Ohr Young Gie
Paichai University, Daejeon 35345, Republic of Korea.
Phys Rev E. 2023 Sep;108(3-2):035301. doi: 10.1103/PhysRevE.108.035301.
The total cross section of binary collision is, in general, unbounded due to the long-range interations of molecules. It is conventional to truncate the small angle deflections of collisions. The present work suggests an alternative way of avoiding the difficulty of unboundedness. We employ the mean value theorem of definite integral over the deflection angle for the cross section. A series of numerical experiments were carried out to look for the representative collision cross section through which the single-angle simulation is amenable to the solution of the Boltzmann equation. Results show that the cross section should be 〈Σ〉=Σ_{D}^{2}/(2Σ_{D}-Σ_{μ}), and the representative deflection for the single-angle simulation be cos〈χ〉=Σ_{μ}/Σ_{D}-1, where Σ_{D} is the diffusion cross section and Σ_{μ} is the viscosity cross section. The single-angle computations for the inverse power law and the Lennard-Jones force law perfectly reproduce the conventional scattering algorithms for one-dimensional (1D) simulations of transport coefficients and 1D shock thickness. The computation costs for Lennard-Jones molecules are comparable to the costs for inverse power-law models.
由于分子间的长程相互作用，二元碰撞的总截面通常是无界的。截断碰撞的小角度偏转是一种常规做法。本文提出了一种避免无界性困难的替代方法。我们利用截面偏转角的定积分中值定理。进行了一系列数值实验，以寻找适用于玻尔兹曼方程解的单角度模拟的代表性碰撞截面。结果表明，该截面应为〈Σ〉=Σ_D^2/(2Σ_D - Σ_μ)，单角度模拟的代表性偏转为cos〈χ〉=Σ_μ/Σ_D - 1，其中Σ_D是扩散截面，Σ_μ是粘性截面。对于逆幂律和 Lennard-Jones 力律的单角度计算完美地再现了用于一维（1D）输运系数模拟和 1D 激波厚度的传统散射算法。Lennard-Jones 分子的计算成本与逆幂律模型的成本相当。
