We study formation of the Vibrational Distribution Function (VDF) in a molecular gas at low pressure, when vibrational levels are excited by electron impact and deactivated in collisions with walls and show that this problem has a convenient analytical solution that can be used to obtain VDF and its dependence on external parameters. The VDF is determined by excitation of vibrational levels by an external source and deactivation in collisions with the wall. Deactivation in wall collisions is little known process. However, we found that the VDF is weakly dependent on the functional form of the actual form of probability gamma_v'->v for a vibrational number v' to transfer into a lower level v at the wall. Because for a given excitation source of vibrational states, the problem is linear the solution for VDF involves solving linear matrix equation. The matrix equation can be easily solved if we approximate probability, in the form: gamma_v'->v=(1/v')theta(v'->v). In this case, the steady-state solution for VDF(v) simply involves a sum of source rates for levels above , with a factor of 1/(v+1). As an example of application, we study the vibrational kinetics in a hydrogen gas and verify the analytical solution by comparing with a full model.

中文翻译：

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分子与壁碰撞的振动分布函数的便捷解析解

我们研究了在低压下分子气体中振动分布函数 (VDF) 的形成，当振动能级被电子撞击激发并在与壁碰撞时失活，并表明该问题有一个方便的解析解，可用于获得 VDF及其对外部参数的依赖。VDF 由外部源激发振动水平和在与壁碰撞时失活来确定。壁面碰撞中的失活是鲜为人知的过程。然而，我们发现 VDF 弱依赖于实际形式的概率 gamma_v'->v 的函数形式，以便振动数 v' 转移到壁上的较低水平 v。因为对于给定的振动状态激发源，问题是线性的 VDF 的解决方案涉及求解线性矩阵方程。如果我们近似概率，矩阵方程可以很容易地求解，形式为：gamma_v'->v=(1/v')theta(v'->v)。在这种情况下，VDF(v) 的稳态解只涉及高于 的源速率之和，因子为 1/(v+1)。作为应用示例，我们研究了氢气中的振动动力学，并通过与完整模型进行比较来验证解析解。