论文标题
从线性化的玻尔兹曼方程到兰道方程的放牧碰撞限制了短距离电势
The grazing collisions limit from the linearized Boltzmann equation to the Landau equation for short-range potentials
论文作者
论文摘要
Landau方程式和Boltzmann方程是通过放牧碰撞的极限连接的。对于某些专注于放牧碰撞的玻尔兹曼经营者家属而言，这已被证明是严格的。在此贡献中，我们研究了三个维度的有限范围电位$φ（x）$与两粒子散射相关的碰撞内核。然后，我们考虑$φ_ε（x）=εφ（x）$给出的弱相互作用的极限。这里$ε\ rightarrow 0 $是放牧参数，碰撞率被重新定制以获得非平凡的限制。放牧碰撞限制特别感兴趣，对于原始订单$ s \ geq 0 $的潜在，即$ ϕ（x）\ sim | x | x |^{ - s} $ as $ | x | x | \ rightarrow 0 $。对于[0,1] $中的$ s \，我们证明了与Born近似值给出的扩散系数与Landau方程式的收敛，如Landau和Balescu的作品所预测的那样。另一方面，对于具有$ s> 1 $的电势，我们获得了极限的非切割玻尔兹曼方程。库仑奇异性$ s = 1 $作为阈值值以对数校正对扩散时间尺度的对数校正，即所谓的库仑对数。
The Landau equation and the Boltzmann equation are connected through the limit of grazing collisions. This has been proved rigorously for certain families of Boltzmann operators concentrating on grazing collisions. In this contribution, we study the collision kernels associated to the two-particle scattering via a finite range potential $Φ(x)$ in three dimensions. We then consider the limit of weak interaction given by $Φ_ε(x) = εΦ(x)$. Here $ε\rightarrow 0$ is the grazing parameter, and the rate of collisions is rescaled to obtain a non-trivial limit. The grazing collisions limit is of particular interest for potentials with a singularity of order $s\geq 0$ at the origin, i.e. $ϕ(x) \sim |x|^{-s}$ as $|x|\rightarrow 0$. For $s\in [0,1]$, we prove the convergence to the Landau equation with diffusion coefficient given by the Born approximation, as predicted in the works of Landau and Balescu. On the other hand, for potentials with $s>1$ we obtain the non-cutoff Boltzmann equation in the limit. The Coulomb singularity $s=1$ appears as a threshold value with a logarithmic correction to the diffusive timescale, the so-called Coulomb logarithm.
