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第83行： − − − − 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 − − f \left (\mathbf{r}+\frac{\mathbf{p}}{m} \, \Delta t,\mathbf{p}+\mathbf{F} \, \Delta t, t+\Delta t \right )\,d^3\mathbf{r}\,d^3\mathbf{p} = f(\mathbf{r}, \mathbf{p},t) \, d^3\mathbf{r} \, d^3\mathbf{p} − − </math> − − <math>\int A \frac{\partial f}{\partial t} \,d^3p = \frac{\partial }{\partial t} (n \langle A \rangle),</math> − − (n langle a rangle) ，</math > − − <math>\int \frac{p_j A}{m}\frac{\partial f}{\partial x_j} \,d^3p = \frac{1}{m}\frac{\partial}{\partial x_j}(n\langle A p_j \rangle),</math> − − <math> {m}\Delta t,\mathbf{p} + \mathbf{F}\Delta t, t+\Delta t \right)d^3\mathbf{r}d^3\mathbf{p} - f(\mathbf{r}, \mathbf{p}, t) \, d^3\mathbf{r} \, d^3\mathbf{p} \\[5pt] & = \Delta f \, d^3\mathbf{r} \, d^3\mathbf{p}</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: including the formation of the light elements in Big Bang nucleosynthesis, the production of dark matter and baryogenesis. It is not a priori clear that the state of a quantum system can be characterized by a classical phase space density f. However, for a wide class of applications a well-defined generalization of f exists which is the solution of an effective Boltzmann equation that can be derived from first principles of quantum field theory. − − 最后一项是零，因为 a 在碰撞中守恒。让粒子的质量，积分玻尔兹曼方程成为质量守恒方程: 包括太初核合成中轻元素的形成，暗物质的产生和重子形成。量子系统的状态是否可以用经典的相空间密度 f 来表示，这一点先验上并不清楚。然而，对于广泛的应用来说，f 的一个定义明确的推广是存在的，它是一个有效的拥有属性玻尔兹曼方程的解，可以从量子场论的第一原理中推导出来。 − − \end{align}</math> − − |{{EquationRef|1}}}} − − − − The Boltzmann equation is of use in galactic dynamics. A galaxy, under certain assumptions, may be approximated as a continuous fluid; its mass distribution is then represented by f; in galaxies, physical collisions between the stars are very rare, and the effect of gravitational collisions can be neglected for times far longer than the age of the universe. − − 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。 − − − − {{NumBlk|:| − + − − − − − + − − 第196行：
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→The force and diffusion terms “force”项与“diff”项
Suppose at time ''t'' some number of particles all have position '''r''' within element <math> d^3\bf{r}</math> and momentum '''p''' within <math> d^3\bf{p}</math>. If a force '''F''' instantly acts on each particle, then at time ''t'' + Δ''t'' their position will be '''r''' + Δ'''r''' = '''r''' + '''p'''Δ''t''/''m'' and momentum '''p''' + Δ'''p''' = '''p''' + '''F'''Δ''t''. Then, in the absence of collisions, ''f'' must satisfy
Suppose at time ''t'' some number of particles all have position '''r''' within element <math> d^3\bf{r}</math> and momentum '''p''' within <math> d^3\bf{p}</math>. If a force '''F''' instantly acts on each particle, then at time ''t'' + Δ''t'' their position will be '''r''' + Δ'''r''' = '''r''' + '''p'''Δ''t''/''m'' and momentum '''p''' + Δ'''p''' = '''p''' + '''F'''Δ''t''. Then, in the absence of collisions, ''f'' must satisfy
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
Its generalization in general relativity. is
Its generalization in general relativity. is
它在广义相对论的推广。是
它在广义相对论的推广。是{{NumBlk|2=<math>\frac{d f}{d t} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>|3={{EquationRef|2}}}}
<math>\frac{d f}{d t} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>
|{{EquationRef|2}}}}
<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>
<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>
{ mathbf { l } mathrm { GR } = p ^ alpha frac { partial }{ x ^ alpha }-Gamma ^ alpha {}{ beta Gamma } p ^ p ^ frac { partial }{ partial p ^ alpha } ，</math >
{ mathbf { l } mathrm { GR } = p ^ alpha frac { partial }{ x ^ alpha }-Gamma ^ alpha {}{ beta Gamma } p ^ p ^ frac { partial }{ partial p ^ alpha } ，<nowiki></math ></nowiki>
The total [[differential of a function|differential]] of ''f'' is:
The total [[differential of a function|differential]] of ''f'' is:
where ∇ is the [[gradient]] operator, '''·''' is the [[dot product]],
where ∇ is the [[gradient]] operator, '''·''' is the [[dot product]],
is a shorthand for the momentum analogue of ∇, and '''ê'''<sub>''x''</sub>, '''ê'''<sub>''y''</sub>, '''ê'''<sub>''z''</sub> are [[cartesian coordinates|Cartesian]] [[unit vector]]s.
is a shorthand for the momentum analogue of ∇, and '''ê'''<sub>''x''</sub>, '''ê'''<sub>''y''</sub>, '''ê'''<sub>''z''</sub> are [[cartesian coordinates|Cartesian]] [[unit vector]]s.
1 = Harris
1 = Harris
| first1= Stewart
| first1= Stewart
1-link =
1-link =
| title= An introduction to the theory of the Boltzmann equation | publisher=Dover Books|pages=221 | year= 1971 | isbn=978-0-486-43831-3|url=https://books.google.com/books?id=KfYK1lyq3VYC}}. Very inexpensive introduction to the modern framework (starting from a formal deduction from Liouville and the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy (BBGKY) in which the Boltzmann equation is placed). Most statistical mechanics textbooks like Huang still treat the topic using Boltzmann's original arguments. To derive the equation, these books use a heuristic explanation that does not bring out the range of validity and the characteristic assumptions that distinguish Boltzmann's from other transport equations like Fokker–Planck or Landau equations.
| title= An introduction to the theory of the Boltzmann equation | publisher=Dover Books|pages=221 | year= 1971 | isbn=978-0-486-43831-3|url=https://books.google.com/books?id=KfYK1lyq3VYC}}. Very inexpensive introduction to the modern framework (starting from a formal deduction from Liouville and the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy (BBGKY) in which the Boltzmann equation is placed). Most statistical mechanics textbooks like Huang still treat the topic using Boltzmann's original arguments. To derive the equation, these books use a heuristic explanation that does not bring out the range of validity and the characteristic assumptions that distinguish Boltzmann's from other transport equations like Fokker–Planck or Landau equations.
This equation is more useful than the principal one above, yet still incomplete, since ''f'' cannot be solved unless the collision term in ''f'' is known. This term cannot be found as easily or generally as the others – it is a statistical term representing the particle collisions, and requires knowledge of the statistics the particles obey, like the [[Maxwell–Boltzmann distribution|Maxwell–Boltzmann]], [[Fermi–Dirac distribution|Fermi–Dirac]] or [[Bose–Einstein distribution|Bose–Einstein]] distributions.
This equation is more useful than the principal one above, yet still incomplete, since ''f'' cannot be solved unless the collision term in ''f'' is known. This term cannot be found as easily or generally as the others – it is a statistical term representing the particle collisions, and requires knowledge of the statistics the particles obey, like the [[Maxwell–Boltzmann distribution|Maxwell–Boltzmann]], [[Fermi–Dirac distribution|Fermi–Dirac]] or [[Bose–Einstein distribution|Bose–Einstein]] distributions.
| last1= Arkeryd
| last1= Arkeryd
| last1= Arkeryd
| last1= Arkeryd
1 = Leif
1 = Leif
|author1-link= Leif Arkeryd
|author1-link= Leif Arkeryd
<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
<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
<math>\int A \frac{\partial f}{\partial t} \,d^3p = \frac{\partial }{\partial t} (n \langle A \rangle),</math>
<math>\int \frac{p_j A}{m}\frac{\partial f}{\partial x_j} \,d^3p = \frac{1}{m}\frac{\partial}{\partial x_j}(n\langle A p_j \rangle),</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:
最后一项是零，因为 a 在碰撞中守恒。让粒子的质量，积分玻尔兹曼方程成为质量守恒方程:
量子理论和粒子数守恒的违背
including the formation of the light elements in Big Bang nucleosynthesis, the production of dark matter and baryogenesis. It is not a priori clear that the state of a quantum system can be characterized by a classical phase space density f. However, for a wide class of applications a well-defined generalization of f exists which is the solution of an effective Boltzmann equation that can be derived from first principles of quantum field theory.
包括太初核合成中轻元素的形成，暗物质的产生和重子形成。量子系统的状态是否可以用经典的相空间密度 f 来表示，这一点先验上并不清楚。然而，对于广泛的应用来说，f 的一个定义明确的推广是存在的，它是一个有效的拥有属性玻尔兹曼方程的解，可以从量子场论的第一原理中推导出来。
广义相对论与天文学
The Boltzmann equation is of use in galactic dynamics. A galaxy, under certain assumptions, may be approximated as a continuous fluid; its mass distribution is then represented by f; in galaxies, physical collisions between the stars are very rare, and the effect of gravitational collisions can be neglected for times far longer than the age of the universe.
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。
== 方程求解 ==
== 方程求解 ==