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===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_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-[[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  

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：

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,

<math>\text{d}N = f (\mathbf{r},\mathbf{p},t)\,\text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p}</math>

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

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：

 

 

 

 

            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:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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'''₁, '''p'''₁ for particle 1, '''r'''₂, '''p'''₂ for particle 2, etc. up to '''r'''_''N'', '''p'''_''N'' 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'''₁, '''p'''₁ for particle 1, '''r'''₂, '''p'''₂ for particle 2, etc. up to '''r'''_''N'', '''p'''_''N'' for particle ''N''.

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>