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The set of all possible positions r and momenta p is called the phase space of the system; in other words a set of three coordinates for each position coordinate x, y, z, and three more for each momentum component p_x, p_y, p_z. The entire space is 6-dimensional: a point in this space is (r, p) = (x, y, z, p_x, p_y, p_z), and each coordinate is parameterized by time t. The small volume ("differential volume element") is written  

The set of all possible positions r and momenta p is called the phase space of the system; in other words a set of three coordinates for each position coordinate x, y, z, and three more for each momentum component p_x, p_y, p_z. The entire space is 6-dimensional: a point in this space is (r, p) = (x, y, z, p_x, p_y, p_z), and each coordinate is parameterized by time t. The small volume ("differential volume element") is written  

系统中所有可能的位置'''r'''和动量'''p'''的集合称为系统的相空间，集合中位置坐标记为 x，y，z，动量坐标记为''p_x，p_y，p_z。''整个空间是6维的：空间中一点可以表示为('''r''', '''p''') = ( ''x, y, z, p_x, p_y, p_z'' )，每个坐标由时间 t 参数化。小体积元(“微分体积元”)写作

系统中所有可能的位置'''r'''和动量'''p'''的集合称为系统的相空间，集合中位置坐标记为 x，y，z，动量坐标记为''p_x，p_y，p_z。''整个空间是6维的：空间中一点可以表示为('''r''', '''p''') = ( ''x, y, z, p_x, p_y, p_z'' )，每个坐标由时间 t 参数化。微元(即微分体积元)写作：

<math> \text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p} = \text{d}x\,\text{d}y\,\text{d}z\,\text{d}p_x\,\text{d}p_y\,\text{d}p_z. </math>.

<math> \text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p} = \text{d}x\,\text{d}y\,\text{d}z\,\text{d}p_x\,\text{d}p_y\,\text{d}p_z. </math>.

Since the probability of N molecules which all have r and p within <math> \mathrm{d}^3\bf{r}</math>&nbsp;<math> \mathrm{d}^3\bf{p}</math> is in question, at the heart of the equation is a quantity f which gives this probability per unit phase-space volume, or probability per unit length cubed per unit momentum cubed, at an instant of time t. This is a probability density function: f(r, p, t), defined so that,

Since the probability of N molecules which all have r and p within <math> \mathrm{d}^3\bf{r}</math>&nbsp;<math> \mathrm{d}^3\bf{p}</math> is in question, at the heart of the equation is a quantity f which gives this probability per unit phase-space volume, or probability per unit length cubed per unit momentum cubed, at an instant of time t. This is a probability density function: f(r, p, t), defined so that,

由于 n 分子的概率都有 r 和 p 在 < math > mathrm { d } ^ 3 bf { r } </math > < math > < mathrm { d } ^ 3 bf { p } </math > 存在疑问，方程的核心是一个量 f，它给出了单位相空间体积的概率，或单位长度立方的概率，在一瞬间。这是一个概率密度函数: f (r，p，t) ，定义为,

由于在<math> \mathrm{d}^3\bf{r}</math><math> \mathrm{d}^3\bf{p}</math>的N个分子都具有的概率都位置'''r'''和动量'''p'''存在疑问，玻尔兹曼方程的核心是量f，它可以给出在某一时刻t单位相空间体积的概率。这一概率密度函数: f (r，p，t) 定义为,

The set of all possible positions '''r''' and momenta '''p''' is called the [[phase space]] of the system; in other words a set of three [[coordinates]] for each position coordinate ''x, y, z'', and three more for each momentum component ''p_x, p_y, p_z''. The entire space is 6-[[dimension]]al: a point in this space is ('''r''', '''p''') = (''x, y, z, p_x, p_y, p_z''), and each coordinate is [[Parametric equation|parameterized]] by time ''t''. The small volume ("differential [[volume element]]") is written  

The set of all possible positions '''r''' and momenta '''p''' is called the [[phase space]] of the system; in other words a set of three [[coordinates]] for each position coordinate ''x, y, z'', and three more for each momentum component ''p_x, p_y, p_z''. The entire space is 6-[[dimension]]al: a point in this space is ('''r''', '''p''') = (''x, y, z, p_x, p_y, p_z''), and each coordinate is [[Parametric equation|parameterized]] by time ''t''. The small volume ("differential [[volume element]]") is written  

  <math>

  <math>

          《数学》

            《数学》

          \begin{align}

            \begin{align}

          开始{ align }

            开始{ align }

          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:

            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]