Asymptotic theory of the linear transport equation for small mean free paths. I
Asymptotic theory of the linear transport equation for small mean free paths. I
复制标题
小平均自由程线性输运方程的渐近理论。
DOI：
10.1103/physreva.13.1933
复制
发表时间：
1976
影响因子：
2.9
通讯作者：
Jose J. D'Arruda
中科院分区：
文献类型：
--
作者：
E. Larsen;Jose J. D'Arruda
我们考虑线性输运方程，它描述了稀薄气体与较稠密气体的相互作用。我们要求这个方程中所有量的空间变化在典型的平均自由程的距离上很小，并且在典型的平均自由时间内所有量的时间变化很小。我们还要求密度较大的气体的平均速度、外部电场和内部源的平均速度都很小。这些小问题是通过一个小参数$\ensureath{\epsilon}$来解析表示的。构造了满足上述约束条件的输运方程的解，该解关于$ensureath{epsilon}$渐近。对于前序，解具有形式$\ensuremath{\psi}\ensuremath{\sim}{A}_{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}，t)\ensuremath{\varphi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}，v，t)$，其中$是某个算子的本征函数，并且${A}{0}$满足扩散方程，该扩散方程包含由于较稠密的气体的运动、较稠密的气体中的不均匀、以及外部电场和磁场。我们讨论了这个方程的物理解释，并给出了稀薄粒子密度和平均速度的表达式。
We consider the linear transport equation, which describes the interaction of a rarefied gas with a denser gas. We require the spatial variation of all quantities in this equation to be small over the distance of a typical mean free path, and the time variation of all quantities to be small during a typical mean free time. We also require the average velocity of the denser gas, the external electric field, and the internal sources to be small. These smallnesses are expressed analytically by means of a small parameter $\ensuremath{\epsilon}$. A solution of the transport equation, subject to the above restrictions, is constructed which is asymptotic with respect to $\ensuremath{\epsilon}$. To leading order the solution has the form $\ensuremath{\psi}\ensuremath{\sim}{A}_{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)\ensuremath{\varphi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},v,t)$, where $\ensuremath{\varphi}$ is the eigenfunction of a certain operator and ${A}_{0}$ satisfies a diffusion equation containing effects due to the motion of the denser gas, inhomogeneities in the denser gas, and the external electric and magnetic fields. We discuss the physical interpretation of this equation and give expressions for the density and average velocity of the rarefied particles.