Singular Behavior of the Macroscopic Quantity Near the Boundary for a Lorentz-Gas Model with the Infinite-Range Potential

TL;DR

This paper investigates how grazing collisions in a Lorentz-gas model with infinite-range potential affect the boundary behavior of macroscopic quantities, revealing divergence in their gradients despite regularization of the velocity distribution function.

Contribution

It demonstrates that grazing collisions regularize the velocity distribution function but cause divergence in the macroscopic quantity gradients near the boundary, a novel insight for infinite-range potentials.

Findings

Grazing collisions suppress boundary discontinuities in the VDF.

The collision integral becomes infinite in the parallel velocity direction.

The gradient of the macroscopic quantity diverges more strongly than with finite-range potentials.

Abstract

Possibility of the diverging gradient of the macroscopic quantity near the boundary is investigated by a mono-speed Lorentz-gas model, with a special attention to the regularizing effect of the grazing collision for the infinite-range potential on the velocity distribution function (VDF) and its influence on the macroscopic quantity. By careful numerical analyses of the steady one-dimensional boundary-value problem, it is confirmed that the grazing collision suppresses the occurrence of a jump discontinuity of the VDF on the boundary. However, as the price for that regularization, the collision integral becomes no longer finite in the direction of the molecular velocity parallel to the boundary. Consequently, the gradient of the macroscopic quantity diverges, even stronger than the case of the finite-range potential. A conjecture about the diverging rate in approaching the boundary is made as well for a wide range of the infinite-range potentials, accompanied by the numerical evidence.