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{{other uses|Boltzmann's entropy formula|Stefan–Boltzmann law|Maxwell–Boltzmann distribution}}

{{other uses|Boltzmann's entropy formula|Stefan–Boltzmann law|Maxwell–Boltzmann distribution}}

{{redirect|BTE}}

{{redirect|BTE}}[[File:StairsOfReduction.svg|thumb|The place of the Boltzmann kinetic equation on the stairs of model reduction from microscopic dynamics to macroscopic continuum dynamics (illustration to the content of the book<ref>

 

 

 

[[File:StairsOfReduction.svg|thumb|The place of the Boltzmann kinetic equation on the stairs of model reduction from microscopic dynamics to macroscopic continuum dynamics (illustration to the content of the book<ref>

 

The place of the Boltzmann kinetic equation on the stairs of model reduction from microscopic dynamics to macroscopic continuum dynamics (illustration to the content of the book)

The place of the Boltzmann kinetic equation on the stairs of model reduction from microscopic dynamics to macroscopic continuum dynamics (illustration to the content of the book)

玻耳兹曼动力学方程在从微观动力学到宏观连续动力学的模型简化阶梯上的位置(本书内容的说明)

玻耳兹曼动力学方程在从微观动力学到宏观连续动力学的模型简化阶梯上的位置(本书内容的说明)

{{{cite book |last1=Gorban |first1= Alexander N.|last2= Karlin |first2= Ilya V. |date=2005 |title= Invariant Manifolds for Physical and Chemical Kinetics|url= https://www.academia.edu/17378865|url-access=| location= Berlin, Heidelberg |publisher= Springer|series= Lecture Notes in Physics (LNP, vol. 660)| isbn= 978-3-540-22684-0|doi= 10.1007/b98103|via= |quote=}} [https://archive.org/details/gorban-karlin-lnp-2005 Alt URL]</ref>)]]

{{{cite book |last1=Gorban |first1= Alexander N.|last2= Karlin |first2= Ilya V. |date=2005 |title= Invariant Manifolds for Physical and Chemical Kinetics|url= https://www.academia.edu/17378865|url-access=| location= Berlin, Heidelberg |publisher= Springer|series= Lecture Notes in Physics (LNP, vol. 660)| isbn= 978-3-540-22684-0|doi= 10.1007/b98103|via= |quote=}} [https://archive.org/details/gorban-karlin-lnp-2005 Alt URL]</ref>)|链接=Special:FilePath/StairsOfReduction.svg]]

 

 

The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.

The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.

  <math>

  <math>

 

《数学》

《数学》

 

 

 

\begin{align}

\begin{align}

 

开始{ align }

开始{ align }

 

is the number of molecules which ''all'' have positions lying within a volume element <math> d^3\bf{r}</math> about '''r''' and momenta lying within a [[momentum space]] element <math> \mathrm{d}^3\bf{p}</math> about '''p''', at time ''t''.<ref>{{Cite book |last=Huang |first=Kerson |year=1987 |title=Statistical Mechanics |url=https://archive.org/details/statisticalmecha00huan_475 |url-access=limited |location=New York |publisher=Wiley |isbn=978-0-471-81518-1 |page=[https://archive.org/details/statisticalmecha00huan_475/page/n65 53] |edition=Second }}</ref> [[Integration (calculus)|Integrating]] over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:

is the number of molecules which ''all'' have positions lying within a volume element <math> d^3\bf{r}</math> about '''r''' and momenta lying within a [[momentum space]] element <math> \mathrm{d}^3\bf{p}</math> about '''p''', at time ''t''.<ref>{{Cite book |last=Huang |first=Kerson |year=1987 |title=Statistical Mechanics |url=https://archive.org/details/statisticalmecha00huan_475 |url-access=limited |location=New York |publisher=Wiley |isbn=978-0-471-81518-1 |page=[https://archive.org/details/statisticalmecha00huan_475/page/n65 53] |edition=Second }}</ref> [[Integration (calculus)|Integrating]] over a region of position space and momentum space gives the total number of particles which have positions and momenta in that region:

N & = \int\limits_\mathrm{momenta} \text{d}^3\mathbf{p} \int\limits_\mathrm{positions} \text{d}^3\mathbf{r}\,f (\mathbf{r},\mathbf{p},t) \\[5pt]

N & = \int\limits_\mathrm{momenta} \text{d}^3\mathbf{p} \int\limits_\mathrm{positions} \text{d}^3\mathbf{r}\,f (\mathbf{r},\mathbf{p},t) \\[5pt]

which is a 6-fold integral. While f is associated with a number of particles, the phase space is for one-particle (not all of them, which is usually the case with deterministic many-body systems), since only one r and p is in question. It is not part of the analysis to use r₁, p₁ for particle 1, r₂, p₂ for particle 2, etc. up to r_N, p_N for particle N.

which is a 6-fold integral. While f is associated with a number of particles, the phase space is for one-particle (not all of them, which is usually the case with deterministic many-body systems), since only one r and p is in question. It is not part of the analysis to use r₁, p₁ for particle 1, r₂, p₂ for particle 2, etc. up to r_N, p_N for particle N.

这是一个6重积分。虽然 f 与许多粒子相关联，但相空间是单粒子的(不是所有粒子，通常是确定性多体系统的情况) ，因为只有一个 r 和 p 存在问题。用 r < sub > 1 </sub > 、 p < sub > 1 </sub > 表示粒子1、 r < sub > 2 </sub > 、 p < sub > 2 </sub > 表示粒子2等不属于分析范围。粒子 n 可达 r < sub > n </sub > ，p < sub > n </sub > 。

这是一个6重积分。虽然 f 与许多粒子相关联，但相空间是单粒子的(不是所有粒子，通常是确定性多体系统的情况) ，因为只有一个 r 和 p 存在问题。用 r < sub > 1 、 p < sub > 1 表示粒子1、 r < sub > 2 、 p < sub > 2 表示粒子2等不属于分析范围。粒子 n 可达 r < sub > n ，p < sub > n 。

\end{align}

\end{align}

where Γ^α_βγ is the Christoffel symbol of the second kind (this assumes there are no external forces, so that particles move along geodesics in the absence of collisions), with the important subtlety that the density is a function in mixed contravariant-covariant (xⁱ, p_i) phase space as opposed to fully contravariant (xⁱ, pⁱ) phase space.

where Γ^α_βγ is the Christoffel symbol of the second kind (this assumes there are no external forces, so that particles move along geodesics in the absence of collisions), with the important subtlety that the density is a function in mixed contravariant-covariant (xⁱ, p_i) phase space as opposed to fully contravariant (xⁱ, pⁱ) phase space.

其中 γ < sup > α </sup > < βγ </sub > </sub > 是第二类 Christoffel 符号(假设没有外力，因此粒子在没有碰撞的情况下沿测地线运动) ，其中重要的微妙之处在于密度是混合逆变-协变(x < sup > i </sup > ，p < sub > i </sub >)相空间中的函数，而不是完全逆变(x < sup > i </sup > ，p </sup >)相空间中的函数。

其中 γ < sup > α < βγ   是第二类 Christoffel 符号(假设没有外力，因此粒子在没有碰撞的情况下沿测地线运动) ，其中重要的微妙之处在于密度是混合逆变-协变(x < sup > i ，p < sub > i )相空间中的函数，而不是完全逆变(x < sup > i ，p )相空间中的函数。