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→Two-body collision term 双体碰撞项
A key insight applied by [[Ludwig Boltzmann|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 "''Stosszahlansatz'' " 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:<ref name="Encyclopaediaof" />
A key insight applied by [[Ludwig Boltzmann|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 "''Stosszahlansatz'' " 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:<ref name="Encyclopaediaof" />
[[路德维希·玻尔兹曼|玻尔兹曼]]在确定碰撞项时所应用到的关键见解就是：他假设的碰撞项完全是由假定在碰撞前不相关的两个粒子的相互碰撞得到的。这个假设被波尔兹曼称为“Stosszahlansatz”，也叫做“[[分子混沌假设 Molecular chaos]]”。根据这一假设，碰撞项可以被写作单粒子分布函数的乘积在动量空间上的积分：<ref name="Encyclopaediaof" />
[[路德维希·玻尔兹曼|玻尔兹曼]]在确定碰撞项时所应用到的关键见解就是：他假设的碰撞项完全是由假定在碰撞前不相关的两个粒子的相互碰撞得到的。这个假设被波尔兹曼称为“Stosszahlansatz”，也叫做“[[Index.php?title=分子混沌假设|分子混沌假设 ]]”。根据这一假设，碰撞项可以被写作单粒子分布函数的乘积在动量空间上的积分：<ref name="Encyclopaediaof" />
:<math>
:<math>
===Simplifications to the collision term===
===Simplifications to the collision term 对碰撞项的简化===
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.<ref>{{Cite journal|title=A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems|journal=Physical Review|date=1954-05-01|pages=511–525|volume=94|issue=3|doi=10.1103/PhysRev.94.511|first1=P. L.|last1=Bhatnagar|first2=E. P.|last2=Gross|first3=M.|last3=Krook|bibcode=1954PhRv...94..511B}}</ref> 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.<ref>{{Cite journal|title=A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems|journal=Physical Review|date=1954-05-01|pages=511–525|volume=94|issue=3|doi=10.1103/PhysRev.94.511|first1=P. L.|last1=Bhatnagar|first2=E. P.|last2=Gross|first3=M.|last3=Krook|bibcode=1954PhRv...94..511B}}</ref> 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:
求解波尔兹曼方程时，许多挑战都来自于其复杂的碰撞项；因此我们会做一些对碰撞项“建模”和简化的尝试。现知最好的模型是由Bhatnagar，Gross和Krook作出的（BGK近似）<ref><cite class="citation journal">Bhatnagar, P. L.; Gross, E. P.; Krook, M. (1954-05-01). </cite></ref>。BGK近似中假设分子的碰撞会迫使一个物理空间中的某一点的非平衡分布函数回到麦克斯韦平衡分布函数，且其发生率正比于分子碰撞频率。于是，波尔兹曼方程可被写作以下的BGK形式：
:<math>\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>\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>
where <math>\nu</math> is the molecular collision frequency, and <math>f_0</math> is the local Maxwellian distribution function given the gas temperature at this point in space.
where <math>\nu</math> is the molecular collision frequency, and <math>f_0</math> is the local Maxwellian distribution function given the gas temperature at this point in space.
其中 <math>\nu</math> 是分子碰撞频率，和[[驰豫时间]] <math>\tau</math> 具有倒数关系：<math>\nu = 1/\tau</math>。<math>f_0</math>是此处局域的麦克斯韦分布函数，由空间中这一点的气体温度给定。
==通用方程（对于混合物） ==
==通用方程（对于混合物） ==
For a mixture of chemical species labelled by indices ''i'' = 1, 2, 3, ..., ''n'' the equation for species ''i'' is<ref name="Encyclopaediaof" />
For a mixture of chemical species labelled by indices ''i'' = 1, 2, 3, ..., ''n'' the equation for species ''i'' is<ref name="Encyclopaediaof" />
对于具有多种化学组分的[[混合物]]，我们以 i =1，2，3，……，n 标记各种成分。则对于组分i的方程是：<ref name="Encyclopaediaof"/>
:<math>\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>\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>
where ''f<sub>i</sub>'' = ''f<sub>i</sub>''('''r''', '''p'''<sub>''i''</sub>, ''t''), and the collision term is
where ''f<sub>i</sub>'' = ''f<sub>i</sub>''('''r''', '''p'''<sub>''i''</sub>, ''t''), and the collision term is
其中 <math>f_i = f_i(\mathbf{r}, \mathbf{p_i}, t)</math>。碰撞项为
:<math>
:<math>
where ''f′'' = ''f′''('''p′'''<sub>''i''</sub>, ''t''), the magnitude of the relative momenta is
where ''f′'' = ''f′''('''p′'''<sub>''i''</sub>, ''t''), the magnitude of the relative momenta is
其中 <math>f' = f'(\mathbf{p_i'}, t)</math>，相对动量的大小是
:<math>g_{ij} = |\mathbf{p}_i - \mathbf{p}_j| = |\mathbf{p'}_i - \mathbf{p'}_j|,</math>
:<math>g_{ij} = |\mathbf{p}_i - \mathbf{p}_j| = |\mathbf{p'}_i - \mathbf{p'}_j|,</math>
and ''I<sub>ij</sub>'' 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<sub>ij</sub>'' 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<sub>ij</sub> 是粒子i和粒子j之间的微分散射截面。此积分的和描述的是某一相空间元中，组分i粒子的进出。
==应用与推广==
==应用与推广==
===Conservation equations===
===Conservation equations 守恒方程===
The Boltzmann equation can be used to derive the fluid dynamic conservation laws for mass, charge, momentum, and energy.<ref name="dG1984">{{cite book|last1=de Groot|first1=S. R.|last2=Mazur|first2=P.|title=Non-Equilibrium Thermodynamics|year=1984|publisher=Dover Publications Inc.|location=New York|isbn=978-0-486-64741-8}}</ref>{{rp|p 163}} 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.<ref name="dG1984">{{cite book|last1=de Groot|first1=S. R.|last2=Mazur|first2=P.|title=Non-Equilibrium Thermodynamics|year=1984|publisher=Dover Publications Inc.|location=New York|isbn=978-0-486-64741-8}}</ref>{{rp|p 163}} For a fluid consisting of only one kind of particle, the number density ''n'' is given by
玻尔兹曼方程可用于推导流体动力学中的质量守恒，电量守恒，动量守恒，以及能量守恒定律<ref name="dG1984">{{cite book |last1=de Groot |first1=S.R. |last2=Mazur |first2=P. |title=Non-Equilibrium Thermodynamics |url=http://www.amazon.com/Non-Equilibrium-Thermodynamics-Dover-Books-Physics/dp/0486647412 |accessdate=2013-01-31 |year=1984 |publisher=Dover Publications Inc. |location=New York |isbn=0-486-64741-2 |archive-date=2013-03-28 |archive-url=https://web.archive.org/web/20130328114445/http://www.amazon.com/Non-Equilibrium-Thermodynamics-Dover-Books-Physics/dp/0486647412 |dead-url=no }}</ref>{{rp|p 163}}。对于只含有一种粒子的流体，粒子数密度 <math>n</math> 为：
:<math>n = \int f \,d^3p.</math>
:<math>n = \int f \,d^3p.</math>
The average value of any function ''A'' is
The average value of any function ''A'' is
算符 A 的期望值由下式给出：
:<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>\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
:<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>
:<math>\int A \left(\frac{\partial f}{\partial t}\right)_\text{coll} \,d^3p = 0,</math>
:<math>\int A \left(\frac{\partial f}{\partial t}\right)_\text{coll} \,d^3p = 0,</math>
where the last term is zero, since ''A'' is conserved in a collision. Letting <math>A = m</math>, the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation:<ref name="dG1984" />{{rp|pp 12,168}}
where the last term is zero, since ''A'' is conserved in a collision.
因为 <math>g</math> 在碰撞中守恒，所以最后一项为零。
Letting <math>A = m</math>, the mass of the particle, the integrated Boltzmann equation becomes the conservation of mass equation:<ref name="dG1984" />{{rp|pp 12,168}}
令 <math>A=m</math>，即粒子质量，积分后的玻尔兹曼方程化为质量守恒方程<ref name="dG1984" />{{rp|pp 12,168}}：
:<math>\frac{\partial}{\partial t}\rho + \frac{\partial}{\partial x_j}(\rho V_j) = 0,</math>
:<math>\frac{\partial}{\partial t}\rho + \frac{\partial}{\partial x_j}(\rho V_j) = 0,</math>
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.
===Hamiltonian mechanics ===
===Hamiltonian mechanics 哈密顿力学===
In [[wikipedia:Hamiltonian_mechanics|Hamiltonian mechanics]], the Boltzmann equation is often written more generally as
In [[wikipedia:Hamiltonian_mechanics|Hamiltonian mechanics]], the Boltzmann equation is often written more generally as