玻尔兹曼方程

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。

"BTE" redirects here. For other uses, see BTE (disambiguation).

The place of the Boltzmann kinetic equation on the stairs of model reduction from microscopic dynamics to macroscopic continuum dynamics (illustration to the content of the book)

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The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.

The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.

玻尔兹曼方程或玻尔兹曼输运方程(BTE)描述了非平衡态热力学系统的统计行为，该方程由路德维希·玻尔兹曼于1872年提出。

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.

这种系统的典型例子是一种流体，其空间温度梯度导致热量从较热的区域流向较冷的区域，这种流体是由组成这种流体的粒子的随机但有偏差的输送引起的。在现代文献中，玻尔兹曼方程一词通常用于更广泛的意义上，指的是任何描述热力学系统中宏观量变化的动力学方程，例如能量、电荷或粒子数。

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]\displaystyle{ \mathrm{d}^3 \bf{r} }[/math]) centered at the position [math]\displaystyle{ \bf{r} }[/math], and has momentum nearly equal to a given momentum vector [math]\displaystyle{ \bf{p} }[/math] (thus occupying a very small region of momentum space [math]\displaystyle{ \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]\displaystyle{ \mathrm{d}^3 \bf{r} }[/math]) centered at the position [math]\displaystyle{ \bf{r} }[/math], and has momentum nearly equal to a given momentum vector [math]\displaystyle{ \bf{p} }[/math] (thus occupying a very small region of momentum space [math]\displaystyle{ \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(数学上是体积元数学公式 ^ 3) ，以位置数学公式 / 数学公式为中心，动量几乎等于给定的动量矢量数学公式(因此占据了动量空间数学公式 ^ 3的一个非常小的区域) ，在时间的瞬间。

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). 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). See also convection–diffusion equation.

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)。参见对流-扩散方程。

The equation is a nonlinear integro-differential equation, and the unknown function in the equation is a probability density function in six-dimensional space of a particle position and momentum. The problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising.

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Overview

The phase space and density function

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 有三个坐标，每个动量分量 p 子 x / sub，p 子 y / sub，p 子 z / sub 有三个坐标。整个空间是6维的: 这个空间中的一个点是(r，p)(x，y，z，p 子 x / sub，p 子 y / sub，p 子 z / sub) ，每个坐标都用时间 t 来参数化。 写入小体积(“微分体积元”)

[math]\displaystyle{ \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]\displaystyle{ \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]

数学文本{ d } ^ 3 mathbf { r } ， text { d } text { d } y ， text { d } z ， text { d } p ， text { d } p y， text { d } p z。数学

Since the probability of N molecules which all have r and p within [math]\displaystyle{ \mathrm{d}^3\bf{r} }[/math] [math]\displaystyle{ \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]\displaystyle{ \mathrm{d}^3\bf{r} }[/math] [math]\displaystyle{ \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 分子在数学上都有 r 和 p 的概率存在问题，因此方程的核心是一个量 f，它给出单位相空间体积的概率，或单位长度立方体单位动量的概率。 这是一个概率密度函数: f (r，p，t) ，定义为,

[math]\displaystyle{ \text{d}N = f (\mathbf{r},\mathbf{p},t)\,\text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p} }[/math]

[math]\displaystyle{ \text{d}N = f (\mathbf{r},\mathbf{p},t)\,\text{d}^3\mathbf{r}\,\text{d}^3\mathbf{p} }[/math]

数学文本{ 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]\displaystyle{ d^3\bf{r} }[/math] about r and momenta lying within a momentum space element [math]\displaystyle{ \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]\displaystyle{ d^3\bf{r} }[/math] about r and momenta lying within a momentum space element [math]\displaystyle{ \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:

是所有位置都在一个体积元数学 d ^ 3 / 关于 r 和动量的分子数，位于一个动量空间元数学 d ^ 3 / 关于 p 的数学，在时间 t 上。在一个位置空间和动量空间的区域上积分，得出在该区域中具有位置和动量的粒子总数:

[math]\displaystyle{ \lt math\gt 数学 \begin{align} \begin{align} Begin { align } 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] N & int 限制[ momenta } text { d } ^ 3 mathem { 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 & = \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 (x，y，z，p x，p y，p z，t) ， text { d } y， text { d } z， text { d } ， p p p p p p \end{align} \end{align} End { align } }[/math]

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数学

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 r₁, p₁ for particle 1, r₂, p₂ for particle 2, etc. up to r_N, p_N for particle N.

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 r₁, p₁ for particle 1, r₂, p₂ for particle 2, etc. up to r_N, p_N 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。对粒子1使用 r 子1 / 子，对粒子1使用 p 子1 / 子，对粒子2使用 p 子2 / 子，等等。粒子 n 可达 r 次 n / sub，p 次 n / sub。

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.

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。

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

[math]\displaystyle{ \lt math\gt 数学 \frac{df}{dt} = \frac{df}{dt} = 我不知道你在说什么 \left(\frac{\partial f}{\partial t}\right)_\text{force} + \left(\frac{\partial f}{\partial t}\right)_\text{force} + 左( frac { partial t } right) text { force } + \left(\frac{\partial f}{\partial t}\right)_\text{diff} + \left(\frac{\partial f}{\partial t}\right)_\text{diff} + 左( frac { partial t }右) text { diff } + \left(\frac{\partial f}{\partial t}\right)_\text{coll}, \left(\frac{\partial f}{\partial t}\right)_\text{coll}, 左( frac { partial t } right) text { coll } , }[/math]

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数学

where the "force" term corresponds to the forces exerted on the particles by an external influence (not by the particles themselves), the "diff" term represents the diffusion of particles, and "coll" is the collision term – accounting for the forces acting between particles in collisions. Expressions for each term on the right side are provided below.

where the "force" term corresponds to the forces exerted on the particles by an external influence (not by the particles themselves), the "diff" term represents the diffusion of particles, and "coll" is the collision term – accounting for the forces acting between particles in collisions. Expressions for each term on the right side are provided below.

如果“力”这个术语对应于外部影响(而不是粒子本身)施加在粒子上的力，“ diff”这个术语代表粒子的扩散，“ coll”是碰撞术语——解释粒子碰撞中的作用力。下面提供了右边每个术语的表达式。

Note that some authors use the particle velocity v instead of momentum p; they are related in the definition of momentum by p = mv.

Note that some authors use the particle velocity v instead of momentum p; they are related in the definition of momentum by p = mv.

请注意，有些作者用粒子速度 v 代替动量 p，它们在动量的定义中与 p / mv 有关。

The force and diffusion terms

Consider particles described by f, each experiencing an external force F not due to other particles (see the collision term for the latter treatment).

Consider particles described by f, each experiencing an external force F not due to other particles (see the collision term for the latter treatment).

考虑由 f 描述的粒子，每个粒子都受到一个外力 f，而不是由于其他粒子(后一种处理方法见碰撞术语)。

Suppose at time t some number of particles all have position r within element [math]\displaystyle{ d^3\bf{r} }[/math] and momentum p within [math]\displaystyle{ d^3\bf{p} }[/math]. If a force F instantly acts on each particle, then at time t + Δt their position will be r + Δr = r + pΔt/m and momentum p + Δp = p + FΔt. Then, in the absence of collisions, f must satisfy

Suppose at time t some number of particles all have position r within element [math]\displaystyle{ d^3\bf{r} }[/math] and momentum p within [math]\displaystyle{ d^3\bf{p} }[/math]. If a force F instantly acts on each particle, then at time t + Δt their position will be r + Δr = r + pΔt/m and momentum p + Δp = p + FΔt. Then, in the absence of collisions, f must satisfy

假设在 t 时，某些粒子的位置都在元素数学 d ^ 3{ r } / math 中，动量 p 在数学 d ^ 3{ p } / math 中。如果一个力 f 立刻作用在每个粒子上，那么在时间 t + t 时，它们的位置是 r + r + pt / m，动量 p + p + f。那么，在没有碰撞的情况下，f 必须满足

[math]\displaystyle{ \lt math\gt 数学 f \left (\mathbf{r}+\frac{\mathbf{p}}{m} \, \Delta t,\mathbf{p}+\mathbf{F} \, \Delta t, t+\Delta t \right )\,d^3\mathbf{r}\,d^3\mathbf{p} = f(\mathbf{r}, \mathbf{p},t) \, d^3\mathbf{r} \, d^3\mathbf{p} f \left (\mathbf{r}+\frac{\mathbf{p}}{m} \, \Delta t,\mathbf{p}+\mathbf{F} \, \Delta t, t+\Delta t \right )\,d^3\mathbf{r}\,d^3\mathbf{p} = f(\mathbf{r}, \mathbf{p},t) \, d^3\mathbf{r} \, d^3\mathbf{p} F 左(r) + f 右(d ^ 3) ，d ^ 3(d ^ 3) ，d ^ 3(d ^ 3) ，d ^ 3(d ^ 3) }[/math]

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数学

Note that we have used the fact that the phase space volume element [math]\displaystyle{ d^3\bf{r} }[/math] [math]\displaystyle{ d^3\bf{p} }[/math] is constant, which can be shown using Hamilton's equations (see the discussion under Liouville's theorem). However, since collisions do occur, the particle density in the phase-space volume [math]\displaystyle{ d^3\bf{r} }[/math] '[math]\displaystyle{ d^3\bf{p} }[/math] changes, so

Note that we have used the fact that the phase space volume element [math]\displaystyle{ d^3\bf{r} }[/math] [math]\displaystyle{ d^3\bf{p} }[/math] is constant, which can be shown using Hamilton's equations (see the discussion under Liouville's theorem). However, since collisions do occur, the particle density in the phase-space volume [math]\displaystyle{ d^3\bf{r} }[/math] '[math]\displaystyle{ d^3\bf{p} }[/math] changes, so

注意，我们已经使用了相空间体元数学 d ^ 3 bf { r } / math d ^ 3 bf { p } / math 是常数这一事实，这可以用 Hamilton 方程表示(参见刘维尔定理下的讨论)。然而，由于碰撞确实存在，相空间体积数学 d ^ 3 / math 数学 d ^ 3 / math 的粒子密度发生了变化，所以呢

[math]\displaystyle{ \begin{align} \lt math\gt \begin{align} 数学 begin { align } dN_\mathrm{coll} & = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}\Delta td^3\mathbf{r} d^3\mathbf{p} \\[5pt] dN_\mathrm{coll} & = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}\Delta td^3\mathbf{r} d^3\mathbf{p} \\[5pt] [2][3][4][4][4][5][5][5][4][5][5][5][5][5][5][5][5][5][5][5][5][5][5][5][5][5][5][5][5][5][5][5][5] & = f \left (\mathbf{r}+\frac{\mathbf{p}}{m}\Delta t,\mathbf{p} + \mathbf{F}\Delta t, t+\Delta t \right)d^3\mathbf{r}d^3\mathbf{p} - f(\mathbf{r}, \mathbf{p}, t) \, d^3\mathbf{r} \, d^3\mathbf{p} \\[5pt] & = f \left (\mathbf{r}+\frac{\mathbf{p}}{m}\Delta t,\mathbf{p} + \mathbf{F}\Delta t, t+\Delta t \right)d^3\mathbf{r}d^3\mathbf{p} - f(\mathbf{r}, \mathbf{p}, t) \, d^3\mathbf{r} \, d^3\mathbf{p} \\[5pt] 左(右)数学(r) + 数学(t) ，数学(d) ，数学(d) ，数学(d) ，数学(d) ，数学(d) ，数学(d) ，数学(d) ，数学(d) ，数学(d) & = \Delta f \, d^3\mathbf{r} \, d^3\mathbf{p} & = \Delta f \, d^3\mathbf{r} \, d^3\mathbf{p} 增量 f，d ^ 3，d ^ 3 \end{align} }[/math] \end{align}</math> End { align } / math (1) ()

()

where Δf is the total change in f. Dividing (1) by [math]\displaystyle{ d^3\bf{r} }[/math] [math]\displaystyle{ d^3\bf{p} }[/math] Δt and taking the limits Δt → 0 and Δf → 0, we have

where Δf is the total change in f. Dividing () by [math]\displaystyle{ d^3\bf{r} }[/math] [math]\displaystyle{ d^3\bf{p} }[/math] Δt and taking the limits Δt → 0 and Δf → 0, we have

F 除以数学 d ^ 3 / 数学 d ^ 3 / 数学 t 并取极限 t →0和 f →0的总变化是多少

[math]\displaystyle{ \frac{d f}{d t} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll} }[/math] [math]\displaystyle{ \frac{d f}{d t} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll} }[/math] 左边(左边部分 t 右边)右边(左边) (2) ()

()

The total differential of f is:

The total differential of f is:

F 的全微分是:

[math]\displaystyle{ \begin{align} \lt math\gt \begin{align} 数学 begin { align } d f & = \frac{\partial f}{\partial t} \, dt d f & = \frac{\partial f}{\partial t} \, dt 部分的部分的部分的部分的部分的，dt +\left(\frac{\partial f}{\partial x} \, dx +\left(\frac{\partial f}{\partial x} \, dx + 左( frac { partial f } x，dx +\frac{\partial f}{\partial y} \, dy +\frac{\partial f}{\partial y} \, dy +-frac { partial f }{ partial y } ，dy +\frac{\partial f}{\partial z} \, dz +\frac{\partial f}{\partial z} \, dz 部分的，部分的，部分的 \right) \right) 对) +\left(\frac{\partial f}{\partial p_x} \, dp_x +\left(\frac{\partial f}{\partial p_x} \, dp_x + 左( frac { partial f }{ partial p x } ，dp x +\frac{\partial f}{\partial p_y} \, dp_y +\frac{\partial f}{\partial p_y} \, dp_y + frac { partial f } partial py } ，dp y +\frac{\partial f}{\partial p_z} \, dp_z +\frac{\partial f}{\partial p_z} \, dp_z + frac { partial f } partial pz } ，dp z \right)\\[5pt] \right)\\[5pt] 右)[5 pt ] & = \frac{\partial f}{\partial t}dt +\nabla f \cdot d\mathbf{r} + \frac{\partial f}{\partial \mathbf{p}}\cdot d\mathbf{p} \\[5pt] & = \frac{\partial f}{\partial t}dt +\nabla f \cdot d\mathbf{r} + \frac{\partial f}{\partial \mathbf{p}}\cdot d\mathbf{p} \\[5pt] 部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的 & = \frac{\partial f}{\partial t}dt +\nabla f \cdot \frac{\mathbf{p}}{m}dt + \frac{\partial f}{\partial \mathbf{p}}\cdot \mathbf{F} \, dt & = \frac{\partial f}{\partial t}dt +\nabla f \cdot \frac{\mathbf{p}}{m}dt + \frac{\partial f}{\partial \mathbf{p}}\cdot \mathbf{F} \, dt 部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的部分函数的 \end{align} }[/math] \end{align}</math> End { align } / math (3) ()

()

where ∇ is the gradient operator, · is the dot product,

where ∇ is the gradient operator, · is the dot product,

其中 something 是梯度算符，是点积,

[math]\displaystyle{ \lt math\gt 数学 \frac{\partial f}{\partial \mathbf{p}} = \mathbf{\hat{e}}_x\frac{\partial f}{\partial p_x} + \mathbf{\hat{e}}_y\frac{\partial f}{\partial p_y} + \mathbf{\hat{e}}_z \frac{\partial f}{\partial p_z}= \nabla_\mathbf{p}f \frac{\partial f}{\partial \mathbf{p}} = \mathbf{\hat{e}}_x\frac{\partial f}{\partial p_x} + \mathbf{\hat{e}}_y\frac{\partial f}{\partial p_y} + \mathbf{\hat{e}}_z \frac{\partial f}{\partial p_z}= \nabla_\mathbf{p}f 部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分的，部分, }[/math]

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数学

is a shorthand for the momentum analogue of ∇, and ê_x, ê_y, ê_z are Cartesian unit vectors.

is a shorthand for the momentum analogue of ∇, and ê_x, ê_y, ê_z are Cartesian unit vectors.

是 something 的动量模拟的简写，子 x / sub，子 y / sub，子 z / sub 是笛卡尔单位向量。

Final statement

Dividing (3) by dt and substituting into (2) gives:

Dividing () by dt and substituting into () gives:

用 dt 除以()并代入()得出:

[math]\displaystyle{ \frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll} }[/math]

[math]\displaystyle{ \frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll} }[/math]

数学部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数右)

In this context, F(r, t) is the force field acting on the particles in the fluid, and m is the mass of the particles. The term on the right hand side is added to describe the effect of collisions between particles; if it is zero then the particles do not collide. The collisionless Boltzmann equation, where individual collisions are replaced with long-range aggregated interactions, e.g. Coulomb interactions, is often called the Vlasov equation.

In this context, F(r, t) is the force field acting on the particles in the fluid, and m is the mass of the particles. The term on the right hand side is added to describe the effect of collisions between particles; if it is zero then the particles do not collide. The collisionless Boltzmann equation, where individual collisions are replaced with long-range aggregated interactions, e.g. Coulomb interactions, is often called the Vlasov equation.

在这种情况下，f (r，t)是作用于流体中粒子的力场，m 是粒子的质量。右边的术语是用来描述粒子间碰撞的效果; 如果它是零，那么粒子就不会碰撞。无碰撞玻尔兹曼方程，其中个体碰撞被长程聚合的相互作用所取代，例如:。库仑相互作用，通常被称为弗拉索夫方程。

This equation is more useful than the principal one above, yet still incomplete, since f cannot be solved unless the collision term in f is known. This term cannot be found as easily or generally as the others – it is a statistical term representing the particle collisions, and requires knowledge of the statistics the particles obey, like the Maxwell–Boltzmann, Fermi–Dirac or Bose–Einstein distributions.

This equation is more useful than the principal one above, yet still incomplete, since f cannot be solved unless the collision term in f is known. This term cannot be found as easily or generally as the others – it is a statistical term representing the particle collisions, and requires knowledge of the statistics the particles obey, like the Maxwell–Boltzmann, Fermi–Dirac or Bose–Einstein distributions.

这个方程比上面的主方程更有用，但仍然不完整，因为除非知道 f 中的碰撞项，否则 f 就不能求解。这个术语不能像其他术语那样容易或普遍地找到——它是一个代表粒子碰撞的统计术语，需要粒子服从的统计知识，如麦克斯韦-玻尔兹曼、费米-狄拉克或玻色-爱因斯坦分布。

The collision term (Stosszahlansatz) and molecular chaos

Two-body collision term

A key insight applied by Boltzmann was to determine the collision term resulting solely from two-body collisions between particles that are assumed to be uncorrelated prior to the collision. This assumption was referred to by Boltzmann as the "" and is also known as the "molecular chaos assumption". Under this assumption the collision term can be written as a momentum-space integral over the product of one-particle distribution functions:

A key insight applied by Boltzmann was to determine the collision term resulting solely from two-body collisions between particles that are assumed to be uncorrelated prior to the collision. This assumption was referred to by Boltzmann as the "" and is also known as the "molecular chaos assumption". Under this assumption the collision term can be written as a momentum-space integral over the product of one-particle distribution functions:

Boltzmann 应用的一个关键洞察力是确定仅仅由粒子之间的两体碰撞产生的碰撞项，而这种碰撞被认为是碰撞前不相关的。这个假设被玻尔兹曼称为“” ，也被称为“分子混沌假设”。在这种假设下，碰撞项可以写成单粒子分布函数乘积上的动量空间积分:

[math]\displaystyle{ \lt math\gt 数学 \left(\frac{\partial f}{\partial t}\right)_\text{coll} = \left(\frac{\partial f}{\partial t}\right)_\text{coll} = 左( frac { partial t } right) text { coll } \iint gI(g, \Omega)[f(\mathbf{r},\mathbf{p'}_A, t) f(\mathbf{r},\mathbf{p'}_B,t) - f(\mathbf{r},\mathbf{p}_A,t) f(\mathbf{r},\mathbf{p}_B,t)] \,d\Omega \,d^3\mathbf{p}_A \,d^3\mathbf{p}_B, \iint gI(g, \Omega)[f(\mathbf{r},\mathbf{p'}_A, t) f(\mathbf{r},\mathbf{p'}_B,t) - f(\mathbf{r},\mathbf{p}_A,t) f(\mathbf{r},\mathbf{p}_B,t)] \,d\Omega \,d^3\mathbf{p}_A \,d^3\mathbf{p}_B, [ f (mathbf { r } ， mathbf { p’} a，t) f ( mathbf { r } ， mathbf { p’} b，t)-f ( mathbf { r } ，mathba，t) f ( mathbf { r } ， mathbf { p } ，t)] ，d ^ 3 | mathomega，d ^ 3 | mathbf { p } , }[/math]

</math>

数学

where p_A and p_B are the momenta of any two particles (labeled as A and B for convenience) before a collision, p′_A and p′_B are the momenta after the collision,

where p_A and p_B are the momenta of any two particles (labeled as A and B for convenience) before a collision, p′_A and p′_B are the momenta after the collision,

其中 p 子 a / sub 和 p 子 b / sub 是碰撞前任意两个粒子(为方便起见，分别标为 a 和 b)的动量，p & prime; 子 a / sub 和 p & prime; 子 b / sub 是碰撞后的动量,

[math]\displaystyle{ \lt math\gt 数学 g = |\mathbf{p}_B - \mathbf{p}_A| = |\mathbf{p'}_B - \mathbf{p'}_A| g = |\mathbf{p}_B - \mathbf{p}_A| = |\mathbf{p'}_B - \mathbf{p'}_A| 请看下面的链接: http: / / www.ecauses.org }[/math]

</math>

数学

is the magnitude of the relative momenta (see relative velocity for more on this concept), and I(g, Ω) is the differential cross section of the collision, in which the relative momenta of the colliding particles turns through an angle θ into the element of the solid angle dΩ, due to the collision.

is the magnitude of the relative momenta (see relative velocity for more on this concept), and I(g, Ω) is the differential cross section of the collision, in which the relative momenta of the colliding particles turns through an angle θ into the element of the solid angle dΩ, due to the collision.

是相对动量的大小(更多关于这个概念的相对速度) ，而 i (g,)是碰撞的微分截面，碰撞中粒子的相对动量通过一个角度转变为固体角度 d 的元素，由于碰撞。

Simplifications to the collision term

Since much of the challenge in solving the Boltzmann equation originates with the complex collision term, attempts have been made to "model" and simplify the collision term. The best known model equation is due to Bhatnagar, Gross and Krook. The assumption in the BGK approximation is that the effect of molecular collisions is to force a non-equilibrium distribution function at a point in physical space back to a Maxwellian equilibrium distribution function and that the rate at which this occurs is proportional to the molecular collision frequency. The Boltzmann equation is therefore modified to the BGK form:

Since much of the challenge in solving the Boltzmann equation originates with the complex collision term, attempts have been made to "model" and simplify the collision term. The best known model equation is due to Bhatnagar, Gross and Krook. The assumption in the BGK approximation is that the effect of molecular collisions is to force a non-equilibrium distribution function at a point in physical space back to a Maxwellian equilibrium distribution function and that the rate at which this occurs is proportional to the molecular collision frequency. The Boltzmann equation is therefore modified to the BGK form:

由于解决玻尔兹曼方程冲突的大部分挑战来自于复杂的冲突术语，因此人们尝试对冲突术语进行“模型化”和简化。最著名的模型方程是由于 Bhatnagar，Gross 和 Krook。Bgk 近似中的假设是，分子碰撞的效果是迫使物理空间中某一点的非平衡分布函数回到马克斯韦尔平衡分布函数，而这种情况发生的速率与分子碰撞频率成正比。因此，《玻尔兹曼方程修改为 BGK 格式:

[math]\displaystyle{ \frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = \nu (f_0 - f), }[/math]

[math]\displaystyle{ \frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m}\cdot\nabla f + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = \nu (f_0 - f), }[/math]

数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学

where [math]\displaystyle{ \nu }[/math] is the molecular collision frequency, and [math]\displaystyle{ f_0 }[/math] is the local Maxwellian distribution function given the gas temperature at this point in space.

where [math]\displaystyle{ \nu }[/math] is the molecular collision frequency, and [math]\displaystyle{ f_0 }[/math] is the local Maxwellian distribution function given the gas temperature at this point in space.

其中 math / math 是分子碰撞频率，math f0 / math 是给定空间此点气体温度的局部 Maxwellian 分布函数。

General equation (for a mixture)

For a mixture of chemical species labelled by indices i = 1, 2, 3, ..., n the equation for species i is

For a mixture of chemical species labelled by indices i = 1, 2, 3, ..., n the equation for species i is

对于以指数 i 1,2,3，... ，n 标记的化学物种的混合物，物种 i 的方程是

[math]\displaystyle{ \frac{\partial f_i}{\partial t} + \frac{\mathbf{p}_i}{m_i} \cdot \nabla f_i + \mathbf{F} \cdot \frac{\partial f_i}{\partial \mathbf{p}_i} = \left(\frac{\partial f_i}{\partial t} \right)_\text{coll}, }[/math]

[math]\displaystyle{ \frac{\partial f_i}{\partial t} + \frac{\mathbf{p}_i}{m_i} \cdot \nabla f_i + \mathbf{F} \cdot \frac{\partial f_i}{\partial \mathbf{p}_i} = \left(\frac{\partial f_i}{\partial t} \right)_\text{coll}, }[/math]

数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学部分数学

where f_i = f_i(r, p_i, t), and the collision term is

where f_i = f_i(r, p_i, t), and the collision term is

F 子 i / sub f 子 i / sub (r，p 子 i / sub，t)碰撞项是什么

[math]\displaystyle{ \lt math\gt 数学 \left(\frac{\partial f_i}{\partial t} \right)_{\mathrm{coll}} = \sum_{j=1}^n \iint g_{ij} I_{ij}(g_{ij}, \Omega)[f'_i f'_j - f_if_j] \,d\Omega\,d^3\mathbf{p'}, \left(\frac{\partial f_i}{\partial t} \right)_{\mathrm{coll}} = \sum_{j=1}^n \iint g_{ij} I_{ij}(g_{ij}, \Omega)[f'_i f'_j - f_if_j] \,d\Omega\,d^3\mathbf{p'}, 左(部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部分函数部 }[/math]

</math>

数学

where f′ = f′(p′_i, t), the magnitude of the relative momenta is

where f′ = f′(p′_i, t), the magnitude of the relative momenta is

其中 f & prime; f & prime; (p & prime; sub i / sub，t) ，相对动量的大小是

[math]\displaystyle{ g_{ij} = |\mathbf{p}_i - \mathbf{p}_j| = |\mathbf{p'}_i - \mathbf{p'}_j|, }[/math]

[math]\displaystyle{ g_{ij} = |\mathbf{p}_i - \mathbf{p}_j| = |\mathbf{p'}_i - \mathbf{p'}_j|, }[/math]

数学，数学，数学，数学

and I_ij is the differential cross-section, as before, between particles i and j. The integration is over the momentum components in the integrand (which are labelled i and j). The sum of integrals describes the entry and exit of particles of species i in or out of the phase-space element.

and I_ij is the differential cross-section, as before, between particles i and j. The integration is over the momentum components in the integrand (which are labelled i and j). The sum of integrals describes the entry and exit of particles of species i in or out of the phase-space element.

I 小于 ij / 小于是微分截面，和之前一样，粒子 i 和 j 之间的微分截面。积分在被积函数中的动量分量之上(被标记为 i 和 j)。积分之和描述了粒子 i 进入或退出相空间元的过程。

Applications and extensions

Conservation equations

The Boltzmann equation can be used to derive the fluid dynamic conservation laws for mass, charge, momentum, and energy. For a fluid consisting of only one kind of particle, the number density n is given by

The Boltzmann equation can be used to derive the fluid dynamic conservation laws for mass, charge, momentum, and energy. For a fluid consisting of only one kind of particle, the number density n is given by

玻尔兹曼方程可以用来推导质量、电荷、动量和能量的流体动力学守恒定律。对于仅由一种粒子组成的流体，数密度 n 由

[math]\displaystyle{ n = \int f \,d^3p. }[/math]

[math]\displaystyle{ n = \int f \,d^3p. }[/math]

数学，数学，数学，数学

The average value of any function A is

The average value of any function A is

任何函数 a 的平均值都是

[math]\displaystyle{ \langle A \rangle = \frac 1 n \int A f \,d^3p. }[/math]

[math]\displaystyle{ \langle A \rangle = \frac 1 n \int A f \,d^3p. }[/math]

数学 langle a rangle frac 1 n int a f ，d ^ 3 p. / math

Since the conservation equations involve tensors, the Einstein summation convention will be used where repeated indices in a product indicate summation over those indices. Thus [math]\displaystyle{ \mathbf{x} \mapsto x_i }[/math] and [math]\displaystyle{ \mathbf{p} \mapsto p_i = m w_i }[/math], where [math]\displaystyle{ w_i }[/math] is the particle velocity vector. Define [math]\displaystyle{ A(p_i) }[/math] as some function of momentum [math]\displaystyle{ p_i }[/math] only, which is conserved in a collision. Assume also that the force [math]\displaystyle{ F_i }[/math] is a function of position only, and that f is zero for [math]\displaystyle{ p_i \to \pm\infty }[/math]. Multiplying the Boltzmann equation by A and integrating over momentum yields four terms, which, using integration by parts, can be expressed as

Since the conservation equations involve tensors, the Einstein summation convention will be used where repeated indices in a product indicate summation over those indices. Thus [math]\displaystyle{ \mathbf{x} \mapsto x_i }[/math] and [math]\displaystyle{ \mathbf{p} \mapsto p_i = m w_i }[/math], where [math]\displaystyle{ w_i }[/math] is the particle velocity vector. Define [math]\displaystyle{ A(p_i) }[/math] as some function of momentum [math]\displaystyle{ p_i }[/math] only, which is conserved in a collision. Assume also that the force [math]\displaystyle{ F_i }[/math] is a function of position only, and that f is zero for [math]\displaystyle{ p_i \to \pm\infty }[/math]. Multiplying the Boltzmann equation by A and integrating over momentum yields four terms, which, using integration by parts, can be expressed as

由于守恒方程涉及张量，爱因斯坦总和约定将用于重复索引在一个积表明总和超过这些索引。因此，math mathbf { x }映射到 x i / math 和 math mathbf { p }映射到 p i m w i / math，其中 math w i / math 是粒子速度矢量。定义 math a (pi) / math 为动量 math pi / math 的函数，它在碰撞中是守恒的。假设力的数学公式 f i / math 只是位置的函数，而 f 对于数学公式 i pm infty / math 是0。用玻尔兹曼方程乘以 a，再加上动量积分得到4个术语，用部分积分可以表示为

[math]\displaystyle{ \int A \frac{\partial f}{\partial t} \,d^3p = \frac{\partial }{\partial t} (n \langle A \rangle), }[/math]

[math]\displaystyle{ \int A \frac{\partial f}{\partial t} \,d^3p = \frac{\partial }{\partial t} (n \langle A \rangle), }[/math]

部分 t，d ^ 3p 部分 t (n langle a rangle) ，/ math

[math]\displaystyle{ \int \frac{p_j A}{m}\frac{\partial f}{\partial x_j} \,d^3p = \frac{1}{m}\frac{\partial}{\partial x_j}(n\langle A p_j \rangle), }[/math]

[math]\displaystyle{ \int \frac{p_j A}{m}\frac{\partial f}{\partial x_j} \,d^3p = \frac{1}{m}\frac{\partial}{\partial x_j}(n\langle A p_j \rangle), }[/math]

数学，数学，数学，数学，数学，数学，数学，数学，数学，数学

[math]\displaystyle{ \int A F_j \frac{\partial f}{\partial p_j} \,d^3p = -nF_j\left\langle \frac{\partial A}{\partial p_j}\right\rangle, }[/math]

[math]\displaystyle{ \int A F_j \frac{\partial f}{\partial p_j} \,d^3p = -nF_j\left\langle \frac{\partial A}{\partial p_j}\right\rangle, }[/math]

部分 p j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j

[math]\displaystyle{ \int A \left(\frac{\partial f}{\partial t}\right)_\text{coll} \,d^3p = 0, }[/math]

[math]\displaystyle{ \int A \left(\frac{\partial f}{\partial t}\right)_\text{coll} \,d^3p = 0, }[/math]

A 左( frac partial t } right) text { coll } ，d ^ 3p0，/ math

where the last term is zero, since A is conserved in a collision. Letting [math]\displaystyle{ A = m }[/math], the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation:

where the last term is zero, since A is conserved in a collision. Letting [math]\displaystyle{ A = m }[/math], the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation:

最后一项是零，因为 a 在碰撞中是守恒的。让数学 a / m / math，粒子的质量，积分的玻尔兹曼方程变成了质量守恒方程:

[math]\displaystyle{ \frac{\partial}{\partial t}\rho + \frac{\partial}{\partial x_j}(\rho V_j) = 0, }[/math]

[math]\displaystyle{ \frac{\partial}{\partial t}\rho + \frac{\partial}{\partial x_j}(\rho V_j) = 0, }[/math]

数学部分 t rho + frac 部分 x }( rho v j)0，/ math

where [math]\displaystyle{ \rho = mn }[/math] is the mass density, and [math]\displaystyle{ V_i = \langle w_i\rangle }[/math] is the average fluid velocity.

where [math]\displaystyle{ \rho = mn }[/math] is the mass density, and [math]\displaystyle{ V_i = \langle w_i\rangle }[/math] is the average fluid velocity.

其中数学公式是质量密度，数学公式是流体的平均速度。

Letting [math]\displaystyle{ A = p_i }[/math], the momentum of the particle, the integrated Boltzmann equation becomes the conservation of momentum equation:

Letting [math]\displaystyle{ A = p_i }[/math], the momentum of the particle, the integrated Boltzmann equation becomes the conservation of momentum equation:

让数学运算一个 p / i / 数学，粒子的动量，积分玻尔兹曼方程变成了动量守恒方程:

[math]\displaystyle{ \frac{\partial}{\partial t}(\rho V_i) + \frac{\partial}{\partial x_j}(\rho V_i V_j+P_{ij}) - nF_i = 0, }[/math]

[math]\displaystyle{ \frac{\partial}{\partial t}(\rho V_i) + \frac{\partial}{\partial x_j}(\rho V_i V_j+P_{ij}) - nF_i = 0, }[/math]

数学部分 t + 数学部分 x }(数学部分 v i j + p j)-nF i 0，/ math

where [math]\displaystyle{ P_{ij} = \rho \langle (w_i-V_i) (w_j-V_j) \rangle }[/math] is the pressure tensor (the viscous stress tensor plus the hydrostatic pressure).

where [math]\displaystyle{ P_{ij} = \rho \langle (w_i-V_i) (w_j-V_j) \rangle }[/math] is the pressure tensor (the viscous stress tensor plus the hydrostatic pressure).

其中 math p { ij } rho langle (w i-v i)(w j-v j) rangle / math 是压力张量(粘性应力张量加静水压力)。

Letting [math]\displaystyle{ A =\frac{p_i p_i}{2m} }[/math], the kinetic energy of the particle, the integrated Boltzmann equation becomes the conservation of energy equation:

Letting [math]\displaystyle{ A =\frac{p_i p_i}{2m} }[/math], the kinetic energy of the particle, the integrated Boltzmann equation becomes the conservation of energy equation:

让数学 a frac { pi }{2m } / math，粒子的动能，积分玻尔兹曼方程成为能量守恒方程:

[math]\displaystyle{ \frac{\partial}{\partial t}(u + \tfrac{1}{2}\rho V_i V_i) + \frac{\partial}{\partial x_j} (uV_j + \tfrac{1}{2}\rho V_i V_i V_j + J_{qj} + P_{ij}V_i) - nF_iV_i = 0, }[/math]

[math]\displaystyle{ \frac{\partial}{\partial t}(u + \tfrac{1}{2}\rho V_i V_i) + \frac{\partial}{\partial x_j} (uV_j + \tfrac{1}{2}\rho V_i V_i V_j + J_{qj} + P_{ij}V_i) - nF_iV_i = 0, }[/math]

数学部分 t (u + tfrac {1} rho v i) + frac 部分 x }(uj + tfrac {1} v i j + j { qj } v i)-iv i 0，数学

where [math]\displaystyle{ u = \tfrac{1}{2} \rho \langle (w_i-V_i) (w_i-V_i) \rangle }[/math] is the kinetic thermal energy density, and [math]\displaystyle{ J_{qi} = \tfrac{1}{2} \rho \langle(w_i - V_i)(w_k - V_k)(w_k - V_k)\rangle }[/math] is the heat flux vector.

where [math]\displaystyle{ u = \tfrac{1}{2} \rho \langle (w_i-V_i) (w_i-V_i) \rangle }[/math] is the kinetic thermal energy density, and [math]\displaystyle{ J_{qi} = \tfrac{1}{2} \rho \langle(w_i - V_i)(w_k - V_k)(w_k - V_k)\rangle }[/math] is the heat flux vector.

其中动能密度为动能密度，动能密度为动能密度，动能密度为动能密度(w i-v i)(w k-v k)(w k-v k)。

Hamiltonian mechanics

In Hamiltonian mechanics, the Boltzmann equation is often written more generally as

In Hamiltonian mechanics, the Boltzmann equation is often written more generally as

在20世纪90年代哈密顿力学，玻尔兹曼方程通常被写成

[math]\displaystyle{ \hat{\mathbf{L}}[f]=\mathbf{C}[f], \, }[/math]

[math]\displaystyle{ \hat{\mathbf{L}}[f]=\mathbf{C}[f], \, }[/math]

数学[数学][数学][数学][数学]

where L is the Liouville operator (there is an inconsistent definition between the Liouville operator as defined here and the one in the article linked) describing the evolution of a phase space volume and C is the collision operator. The non-relativistic form of L is

where L is the Liouville operator (there is an inconsistent definition between the Liouville operator as defined here and the one in the article linked) describing the evolution of a phase space volume and C is the collision operator. The non-relativistic form of L is

其中 l 是 Liouville 运算符(这里定义的 Liouville 运算符与本文链接的那个运算符定义不一致) ，描述了相空间体的演化，c 是碰撞运算符。L 的非相对论形式是

[math]\displaystyle{ \hat{\mathbf{L}}_\mathrm{NR} = \frac{\partial}{\partial t} + \frac{\mathbf{p}}{m} \cdot \nabla + \mathbf{F}\cdot\frac{\partial}{\partial \mathbf{p}}\,. }[/math]

[math]\displaystyle{ \hat{\mathbf{L}}_\mathrm{NR} = \frac{\partial}{\partial t} + \frac{\mathbf{p}}{m} \cdot \nabla + \mathbf{F}\cdot\frac{\partial}{\partial \mathbf{p}}\,. }[/math]

数学[数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学][数学]

Quantum theory and violation of particle number conservation

It is possible to write down relativistic quantum Boltzmann equations for relativistic quantum systems in which the number of particles is not conserved in collisions. This has several applications in physical cosmology, including the formation of the light elements in Big Bang nucleosynthesis, the production of dark matter and baryogenesis. It is not a priori clear that the state of a quantum system can be characterized by a classical phase space density f. However, for a wide class of applications a well-defined generalization of f exists which is the solution of an effective Boltzmann equation that can be derived from first principles of quantum field theory.

It is possible to write down relativistic quantum Boltzmann equations for relativistic quantum systems in which the number of particles is not conserved in collisions. This has several applications in physical cosmology, including the formation of the light elements in Big Bang nucleosynthesis, the production of dark matter and baryogenesis. It is not a priori clear that the state of a quantum system can be characterized by a classical phase space density f. However, for a wide class of applications a well-defined generalization of f exists which is the solution of an effective Boltzmann equation that can be derived from first principles of quantum field theory.

在碰撞中粒子数不守恒的相对论量子系统中，可以写出相对论量子玻耳兹曼方程。这在物理宇宙学中有几个应用，包括太初核合成中轻元素的形成，暗物质的产生和重子形成。量子系统的状态是否可以用经典的相空间密度 f 来表示，这一点先验上并不清楚。然而，对于广泛的应用来说，f 的一个明确的推广是存在的，它是一个有效的拥有属性玻尔兹曼方程的解，可以从量子场论的第一原理中推导出来。

General relativity and astronomy

The Boltzmann equation is of use in galactic dynamics. A galaxy, under certain assumptions, may be approximated as a continuous fluid; its mass distribution is then represented by f; in galaxies, physical collisions between the stars are very rare, and the effect of gravitational collisions can be neglected for times far longer than the age of the universe.

The Boltzmann equation is of use in galactic dynamics. A galaxy, under certain assumptions, may be approximated as a continuous fluid; its mass distribution is then represented by f; in galaxies, physical collisions between the stars are very rare, and the effect of gravitational collisions can be neglected for times far longer than the age of the universe.

玻尔兹曼方程星云在银河系动力学中有用。在某些假设下，一个星系可以近似为一个连续的流体; 它的质量分布用 f 表示; 在星系中，恒星之间的物理碰撞是非常罕见的，重力碰撞的影响可以忽略倍于宇宙年龄的时间。

Its generalization in general relativity. is

Ellis G F R, Treciokas R, Matravers D R, (1983) Ann. Phys., 150, 487}</ref> is

特雷西奥卡斯 r，马特拉弗斯 d r，(1983)安。Phys，150,487} / ref 是

[math]\displaystyle{ \hat{\mathbf{L}}_\mathrm{GR}=p^\alpha\frac{\partial}{\partial x^\alpha} - \Gamma^\alpha{}_{\beta\gamma}p^\beta p^\gamma\frac{\partial}{\partial p^\alpha}, }[/math]

[math]\displaystyle{ \hat{\mathbf{L}}_\mathrm{GR}=p^\alpha\frac{\partial}{\partial x^\alpha} - \Gamma^\alpha{}_{\beta\gamma}p^\beta p^\gamma\frac{\partial}{\partial p^\alpha}, }[/math]

数学{部分 x ^ alpha }-γ ^ alpha ^ Gamma } p ^ beta p ^ Gamma frac {部分 p ^ alpha } ，/ math

where Γ^α_βγ is the Christoffel symbol of the second kind (this assumes there are no external forces, so that particles move along geodesics in the absence of collisions), with the important subtlety that the density is a function in mixed contravariant-covariant (xⁱ, p_i) phase space as opposed to fully contravariant (xⁱ, pⁱ) phase space.

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In physical cosmology the fully covariant approach has been used to study the cosmic microwave background radiation. More generically the study of processes in the early universe often attempt to take into account the effects of quantum mechanics and general relativity. In the very dense medium formed by the primordial plasma after the Big Bang, particles are continuously created and annihilated. In such an environment quantum coherence and the spatial extension of the wavefunction can affect the dynamics, making it questionable whether the classical phase space distribution f that appears in the Boltzmann equation is suitable to describe the system. In many cases it is, however, possible to derive an effective Boltzmann equation for a generalized distribution function from first principles of quantum field theory. This includes the formation of the light elements in Big Bang nucleosynthesis, the production of dark matter and baryogenesis.

In physical cosmology the fully covariant approach has been used to study the cosmic microwave background radiation. More generically the study of processes in the early universe often attempt to take into account the effects of quantum mechanics and general relativity. In the very dense medium formed by the primordial plasma after the Big Bang, particles are continuously created and annihilated. In such an environment quantum coherence and the spatial extension of the wavefunction can affect the dynamics, making it questionable whether the classical phase space distribution f that appears in the Boltzmann equation is suitable to describe the system. In many cases it is, however, possible to derive an effective Boltzmann equation for a generalized distribution function from first principles of quantum field theory. This includes the formation of the light elements in Big Bang nucleosynthesis, the production of dark matter and baryogenesis.

20世纪90年代物理宇宙学，完全协变方法被用于研究宇宙微波背景辐射。更一般地说，对早期宇宙过程的研究往往试图考虑量子力学和广义相对论的影响。在宇宙大爆炸后由原始等离子体形成的非常致密的介质中，粒子不断地被创造和湮灭。在这样的环境中，量子相干性和波函数的空间扩展会影响系统的动力学行为，使得玻尔兹曼方程中出现的经典相空间分布 f 是否适合描述系统成为疑问。然而，在许多情况下，从量子场论的第一原理导出广义分布函数的有效玻尔兹曼方程是可能的。这包括太初核合成中轻元素的形成，暗物质的产生和重子形成。

Solving the equation

Exact solutions to the Boltzmann equations have been proven to exist in some cases; this analytical approach provides insight, but is not generally usable in practical problems.

Exact solutions to the Boltzmann equations have been proven to exist in some cases; this analytical approach provides insight, but is not generally usable in practical problems.

波尔兹曼方程的精确解在某些情况下已被证明存在; 这种分析方法提供了洞察力，但在实际问题中通常不能使用。

Instead, numerical methods (including finite elements) are generally used to find approximate solutions to the various forms of the Boltzmann equation. Example applications range from hypersonic aerodynamics in rarefied gas flows to plasma flows.

Instead, numerical methods (including finite elements) are generally used to find approximate solutions to the various forms of the Boltzmann equation. Example applications range from hypersonic aerodynamics in rarefied gas flows to plasma flows.

相反，数值方法(包括有限元)通常被用来寻找各种形式的玻尔兹曼方程的近似解。示例应用范围从稀薄气流中的高超音速空气动力学到等离子体流。

Close to local equilibrium, solution of the Boltzmann equation can be represented by an asymptotic expansion in powers of Knudsen number (the Chapman-Enskog expansion). The first two terms of this expansion give the Euler equations and the Navier-Stokes equations. The higher terms have singularities. The problem of developing mathematically the limiting processes, which lead from the atomistic view (represented by Boltzmann's equation) to the laws of motion of continua, is an important part of Hilbert's sixth problem.

Close to local equilibrium, solution of the Boltzmann equation can be represented by an asymptotic expansion in powers of Knudsen number (the Chapman-Enskog expansion). The first two terms of this expansion give the Euler equations and the Navier-Stokes equations. The higher terms have singularities. The problem of developing mathematically the limiting processes, which lead from the atomistic view (represented by Boltzmann's equation) to the laws of motion of continua, is an important part of Hilbert's sixth problem.

在接近局部均衡的情况下，玻尔兹曼方程的解可以用一个克努森数的渐近展开表示(Chapman-Enskog 展开式)。这个展开式的前两项给出了欧拉方程和纳维－斯托克斯方程。较高的项有奇点。从原子观点(以玻尔兹曼方程为代表)到连续统运动定律的极限过程的数学发展问题，是希尔伯特第六个问题的重要组成部分。

See also

• Vlasov equation

• H-theorem

• Fokker–Planck equation

• Williams-Boltzmann equation

• Navier–Stokes equations

• Vlasov–Poisson equation

• Lattice Boltzmann methods

References

• Harris

1 Harris, Stewart (1971). An introduction to the theory of the Boltzmann equation. Dover Books. pp. 221. ISBN 978-0-486-43831-3. https://books.google.com/?id=KfYK1lyq3VYC. . Very inexpensive introduction to the modern framework (starting from a formal deduction from Liouville and the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy (BBGKY) in which the Boltzmann equation is placed). Most statistical mechanics textbooks like Huang still treat the topic using Boltzmann's original arguments. To derive the equation, these books use a heuristic explanation that does not bring out the range of validity and the characteristic assumptions that distinguish Boltzmann's from other transport equations like Fokker–Planck or Landau equations.

| title= An introduction to the theory of the Boltzmann equation | publisher=Dover Books|pages=221 | year= 1971 | isbn=978-0-486-43831-3|url=https://books.google.com/?id=KfYK1lyq3VYC}}. Very inexpensive introduction to the modern framework (starting from a formal deduction from Liouville and the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy (BBGKY) in which the Boltzmann equation is placed). Most statistical mechanics textbooks like Huang still treat the topic using Boltzmann's original arguments. To derive the equation, these books use a heuristic explanation that does not bring out the range of validity and the characteristic assumptions that distinguish Boltzmann's from other transport equations like Fokker–Planck or Landau equations.

玻尔兹曼方程理论导论 | 多佛图书出版社 | 第221页 | 1971年 | isbn 978-0-486-43831-3 | url https://Books.google.com/?id=kfyk1lyq3vyc。非常便宜的现代框架介绍(从 Liouville 和 Bogoliubov-Born-Green-Kirkwood-伊冯等级(BBGKY)的正式演绎玻尔兹曼方程开始)。大多数统计力学的教科书，比如 Huang，仍然使用 Boltzmann 的原始论点来处理这个话题。为了推导这个方程，这些书使用了一种启发式的解释，这种解释没有提出波尔兹曼方程与其他传输方程(如福克-普朗克方程或兰道方程)区别开来的有效性范围和特征性假设。

• Arkeryd, Leif (1972). "On the Boltzmann equation part I: Existence". Arch. Rational Mech. Anal. 45 (1): 1–16. Bibcode:1972ArRMA..45....1A. doi:10.1007/BF00253392.

• Arkeryd, Leif 1. Leif (1972). "On the Boltzmann equation part II: The full initial value problem". Arch. Rational Mech. Anal. 45 (1

关于玻尔兹曼方程的第二部分: 完整的初始值问题): 17–34. Bibcode:1972ArRMA..45...17A. doi:10.1007/BF00253393. {{cite journal}}: Text "Arch 期刊。Rational Mech.肛交。第45卷，第1期" ignored (help); Text "author1-link Leif Arkeryd" ignored (help); line feed character in |first1= at position 6 (help); line feed character in |issue= at position 3 (help)

| pages= 17–34 | year= 1972 | doi = 10.1007/BF00253393 | bibcode = 1972ArRMA..45...17A }}

17-34 | year 1972 | doi 10.1007 / BF00253393 | bibcode 1972ArRMA. . 45... 17A }

• Arkeryd, Leif (1972). "On the Boltzmann equation part I: Existence". Arch. Rational Mech. Anal. 45 (1): 1–16. Bibcode:1972ArRMA..45....1A. doi:10.1007/BF00253392.

• DiPerna, R. J.; Lions, P.-L. (1989). "On the Cauchy problem for Boltzmann equations: global existence and weak stability". Ann. of Math. 2. 130 (2): 321–366. doi:10.2307/1971423. JSTOR 1971423.

External links

• The Boltzmann Transport Equation by Franz Vesely

• Boltzmann gaseous behaviors solved

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