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第1行： − + − − {{other uses|Boltzmann's entropy formula|Stefan–Boltzmann law|Maxwell–Boltzmann distribution}} − − {{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> 第10行：
第6行： − 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. − − '''玻尔兹曼方程'''或'''玻尔兹曼输运方程(Boltzmann transport equation, BTE)'''是一个描述非热力学平衡状态的热力学系统统计行为的偏微分方程，由'''[[路德维希·玻尔兹曼 Ludwig Edward Boltzmann|路德维希·玻尔兹曼 Ludwig Boltzmann]]'''于1872年提出。 − − The '''Boltzmann equation''' or '''Boltzmann transport equation''' ('''BTE''') describes the statistical behaviour of a [[thermodynamic system]] not in a state of [[Thermodynamic equilibrium|equilibrium]], devised by [[Ludwig Boltzmann]] in 1872.<ref name="Encyclopaediaof">Encyclopaedia of Physics (2nd Edition), R. G. Lerner, G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3.</ref> − − The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number. − − 这种系统的典型例子是具有空间温度梯度的流体，通过流体中粒子的随机但有偏的运动，使得热量从较热的区域流向较冷的区域。在现代文献中，玻尔兹曼方程这个术语通常用于更一般的情形，指的是任何描述热力学系统中宏观量（例如能量、电荷或粒子数）变化的动力学方程。 − − The classic example of such a system is a [[fluid]] with [[temperature gradient]]s in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the [[particle]]s making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number. + + 第88行：
第75行： − + − 《数学》+ − + − + − + − \begin{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:+
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此词条由栗子CUGB翻译整理。
此词条由栗子CUGB翻译整理。[[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)
{{{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]]
{{{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 [[Thermodynamic equilibrium|equilibrium]], devised by [[Ludwig Boltzmann]] in 1872.<ref name="Encyclopaediaof">Encyclopaedia of Physics (2nd Edition), R. G. Lerner, G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3.</ref> The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number.
'''玻尔兹曼方程'''或'''玻尔兹曼输运方程(Boltzmann transport equation, BTE)'''是一个描述非热力学平衡状态的热力学系统统计行为的偏微分方程，由'''[[路德维希·玻尔兹曼 Ludwig Edward Boltzmann|路德维希·玻尔兹曼 Ludwig Boltzmann]]'''于1872年提出。<ref name="Encyclopaediaof" /> 这类系统的经典实例是：在空间中具有温度梯度的流体，组成该流体的粒子通过随机但具有偏向性的传输使得热量从较热的区域流向较冷的区域。在现代文献中，玻尔兹曼方程一词通常用于更一般的意义上，指的是描述热力学系统中宏观量变化的任何动力学方程，如能量、电荷或粒子数。
The equation arises not by analyzing the individual positions and momenta of each particle in the fluid but rather by considering a probability distribution for the position and momentum of a typical particle—that is, the probability that the particle occupies a given very small region of space (mathematically the volume element <math>\mathrm{d}^3 \bf{r}</math>) centered at the position <math>\bf{r}</math>, and has momentum nearly equal to a given momentum vector <math> \bf{p}</math> (thus occupying a very small region of momentum space <math>\mathrm{d}^3 \bf{p}</math>), at an instant of time.
The equation arises not by analyzing the individual positions and momenta of each particle in the fluid but rather by considering a probability distribution for the position and momentum of a typical particle—that is, the probability that the particle occupies a given very small region of space (mathematically the volume element <math>\mathrm{d}^3 \bf{r}</math>) centered at the position <math>\bf{r}</math>, and has momentum nearly equal to a given momentum vector <math> \bf{p}</math> (thus occupying a very small region of momentum space <math>\mathrm{d}^3 \bf{p}</math>), at an instant of time.
<math>
<math>
《数学》
\begin{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:
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]