Institute of Applied Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada.
J Chem Phys. 2013 Jul 28;139(4):044113. doi: 10.1063/1.4816279.
We use the spatially homogeneous linear Boltzmann equation to study the time evolution of an initial non-equilibrium distribution function of an ensemble of test particles dilutely dispersed in a background gas at thermal equilibrium. The systems considered are energetic N in He and Xe in He. We employ the quantum mechanical differential cross section to define the collision operator in the Boltzmann equation. The Boltzmann equation is solved with a moment method based on the expansion of the distribution function in the Sonine (Laguerre) polynomials as well as with a direct simulation Monte Carlo method. The moment method provides the approximate eigenvalues and eigenfunctions of the linear Boltzmann collision operator. The reciprocal of the eigenvalues is a measure of the relaxation times to equilibrium. For hard sphere cross sections, the relaxation of the average energy and the shape of the distribution function can be characterized by a single time scale determined by the momentum transfer cross section. We show that this is also the case for realistic quantum cross sections with dominant small angle scattering contributions.
我们使用空间均匀线性玻尔兹曼方程来研究初始非平衡分布函数在热平衡背景气体中稀散的测试粒子系综的时间演化。所考虑的系统是高能 N 在 He 和 Xe 在 He 中。我们采用量子力学微分截面来定义玻尔兹曼方程中的碰撞算子。玻尔兹曼方程通过基于分布函数在 Sonine（Laguerre）多项式中的展开的矩方法以及直接模拟蒙特卡罗方法来求解。矩方法提供了线性玻尔兹曼碰撞算子的近似本征值和本征函数。本征值的倒数是平衡弛豫时间的度量。对于硬球截面，平均能量和分布函数的形状可以由通过动量转移截面确定的单个时间尺度来描述。我们表明，对于具有主导小角度散射贡献的实际量子截面，情况也是如此。
