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{{other uses|Boltzmann's entropy formula|Stefan–Boltzmann law|Maxwell–Boltzmann distribution}}

{{other uses|Boltzmann's entropy formula|Stefan–Boltzmann law|Maxwell–Boltzmann distribution}}

The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number.

The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number.

The classic example of such a system is a [[fluid]] with [[temperature gradient]]s in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the [[particle]]s making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number.

The classic example of such a system is a [[fluid]] with [[temperature gradient]]s in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the [[particle]]s making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number.

The equation arises not by analyzing the individual positions and 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 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 positions and 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 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.

这个方程不是通过分析流体中每个粒子的单个位置和动量而产生的，而是通过考虑一个典型粒子的位置和动量的概率分布，即粒子占据一个给定的非常小的空间区域的概率(数学上是体积元素 < math > mathrm { d } ^ 3 bf { r } </math >) ，动量几乎等于给定的动量矢量 < math > (因此在瞬间占据了一个非常小的动量空间 mathrm { d }3 bf/math >)。

方程的导出不是通过分析流体中每个粒子的单独位置和动量，而是通过考虑一个典型粒子的位置和动量的概率分布——即粒子某一时刻位于给定位置的小邻域（数学上的体积元<math>\mathrm{d}^3 \bf{r}</math>）、在动量空间占据给定动量矢量<math> \bf{p}</math>的小邻域（<math>\mathrm{d}^3 \bf{p}</math>）的概率。一个给定的非常小的空间区域的概率(数学上是体积元素 < math > mathrm { d } ^ 3 bf { r } </math >) ，动量几乎等于给定的动量矢量 < math > (因此在瞬间占据了一个非常小的动量空间 mathrm { d }3 bf/math >)。

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.

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).

玻尔兹曼方程可以用来确定流体在运输过程中物理量如何变化，比如热能和动量。人们还可以推导出流体的其他特性，如粘度、热导率和电导率(通过将材料中的电荷载流子当作气体来处理)。

玻尔兹曼方程可以用来确定流体在运输过程中物理量如何变化，比如热能和动量。人们还可以推导出流体的其他特性，如粘度、热导率和电导率(通过将材料中的载流子当作气体来处理)。参见[[对流扩散方程]]。

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).<ref name="Encyclopaediaof" /> See also [[convection–diffusion equation]].

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).<ref name="Encyclopaediaof" /> See also [[convection–diffusion equation]].

</ref><ref>{{cite journal |author1=Philip T. Gressman |authorlink1=Philip Gressman  |author2=Robert M. Strain  |name-list-style=amp|year=2010 |title= Global classical solutions of the Boltzmann equation with long-range interactions |journal= Proceedings of the National Academy of Sciences  |volume=107 |pages= 5744–5749 | doi = 10.1073/pnas.1001185107  |bibcode = 2010PNAS..107.5744G |arxiv = 1002.3639 |issue= 13 |pmid=20231489 |pmc=2851887}}</ref>

</ref><ref>{{cite journal |author1=Philip T. Gressman |authorlink1=Philip Gressman  |author2=Robert M. Strain  |name-list-style=amp|year=2010 |title= Global classical solutions of the Boltzmann equation with long-range interactions |journal= Proceedings of the National Academy of Sciences  |volume=107 |pages= 5744–5749 | doi = 10.1073/pnas.1001185107  |bibcode = 2010PNAS..107.5744G |arxiv = 1002.3639 |issue= 13 |pmid=20231489 |pmc=2851887}}</ref>

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  

所有可能的位置 r 和动量 p 的集合称为系统的相空间; 换句话说，每个位置坐标 x，y，z 的集合有三个坐标，每个动量分量 p < sub > x </sub > ，p < sub > y </sub > ，p < sub > z </sub > 。整个空间是6维的: 这个空间中的一个点是(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。写入小体积(“微分体积元”)

所有可能的位置 r 和动量 p 的集合称为系统的相空间; 换句话说，每个位置坐标 x，y，z 的集合有三个坐标，每个动量分量 p < sub > x </sub > ，p < sub > y </sub > ，p < sub > z </sub > 。整个空间是6维的: 这个空间中的一个点是(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。写入小体积(“微分体积元”)

 

<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,