Kinetic Bohm criterion in the Tonks-Langmuir model: Assumption or theorem?

New first integral is found in the collision-free Tonks-Langmuir model. The integral has a clear physical interpretation: the weighted mean inverse kinetic energy of ions, evaluated in the quasineutral approximation, equals $(kTe/2)−1$ at all points in space. This feature is also present in the full (not relying on the assumption of quasineutrality) model: for small values of the Debye length, the weighted mean inverse kinetic energy is with good accuracy equal to $(kTe/2)−1$ in the entire region of quasineutral plasma, including in the vicinity of the space-charge sheath. These results constitute a mathematical proof of the kinetic Bohm criterion and provide a new look at the problem, which has been discussed for several decades. In particular, these results show that the much-debated problem of divergence for slow ions stems from a misinterpretation. Moreover, these results explain why no unique form of kinetic Bohm criterion, modified with the account of ionization and/or collisional and/or geometrical effects in the sheath, has emerged: it cannot be postulated in a nonarbitrary way since there is simply no definite value of the inverse mean kinetic energy with which the ions enter the sheath, if these effects are non-negligible.