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第34行： − + − 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<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub>''. The entire space is 6-[[dimension]]al: a point in this space is ('''r''', '''p''') = (''x, y, z, p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub>''), and each coordinate is [[Parametric equation|parameterized]] by time ''t''. The small volume ("differential [[volume element]]") is written + − − Since the probability of ''N'' molecules which ''all'' have '''r''' and '''p''' within <math> \mathrm{d}^3\bf{r}</math> <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,位于动量空间元素中的 r 和动量的分子数目，在时间 t 上。在位置空间和动量空间的一个区域上积分，得出在该区域中具有位置和动量的粒子总数:+ − − − − :<math>\text{d}N = f (\mathbf{r},\mathbf{p},t)\,\text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p}</math> − − <math> − − 《数学》 − − − − \begin{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: − − 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)[5 pt ] − − − − & = \iiint\limits_\mathrm{momenta} \quad \iiint\limits_\mathrm{positions} f(x,y,z,p_x,p_y,p_z,t) \, \text{d}x \, \text{d}y \, \text{d}z \, \text{d}p_x \, \text{d}p_y \, \text{d}p_z − − 限制，限制，限制，限制，限制，限制，限制，限制，限制 − − : <math> − − \end{align} − − 结束{ align } − − \begin{align} − − </math> − − 数学 − − 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] − − & = \iiint\limits_\mathrm{momenta} \quad \iiint\limits_\mathrm{positions} f(x,y,z,p_x,p_y,p_z,t) \, \text{d}x \, \text{d}y \, \text{d}z \, \text{d}p_x \, \text{d}p_y \, \text{d}p_z − − which is a 6-fold integral. While f is associated with a number of particles, the phase space is for one-particle (not all of them, which is usually the case with deterministic many-body systems), since only one r and p is in question. It is not part of the analysis to use r<sub>1</sub>, p<sub>1</sub> for particle 1, r<sub>2</sub>, p<sub>2</sub> for particle 2, etc. up to r<sub>N</sub>, p<sub>N</sub> for particle N. − − 这是一个6重积分。虽然 f 与许多粒子相关联，但相空间是单粒子的(不是所有粒子，通常是确定性多体系统的情况) ，因为只有一个 r 和 p 存在问题。用 r < sub > 1 、 p < sub > 1 表示粒子1、 r < sub > 2 、 p < sub > 2 表示粒子2等不属于分析范围。粒子 n 可达 r < sub > n ，p < sub > n 。 − − \end{align} − − </math> − − It is assumed the particles in the system are identical (so each has an identical mass m). For a mixture of more than one chemical species, one distribution is needed for each, see below. − − 假设系统中的粒子是相同的(因此每个粒子的质量都是相同的)。对于一种以上化学物质的混合物，需要对每种物质进行一次分配，见下文。 − + + + − It is assumed the particles in the system are identical (so each has an identical [[mass]] ''m''). For a mixture of more than one [[chemical species]], one distribution is needed for each, see below.+ + − − − − ===Principal statement===
无编辑摘要
===The phase space and density function 相空间和密度函数===
===The phase space and density function 相空间和密度函数===
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<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub>. The entire space is 6-dimensional: a point in this space is (r, p) = (x, y, z, p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub>), 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<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub>''. The entire space is 6-[[dimension]]al: a point in this space is ('''r''', '''p''') = (''x, y, z, p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub>''), and each coordinate is [[Parametric equation|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<sub>x，</sub>p<sub>y，</sub>p<sub>z。</sub>''整个空间是6维的：空间中一点可以表示为('''r''', '''p''') = ( ''x, y, z, p<sub>x,</sub> p<sub>y,</sub> p<sub>z</sub>'' )，每个坐标由时间 t 参数化。微元(即微分体积元)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<sub>x，</sub>p<sub>y，</sub>p<sub>z。</sub>''整个空间是6维的：空间中一点可以表示为('''r''', '''p''') = ( ''x, y, z, p<sub>x,</sub> p<sub>y,</sub> p<sub>z</sub>'' )，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> <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> <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,
<math>\text{d}N = f (\mathbf{r},\mathbf{p},t)\,\text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p}</math>
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f，它可以给出在某一时刻t单位相空间体积的概率。定义概率密度函数: f (r，p，t) 得到,
<math>\text{d}N = f (\mathbf{r},\mathbf{p},t)\,\text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p}</math>
<math>\text{d}N = f (\mathbf{r},\mathbf{p},t)\,\text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p}</math>
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. 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. Integrating over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:
dN是在t时刻，关于（r，p）的微体积元<math> d^3\bf{r}</math>和微动量元<math> \mathrm{d}^3\bf{p}</math>内的分子数目。在位置空间和动量空间的一个区域上积分，得出在该区域中具有位置和动量的粒子总数：
'''''（此处缺少公式搬运）'''''
which is a [[multiple integral|6-fold integral]]. While ''f'' is associated with a number of particles, the phase space is for one-particle (not all of them, which is usually the case with [[deterministic]] [[many body problem|many-body]] systems), since only one '''r''' and '''p''' is in question. It is not part of the analysis to use '''r'''<sub>1</sub>, '''p'''<sub>1</sub> for particle 1, '''r'''<sub>2</sub>, '''p'''<sub>2</sub> for particle 2, etc. up to '''r'''<sub>''N''</sub>, '''p'''<sub>''N''</sub> for particle ''N''.
which is a [[multiple integral|6-fold integral]]. While ''f'' is associated with a number of particles, the phase space is for one-particle (not all of them, which is usually the case with [[deterministic]] [[many body problem|many-body]] systems), since only one '''r''' and '''p''' is in question. It is not part of the analysis to use '''r'''<sub>1</sub>, '''p'''<sub>1</sub> for particle 1, '''r'''<sub>2</sub>, '''p'''<sub>2</sub> for particle 2, etc. up to '''r'''<sub>''N''</sub>, '''p'''<sub>''N''</sub> for particle ''N''.
虽然f与一群粒子有关，但相空间是针对单一粒子进行讨论(对于所有粒子的分析通常是确定性多体系统的情况)，因为只有一个r和p是需要考虑的。使用r1, p1代表粒子1,r2, p2代表粒子2，......，直到rN, pN代表粒子N，都不在考虑范围之内。
It is assumed the particles in the system are identical (so each has an identical [[mass]] ''m''). For a mixture of more than one [[chemical species]], one distribution is needed for each, see below.
系统假设粒子都是相同的（因此每个粒子的质量m相同）。对于组成多于一种化学物质的混合物，其中每种物质都需要一种分布，见下文。
===Principal statement===
The general equation can then be written as
The general equation can then be written as
一般的方程式可以写成
一般的方程式可以写成
<math>
<math>