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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'''[[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：

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]]''',乘积中的重复标记表示对这些带有标记量的求和。因此 <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  <math>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>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).

<math>P_{ij}=\rho\langle (w_i-V_i) (w_j-V_j) \rangle</math> 为压强张量（粘性应力张量加上流体静力学压强）。

<math>P_{ij}=\rho\langle (w_i-V_i) (w_j-V_j) \rangle</math> 为压强张量（粘性应力张量加上流体静力学压强）。

Letting <math>A =\frac{p_i p_i}{2m}</math>, the kinetic energy of the particle, the integrated Boltzmann equation becomes the conservation of energy equation:<ref name="dG1984" />{{rp|pp 19,169}}

Letting <math>A =\frac{p_i p_i}{2m}</math>, the kinetic energy of the particle, the integrated Boltzmann equation becomes the conservation of energy equation:<ref name="dG1984" />{{rp|pp 19,169}}

令 <math>A =\frac{p_i p_i}{2m}</math>，即粒子动能，积分后的玻尔兹曼方程化为能量守恒方程<ref name="dG1984" />{{rp|pp 19,169}}：

令 <math>A =\frac{p_i p_i}{2m}</math>，即粒子动能，积分后的玻尔兹曼方程化为能量守恒方程<ref name="dG1984" />{{rp|pp 19,169}}：

where <math>u = \tfrac{1}{2} \rho \langle (w_i-V_i) (w_i-V_i) \rangle</math> is the kinetic thermal energy density, and <math>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>u = \tfrac{1}{2} \rho \langle (w_i-V_i) (w_i-V_i) \rangle</math> is the kinetic thermal energy density, and <math>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.

<math>u=\tfrac{1}{2}\rho\langle (w_i-V_i) (w_i-V_i) \rangle</math> 为动力热能密度（kinetic thermal energy density），<math>J_{qi}=\tfrac{1}{2}\rho\langle (w_i-V_i)(w_k-V_k)(w_k-V_k)\rangle</math> 热通量矢量。

 

其中<math>u=\tfrac{1}{2}\rho\langle (w_i-V_i) (w_i-V_i) \rangle</math> 为动力热能密度（kinetic thermal energy density），<math>J_{qi}=\tfrac{1}{2}\rho\langle (w_i-V_i)(w_k-V_k)(w_k-V_k)\rangle</math> 热通量矢量。

===Hamiltonian mechanics 哈密顿力学===

===Hamiltonian mechanics 哈密顿力学===