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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>\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

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>\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

由于守恒方程中包含[[张量]]，以下使用[[爱因斯坦求和约定]]简化标记，即 <math>\mathbf{x}\rightarrow x_i</math> 且 <math>\mathbf{p}\rightarrow p_i = m w_i</math>，其中 <math>w_i</math> 为粒子速度矢量。定义某函数 <math>A(p_i)</math>，使得其唯一的自变量为动量 <math>p_i</math>（碰撞中动量守恒）。假设力 <math>F_i</math> 为位置的函数，且对于 <math>p_i\rightarrow\pm \infty</math>，<math>f</math> 为0。对玻尔兹曼方程两边同乘 <math>A</math> ，并对动量积分可得如下四项

 

由于守恒方程中包含张量，以下使用'''[[wikipedia:Einstein notation|爱因斯坦求和约定 Einstein Summation Convention]]'''简化标记，即 <math>\mathbf{x}\rightarrow x_i</math> 且 <math>\mathbf{p}\rightarrow p_i = m w_i</math>，其中 <math>w_i</math> 为粒子速度矢量。定义某函数 <math>A(p_i)</math>，使得其唯一的自变量为动量 <math>p_i</math>（碰撞中动量守恒）。假设力 <math>F_i</math> 为位置的函数，且对于 <math>p_i\rightarrow\pm \infty</math>，<math>f</math> 为 0。对玻尔兹曼方程两边同乘 <math>A</math> ，并对动量积分，使用分部积分法可得四项。如下所示：

:<math>\int A \frac{\partial f}{\partial t} \,d^3p = \frac{\partial }{\partial t} (n \langle A \rangle),</math>

:<math>\int A \frac{\partial f}{\partial t} \,d^3p = \frac{\partial }{\partial t} (n \langle A \rangle),</math>

where the last term is zero, since ''A'' is conserved in a collision.

where the last term is zero, since ''A'' is conserved in a collision.

因为 <math>A</math> 在碰撞中守恒，所以最后一项为零。

因为 <math>A</math> 在碰撞中守恒，所以最后一项为零。