Various systems of moment equations---consisting of up to (d+3)(d^2+6d+2)/6 moments---in a general dimension d for a dilute granular gas composed of Maxwell molecules are derived from the inelastic Boltzmann equation by employing the Grad moment method. The Navier--Stokes-level constitutive relations for the stress and heat flux appearing in the system of mass, momentum and energy balance equations are determined from the derived moment equations. It has been shown that the moment equations only for the hydrodynamic field variables (density, velocity and granular temperature), stress and heat flux---along with the time-independent value of the fourth cumulant---are sufficient for determining the Navier--Stokes-level constitutive relations in the case of inelastic Maxwell molecules, and that the other higher-order moment equations do not play any role in this case. The homogeneous cooling state of a freely cooling granular gas is investigated with the system of the Grad (d+3)(d^2+6d+2)/6-moment equations and its various subsystems. By performing a linear stability analysis in the vicinity of the homogeneous cooling state, the critical system size for the onset of instability is estimated through the considered Grad moment systems. The results on critical system size from the presented moment theories are found to be in reasonably good agreement with those from simulations.

中文翻译：

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麦克斯韦分子一维稀颗粒气体的矩理论

由麦克斯韦分子组成的稀颗粒气体的一般维数 d 的各种矩方程组——由最多 (d+3)(d^2+6d+2)/6 个矩组成——从采用梯度矩法的非弹性玻尔兹曼方程。出现在质量、动量和能量平衡方程系统中的应力和热通量的纳维-斯托克斯级本构关系由导出的力矩方程确定。已经表明，仅用于流体动力场变量（密度、速度和颗粒温度）、应力和热通量的矩方程——以及第四个累积量的时间无关值——足以确定 Navier -- 无弹性麦克斯韦分子情况下的斯托克斯级本构关系，并且其他高阶矩方程在这种情况下不起作用。用Grad (d+3)(d^2+6d+2)/6矩方程系统及其各个子系统研究了自由冷却颗粒气体的均匀冷却状态。通过在均匀冷却状态附近执行线性稳定性分析，通过考虑的梯度矩系统估计不稳定开始的临界系统尺寸。发现来自当前矩理论的临界系统尺寸的结果与模拟的结果相当吻合。通过在均匀冷却状态附近执行线性稳定性分析，通过考虑的梯度矩系统估计不稳定开始的临界系统尺寸。发现来自当前矩理论的临界系统尺寸的结果与模拟的结果相当吻合。通过在均匀冷却状态附近执行线性稳定性分析，通过考虑的梯度矩系统估计不稳定开始的临界系统尺寸。发现来自当前矩理论的临界系统大小的结果与模拟的结果相当吻合。