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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.

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。

'''玻尔兹曼方程'''或'''玻尔兹曼输运方程(Boltzmann transport equation, BTE)'''是描述非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，指描述热力学系统中宏观量（如能量、电荷或粒子数变化）的任何动力学方程。

The equation arises not by analyzing the individual [[position vector|position]]s and [[momentum|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 [[infinitesimal|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 [[Index.php?title=Positions vector|positions]] and [[momentum|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 [[infinitesimal|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.

玻尔兹曼方程并不分析流体中每个粒子的单个位置和动量，而是只考虑一类粒子的位置和动量的概率分布，此类粒子某一时刻在几何空间占据以给定位置<math>\bf{r}</math>为中心的小邻域（数学上的体积元<math>\mathrm{d}^3 \bf{r}</math>），且其动量几乎与给定动量矢量<math> \bf{p}</math>相等，在动量空间占据非常小的区域<math>\mathrm{d}^3 \bf{p}</math>。

玻尔兹曼方程并不分析流体中每个粒子的位置和动量，而是考虑特定粒子的位置和动量的概率分布，此类粒子某一时刻在几何空间占据以给定位置<math>\bf{r}</math>为中心的小邻域（数学上的体积元<math>\mathrm{d}^3 \bf{r}</math>），且其动量几乎与给定动量矢量<math> \bf{p}</math>相等，在动量空间占据非常小的区域<math>\mathrm{d}^3 \bf{p}</math>。

The Boltzmann equation can be used to determine how physical quantities change, such as [[heat]] energy and [[momentum]], when a fluid is in transport. One may also derive other properties characteristic to fluids such as [[viscosity]], [[thermal conductivity]], and [[electrical conductivity]] (by treating the charge carriers in a material as a gas).<ref name="Encyclopaediaof" /> See also [[convection–diffusion equation]].

The Boltzmann equation can be used to determine how physical quantities change, such as [[heat]] energy and [[momentum]], when a fluid is in transport. One may also derive other properties characteristic to fluids such as [[viscosity]], [[thermal conductivity]], and [[electrical conductivity]] (by treating the charge carriers in a material as a gas).<ref name="Encyclopaediaof" /> See also [[convection–diffusion equation]].