Bilingual edition: English is preserved and Chinese follows each unit. Terminology uses the confirmed v20260916 glossary; automated semantic review remains traceable.
中英双语版：英文原文完整保留，中文紧随对应单元；术语采用 v20260916 确认表，自动语义审校结果可追溯。

Relation of particle velocity to height

粒子速度与高度的关系

Molecules which are higher in the atmosphere are there ultimately because they had a higher kinetic energy to convert into gravitational potential energy on some time scale. (Never mind the fact of untold billions of collisions on their way to that height!) On a very short time scale between collisions, some will rise incrementally higher than others from their starting points. It is that incremental height gain we consider here.)

大气中较高的分子最终存在是因为它们具有较高的动能，可以转化为重力势能。（尽管在它们到达那高度的过程中经历了无数碰撞！）在两次碰撞之间的极短时间尺度上，一些分子会比其他分子从起点逐渐上升到更高的高度。我们在这里考虑的就是这种逐渐增高的高度。

If we have a population of particles which have energy sufficient to reach height z:

如果我们有一群粒子，其能量足以达到高度 z：

then a population with a little higher velocity can get a little higher

然后一个速度稍高的群体可以获得稍高的

These energy considerations must account for the difference in population with height we observe in the barometric formula.
Back to Discussion of Maxwell speed distribution development
回到Maxwell速度分布的讨论
Forward to Relate particle flux to the velocity distribution.
向前推进：将粒子通量与速度分布联系起来
这些能量考虑必须解释我们在气压公式中观察到的随高度变化的人口差异。
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索引 气体定律概念 气体动理论概念
 
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Relation of particle flux to velocity distribution

粒子通量与速度分布的关系

The number of particles traveling from z to z + dz per unit time may be related to the velocity distribution f(v). The number of particles per unit volume available for the trip is

从z到z + dz的粒子数每单位时间可能与速度分布f(v)有关。可用于这次旅行的单位体积粒子数是

For a given velocity vz, the number which would make it is given by multiplying the number per unit volume times the volume swept out

对于给定的速度v_z，要得到相应的数值，需将单位体积的数值乘以扫过的体积。

The general expression for the change in population (put in differential form) is:

人口变化的一般表达式（用微分形式表示）为：

where df is the differential of the velocity distribution f. So the flux is seen to be proportional to the upward velocity.

Back to Discussion of Maxwell speed distribution development
回到Maxwell速度分布的讨论
Forward to Relate velocity distribution to the barometric formula
将速度分布与巴雷特公式联系起来
其中，df 是速度分布 f 的微分。因此，通量被看出与向上速度成正比。
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索引 气体动理论概念
 
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Relate the velocity distribution to the barometric formula

将速度分布与巴氏公式联系起来

The difference in particle population with height from the barometric formula is

从气压公式来看，高度方向上粒子数量的差异

which must be driven by the available energy for the distribution. The differential change with height is

这必须由可用的能来驱动。高度的微分变化是

and it can be related to the net particle flux

并且可以与净粒子通量相关联

For these small excursions in z, which are chosen small enough that no collisions occur, the height z attained can be related to the particle kinetic energy:

对于这些小范围的z位移，由于其足够小以至于不发生碰撞，所达到的高度z可以与粒子的动能相关联：

The constant C is used to lump together the scaling factors.

常数 C 用于将缩放因子合并在一起。
Back to Discussion of Maxwell speed distribution development
回到Maxwell速度分布的讨论
Forward to Develop and normalize the velocity distribution function.
向前推进并规范化速度分布函数
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索引 气体动理论概念
 
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The form of the velocity distribution

速度分布的形式

The form of the velocity distribution in one dimension can be taken from the Boltzmann distribution in terms of particle kinetic energy. Here we are trying to argue the plausibility of that relationship is by specific correlation with the barometric formula, from which we have obtained:

一维速度分布的形式可以从粒子动能的玻尔兹曼分布中获得。这里我们试图通过与巴蒂公式的具体关联来论证该关系的合理性，从其中我们得到了：

.

Since the integral over all values of velocity must be equal to 1, the expression can be normalized to

由于速度的所有值的积分必须等于1，因此该表达式可以被归一化为
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This is the final step in an exercise to show that one can arrive at the Boltzmann distribution by starting with two experimental observations: the equipartition of energy and the barometric formula. It is in no sense a derivation of the Boltzmann distribution, but one more exercise to show the self-consistency of a major relationship in nature upon which we depend heavily in the study of thermodynamics and kinetic theory.

这是通过从两个实验观察出发推导出玻尔兹曼分布的练习的最后一步。这并不是玻尔兹曼分布的推导，而是一次更多的练习，用以展示我们在热力学和气体动理论研究中依赖的重要自然关系的自洽性。
Discussion of Maxwell speed distribution development
Maxwell 速度分布的发展讨论
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索引 气体动理论概念
 
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The 3-Dimensional Velocity Distribution

三维速度分布

The form of the velocity distribution in one dimension can be taken from the Boltzmann distribution in terms of particle kinetic energy. One way to argue the plausibility of that relationship is by specific correlation with the barometric formula, from which we have obtained:

一维速度分布的形式可以从粒子动能的玻尔兹曼分布中获得。一种论证该关系合理性的方法是通过与巴蒂公式的具体相关性，从其中我们得到了：

which is a Boltzmann distribution for which the probability increases exponentially with decreasing energy. There is no evidence of a peak in probability at some finite value. This is a gaussian distribution around zero velocity, since with purely random motion, the average vector particle velocity is zero. Why, then, is the Maxwell speed distribution

skewed by the velocity squared factor toward higher speeds? The qualitative answer is that there are more ways to achieve the higher speeds when all directions in space are considered. The integration of the velocity distribution function over a range of velocities is essentially a "volume" integral where the velocity vector represents the "radius" of the volume. When you sum over all the probabilities for a given increment dv, it encloses more "volume" if that increment is at higher v.
Discussion of Maxwell speed distribution development
Maxwell 速度分布的发展讨论
Maxwell speed distribution directly from Boltzmann
从玻尔兹曼直接得到麦克斯韦速率分布
这是一条玻尔兹曼分布，其中概率随能量的减小而指数性增加。没有证据表明概率在某个有限值处出现峰值。这是围绕零速度的高斯分布，因为纯随机运动下，粒子的平均速度矢量为零。那么，为什么速度分布函数因速度平方因子而向更高速度倾斜？定性回答是，当考虑所有空间方向时，实现更高速度的方式更多。速度分布函数在速度范围内的积分本质上是一个“体积”积分，其中速度矢量代表体积的“半径”。当你对给定的增量dv进行求和时，如果该增量位于更高的v处，所包含的“体积”会更大。
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