  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.
