非弹性粗糙球颗粒气体碰撞矩的Sonine近似
Sonine approximation for collisional moments of granular gases of inelastic rough spheres
摘要：
我们考虑了一个稀薄的颗粒气体硬球碰撞非弹性系数的正常和切向恢复$α$和$β$，分别。表征线速度（$\mathbf{v}$）和角速度（$\bmω$）的分布函数$f(\mathbf{v},\bmω)$的基本量是定义平移温度（$T^\text{tr}$）和旋转温度（$T^\text{rot}$）的二阶矩。$f$与由$T^\text{tr}$和$T^\text{rot}$参数化的麦克斯韦分布的偏差可以通过与四阶速度矩相关的累积量来测量。本文的主要目的是评价这些第二和第四度的时刻通过Sonine近似的碰撞率的变化。结果随后被应用到计算的温度比$T^\text{rot}/T^\text{tr}$和两个范例状态的累积量：均匀冷却状态和均匀的稳态驱动的白噪声随机恒温器。在这两种情况下发现，麦克斯韦近似的温度比不偏离Sonine预测太多。另一方面，由累积量测量的非麦克斯韦性质不能被忽略，特别是在中等和小粗糙度的均匀冷却状态下。此外，在该状态下，与平移速度直接相关的累积量在准光滑极限$β\to -1$中不同于纯光滑球的准光滑极限（$β=-1$）。这种奇异行为与均匀冷却状态的不稳定特性直接相关，因此在随机恒温器情况下不存在。
英文摘要：
We consider a dilute granular gas of hard spheres colliding inelastically with coefficients of normal and tangential restitution $α$ and $β$, respectively. The basic quantities characterizing the distribution function $f(\mathbf{v},\bmω)$ of linear ($\mathbf{v}$) and angular ($\bmω$) velocities are the second-degree moments defining the translational ($T^\text{tr}$) and rotational ($T^\text{rot}$) temperatures. The deviation of $f$ from the Maxwellian distribution parameterized by $T^\text{tr}$ and $T^\text{rot}$ can be measured by the cumulants associated with the fourth-degree velocity moments. The main objective of this paper is the evaluation of the collisional rates of change of these second- and fourth-degree moments by means of a Sonine approximation. The results are subsequently applied to the computation of the temperature ratio $T^\text{rot}/T^\text{tr}$ and the cumulants of two paradigmatic states: the homogeneous cooling state and the homogeneous steady state driven by a white-noise stochastic thermostat. It is found in both cases that the Maxwellian approximation for the temperature ratio does not deviate much from the Sonine prediction. On the other hand, non-Maxwellian properties measured by the cumulants cannot be ignored, especially in the homogeneous cooling state for medium and small roughness. In that state, moreover, the cumulant directly related to the translational velocity differs in the quasi-smooth limit $β\to -1$ from that of pure smooth spheres ($β=-1$). This singular behavior is directly related to the unsteady character of the homogeneous cooling state and thus it is absent in the stochastic thermostat case.
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