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

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.

Appendices

Appendix A Basis Functions

For the sake of the numerical convenience, the grid points in $\theta$-space are arranged to be symmetric with respect to $\theta =0$ in the region $-\pi /2\le \theta \le \pi /2$ so as to make 2N small intervals in both the positive and negative side:

$$\begin{aligned} 0&=\theta ^{(0)}<\theta ^{(1)}<\cdots<\theta ^{(2N-1)}<\theta ^{(2N)}=\pi /2,\quad \theta ^{(-j)}=-\theta ^{(j)},\quad (j=1,\dots ,2N). \end{aligned}$$

The size of the intervals is not uniform and is smaller near $\theta =0$ so that many grid points are around there. Then the following basis function set $\{Y_{i}(\theta )\}$ ($i=-2N,\dots ,2N$) is used for the piecewise quadratic approximation of a function of $\theta$:

$$\begin{aligned} Y_{2\ell }(\theta )= & {} {\left\{ \begin{array}{ll} {\displaystyle \frac{(\theta -\theta ^{(2\ell +1)})(\theta -\theta ^{(2\ell +2)})}{(\theta ^{(2\ell )}-\theta ^{(2\ell +1)})(\theta ^{(2\ell )}-\theta ^{(2\ell +2)})}}, &{} \theta ^{(2\ell )}<\theta<\theta ^{(2\ell +2)},\ -N\le \ell<N,\\ {\displaystyle \frac{(\theta -\theta ^{(2\ell -1)})(\theta -\theta ^{(2\ell -2)})}{(\theta ^{(2\ell )}-\theta ^{(2\ell -1)})(\theta ^{(2\ell )}-\theta ^{(2\ell -2)})}}, &{} \theta ^{(2\ell -2)}<\theta<\theta ^{(2\ell )},\ -N<\ell \le N,\\ 0, &{} \text{ otherwise }, \end{array}\right. } \\ Y_{2\ell +1}(\theta )= & {} {\left\{ \begin{array}{ll} {\displaystyle \frac{(\theta -\theta ^{(2\ell )})(\theta -\theta ^{(2\ell +2)})}{(\theta ^{(2\ell +1)}-\theta ^{(2\ell )})(\theta ^{(2\ell +1)}-\theta ^{(2\ell +2)})}}, &{} \theta ^{(2\ell )}<\theta<\theta ^{(2\ell +2)},\ -N\le \ell <N,\\ 0, &{} \text{ otherwise }. \end{array}\right. } \end{aligned}$$

By definition, $Y_{j}(\theta )=Y_{-j}(-\theta )$ and that $Y_{0}(\theta )$ is even in $\theta$.

In the direct method, $Y_{\pm 0}(\theta )=Y_{0}(\theta )H(\pm \theta )$ is also prepared to express the jump discontinuity of g at $\theta =0$, where $H(\theta )$ is the Heaviside function. Using the notation $g_{\pm 0}(x_{1})=g(x_{1},\theta =\pm 0)$, the g having a jump discontinuity at $\theta =0$ is approximated by

$$\begin{aligned} g(x_{1},\theta )=\sum _{i=1}^{2N}\{g_{-i}(x_{1})Y_{-i}(\theta )+g_{i}(x_{1})Y_{i}(\theta )\}+g_{-0}(x_{1})Y_{-0}(\theta )+g_{+0}(x_{1})Y_{+0}(\theta ).\nonumber \\ \end{aligned}$$

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If there is no jump discontinuity, g is simply approximated by $g=\sum _{i=-2N}^{2N}g_{i}Y_{i}(\theta )$ with the simplified notation $g_{0}(x_{1})\equiv g_{\pm 0}(x_{1})$. Accordingly, the numerical kernel used in the direct method takes the form $C^{+}[g]=\sum _{i=1}^{2N}\{g_{-i}C^{+}[Y_{-i}]+g_{i}C^{+}[Y_{i}]\}+g_{-0}C^{+}[Y_{-0}]+g_{+0}C^{+}[Y_{+0}]$ or $C^{+}[g]=\sum _{i=-2N}^{2N}g_{i}C^{+}[Y_{i}]$, depending on whether the jump discontinuity exists or not.

The analytical expression of $C^{+}[Y_{i}]$ is available with the aid of the series expansion of $|\sin \frac{\varphi }{2}|^{\gamma }$. Although it is truncated by a finite number of terms, the expression is helpful to perform the accurate numerical computation. The same applies to the Galerkin method, i.e., both $A_{ij}$ and $D_{ij}$ can be obtained analytically as well even for the infinite-range potential. The highly accurate computations with the multiple precision arithmetic are achieved in this way.

Appendix B Acceleration Method for Estimating the Asymptotic Behavior

In the present study, an acceleration method proposed in Ref. [26] that makes use of the Richardson extrapolation is found to be very powerful in estimating the asymptotic behavior of the density in approaching the boundary. The method is briefly explained in this appendix.

Suppose that a function f of $x(\ge X)$ behaves

$$\begin{aligned} f(x)\sim f(X)+a_{\alpha }s^{\alpha }+a_{1}s+o(s), \end{aligned}$$

(36)

for $x\sim X$, where $s=x-X$ and $0<\alpha <1$ is an unknown constant. In the application to the present work, put $X=-1/2$. The idea of the method is composed of killing the third term to clearly pick up the second term on the right-hand side, thereby improving the estimate of the exponent $\alpha$ by the linear regression on the log-log plot.

The straightforward estimate (SE) for the exponent $\alpha$ is just to take

$$\begin{aligned} \mathrm {SE}[f]\equiv f(x)-f(X)\sim a_{\alpha }s^{\alpha }+a_{1}s+o(s), \end{aligned}$$

(37)

and to use the linear regression. As is clear from the most-right-hand side, however, the O(s) term may affect the linear regression unless a clear difference of scale appears in the data at hands. In Ref. [26], the following combination of f that makes use of the Richardson extrapolation is proposed by Koike (the KR method, for short):

$$\begin{aligned} \mathrm {KR}[f]\equiv f(x)-2f(X+s/2)+f(X). \end{aligned}$$

(38)

Then, it behaves

$$\begin{aligned} \mathrm {KR}[f]\sim a_{\alpha }(1-2^{1-\alpha })s^{\alpha }+o(s), \end{aligned}$$

and accordingly there is no longer influence of the term O(s) in the linear regression. Hence, the estimate of $\alpha$ should be improved.

Practically, there is a possible drawback such that $\mathrm {KR}[f]$ would require more significant digits than $\mathrm {SE}[f]$ in order to avoid the influence of the round-off error. Indeed, in Figs. 2c and 3b, the influence can be observed in the results by the direct method but not in the results by the Galerkin method. The difference comes from that the computation code for the former uses the double precision arithmetic, while that for the latter uses the multiple precision arithmetic and does not make a discretization in $x_{1}$.