Convenient analytical solution for vibrational distribution function of molecules colliding with a wall

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 γv′ → 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: γv′ → v = (1/v′) θ(v′ - v) In this case, the steady-state solution for VDF(v) simply involves a sum of source rates for levels above v with a factor of 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.