ<sub>N</sub>, then the partition function of the entire system is the product of the individual partition functions:
这是为了确保我们不会“过多计算”微型状态的数量。虽然这看起来似乎是一个奇怪的要求，但实际上有必要为这样的系统保留一个热力学极限。这就是所谓的[[吉布斯悖论]]。
这是为了确保我们不会“过多计算”微型状态的数量。虽然这看起来似乎是一个奇怪的要求，但实际上有必要为这样的系统保留一个热力学极限。这就是所谓的[[吉布斯悖论]]。
== 影响 ==
It may not be obvious why the partition function, as we have defined it above, is an important quantity. First, consider what goes into it. The partition function is a function of the temperature T and the microstate energies E<sub>1</sub>, E<sub>2</sub>, E<sub>3</sub>, etc. The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles. This dependence on microscopic variables is the central point of statistical mechanics. With a model of the microscopic constituents of a system, one can calculate the microstate energies, and thus the partition function, which will then allow us to calculate all the other thermodynamic properties of the system.
It may not be obvious why the partition function, as we have defined it above, is an important quantity. First, consider what goes into it. The partition function is a function of the temperature T and the microstate energies E<sub>1</sub>, E<sub>2</sub>, E<sub>3</sub>, etc. The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles. This dependence on microscopic variables is the central point of statistical mechanics. With a model of the microscopic constituents of a system, one can calculate the microstate energies, and thus the partition function, which will then allow us to calculate all the other thermodynamic properties of the system.