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→The force and diffusion terms “force”项与“diff”项
Note that we have used the fact that the phase space volume element <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> is constant, which can be shown using [[Hamilton's equations]] (see the discussion under [[Liouville's theorem (Hamiltonian)|Liouville's theorem]]). However, since collisions do occur, the particle density in the phase-space volume <math> d^3\bf{r}</math> '<math> d^3\bf{p}</math> changes, so
Note that we have used the fact that the phase space volume element <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> is constant, which can be shown using [[Hamilton's equations]] (see the discussion under [[Liouville's theorem (Hamiltonian)|Liouville's theorem]]). However, since collisions do occur, the particle density in the phase-space volume <math> d^3\bf{r}</math> '<math> d^3\bf{p}</math> changes, so
where Δ''f'' is the ''total'' change in ''f''. Dividing ({{EquationNote|1}}) by <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> Δ''t'' and taking the limits Δ''t'' → 0 and Δ''f'' → 0, we have
where Δ''f'' is the ''total'' change in ''f''. Dividing ({{EquationNote|1}}) by <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> Δ''t'' and taking the limits Δ''t'' → 0 and Δ''f'' → 0, we have{{NumBlk|2=<math>\frac{d f}{d t} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>|3={{EquationRef|2}}}}
The total [[differential of a function|differential]] of ''f'' is:
The total [[differential of a function|differential]] of ''f'' is:
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 来表示; 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。
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 来表示; 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
它在广义相对论的推广是
<math>\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>
where Γ<sup>α</sup><sub>βγ</sub> 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 (x<sup>i</sup>, p<sub>i</sub>) phase space as opposed to fully contravariant (x<sup>i</sup>, p<sup>i</sup>) phase space.
其中 γ < sup > α < βγ 是第二类 Christoffel 符号(假设没有外力，因此粒子在没有碰撞的情况下沿测地线运动) ，其中重要的微妙之处在于密度是混合逆变-协变(x < sup > i ，p < sub > i )相空间中的函数，而不是完全逆变(x < sup > i ，p )相空间中的函数。
{{NumBlk|:|
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. this analytical approach provides insight, but is not generally usable in practical problems.
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。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。
<math>\begin{align}
d f & = \frac{\partial f}{\partial t} \, dt
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)通常用于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流。
+\left(\frac{\partial f}{\partial x} \, dx
+\frac{\partial f}{\partial y} \, dy
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。
+\frac{\partial f}{\partial z} \, dz
\right)
+\left(\frac{\partial f}{\partial p_x} \, dp_x
+\frac{\partial f}{\partial p_y} \, dp_y
+\frac{\partial f}{\partial p_z} \, dp_z
<!--*BGK equation
< ! -- * BGK 方程
\right)\\[5pt]
& = \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 \frac{\mathbf{p}}{m}dt + \frac{\partial f}{\partial \mathbf{p}}\cdot \mathbf{F} \, dt
\end{align}</math>
|{{EquationRef|3}}}}
== 方程求解 ==
== 方程求解 ==