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→The force and diffusion terms “force”项与“diff”项
===一般形式===
===一般形式===
The general equation can then be written as玻尔兹曼方程的一般形式可以写作：
The general equation can then be written as
玻尔兹曼方程的一般形式可以写作：
<math>
<math>
</math>
</math>
其中“force”一词指外界对粒子施加的力（而不是粒子间的作用），“diff”表示粒子扩散，“coll”表示粒子碰撞，指碰撞中粒子间相互的作用力。上述三项的具体形式将会在下文给出<ref name=":3" />。注意，一些作者会使用 '''v''' 表示粒子的速度，而不是动量 '''p。'''这两个物理量可以通过动量的定义'''p''' = m'''v'''联系起来。
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<ref>McGraw Hill Encyclopaedia of Physics (2nd Edition), C. B. Parker, 1994, <nowiki>ISBN 0-07-051400-3</nowiki>.</ref>.
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<ref name=":3">McGraw Hill Encyclopaedia of Physics (2nd Edition), C. B. Parker, 1994, <nowiki>ISBN 0-07-051400-3</nowiki>.</ref>.
Note that some authors use the particle velocity '''v''' instead of momentum '''p'''; they are related in the definition of momentum by '''p''' = ''m'''''v'''.
Note that some authors use the particle velocity '''v''' instead of momentum '''p'''; they are related in the definition of momentum by '''p''' = ''m'''''v'''.
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，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 a (p _ i) </math > 为动量 < math > p _ i </math > 的某个函数，它在碰撞中是守恒的。还假设力 < math > f _ i </math > 是位置的函数，而且 f 对 < math > p _ i 到 pm </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
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，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 a (p _ i) </math > 为动量 < math > p _ i </math > 的某个函数，它在碰撞中是守恒的。还假设力 < math > f _ i </math > 是位置的函数，而且 f 对 < math > p _ i 到 pm </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>\langle A \rangle = \frac 1 n \int A f \,d^3p.</math>
<math>\langle A \rangle = \frac 1 n \int A f \,d^3p.</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>
<math>\mathbf{x} \mapsto x_i</math> and <math>\mathbf{p} \mapsto p_i = m w_i</math>, where <math>w_i</math> is the particle velocity vector. Define <math>A(p_i)</math> as some function of momentum <math>p_i</math> only, which is conserved in a collision. Assume also that the force <math>F_i</math> is a function of position only, and that f is zero for <math>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
== 方程求解 ==
== 方程求解 ==