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添加31字节 、 2021年10月25日 (一) 16:14
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The equation arises not by analyzing the individual [[position vector|position]]s and [[momentum|momenta]] of each particle in the fluid but rather by considering a probability distribution for the position and momentum of a typical particle—that is, the [[probability]] that the particle occupies a given [[infinitesimal|very small]] region of space (mathematically the [[volume element]] <math>\mathrm{d}^3 \bf{r}</math>) centered at the position <math>\bf{r}</math>, and has momentum nearly equal to a given momentum vector <math> \bf{p}</math> (thus occupying a very small region of [[momentum space]] <math>\mathrm{d}^3 \bf{p}</math>), at an instant of time.
 
The equation arises not by analyzing the individual [[position vector|position]]s and [[momentum|momenta]] of each particle in the fluid but rather by considering a probability distribution for the position and momentum of a typical particle—that is, the [[probability]] that the particle occupies a given [[infinitesimal|very small]] region of space (mathematically the [[volume element]] <math>\mathrm{d}^3 \bf{r}</math>) centered at the position <math>\bf{r}</math>, and has momentum nearly equal to a given momentum vector <math> \bf{p}</math> (thus occupying a very small region of [[momentum space]] <math>\mathrm{d}^3 \bf{p}</math>), at an instant of time.
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玻尔兹曼方程并不分析流体中每个粒子的单个位置和动量,而是着重考虑一个典型粒子的位置和动量的概率分布,即粒子某一时刻在几何空间占据给定位置<math>\bf{r}</math>处小邻域(数学上的体积元<math>\mathrm{d}^3 \bf{r}</math>),以及在动量空间占据给定动量矢量<math> \bf{p}</math>处小邻域(<math>\mathrm{d}^3 \bf{p}</math>)的概率。
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玻尔兹曼方程并不分析流体中每个粒子的单个位置和动量,而是只考虑一类粒子的位置和动量的概率分布,此类粒子某一时刻在几何空间占据以给定位置<math>\bf{r}</math>为中心的小邻域(数学上的体积元<math>\mathrm{d}^3 \bf{r}</math>),且其动量几乎与给定动量矢量<math> \bf{p}</math>相等,在动量空间占据非常小的区域<math>\mathrm{d}^3 \bf{p}</math>
    
The Boltzmann equation can be used to determine how physical quantities change, such as heat energy and momentum, when a fluid is in transport. One may also derive other properties characteristic to fluids such as viscosity, thermal conductivity, and electrical conductivity (by treating the charge carriers in a material as a gas).
 
The Boltzmann equation can be used to determine how physical quantities change, such as heat energy and momentum, when a fluid is in transport. One may also derive other properties characteristic to fluids such as viscosity, thermal conductivity, and electrical conductivity (by treating the charge carriers in a material as a gas).
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  <math>
 
  <math>
       
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        《数学》
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        《数学》
       
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        \begin{align}
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        \begin{align}
       
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        开始{ align }
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        开始{ align }
       
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        is the number of molecules which ''all'' have positions lying within a volume element <math> d^3\bf{r}</math> about '''r''' and momenta lying within a [[momentum space]] element <math> \mathrm{d}^3\bf{p}</math> about '''p''', at time ''t''.<ref>{{Cite book |last=Huang |first=Kerson |year=1987 |title=Statistical Mechanics |url=https://archive.org/details/statisticalmecha00huan_475 |url-access=limited |location=New York |publisher=Wiley |isbn=978-0-471-81518-1 |page=[https://archive.org/details/statisticalmecha00huan_475/page/n65 53] |edition=Second }}</ref> [[Integration (calculus)|Integrating]] over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:
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        is the number of molecules which ''all'' have positions lying within a volume element <math> d^3\bf{r}</math> about '''r''' and momenta lying within a [[momentum space]] element <math> \mathrm{d}^3\bf{p}</math> about '''p''', at time ''t''.<ref>{{Cite book |last=Huang |first=Kerson |year=1987 |title=Statistical Mechanics |url=https://archive.org/details/statisticalmecha00huan_475 |url-access=limited |location=New York |publisher=Wiley |isbn=978-0-471-81518-1 |page=[https://archive.org/details/statisticalmecha00huan_475/page/n65 53] |edition=Second }}</ref> [[Integration (calculus)|Integrating]] over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:
    
N & = \int\limits_\mathrm{momenta} \text{d}^3\mathbf{p} \int\limits_\mathrm{positions} \text{d}^3\mathbf{r}\,f (\mathbf{r},\mathbf{p},t) \\[5pt]
 
N & = \int\limits_\mathrm{momenta} \text{d}^3\mathbf{p} \int\limits_\mathrm{positions} \text{d}^3\mathbf{r}\,f (\mathbf{r},\mathbf{p},t) \\[5pt]
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