In a plasma where the ions and electrons are in thermal equilibrium, in the absence of short-lived induced electromagnetic fields, the densities and can be predicted.
Meanwhile, if we treat the electrons as a gas obeying the ideal gas law , then the pressure leads to another force density :
At equilibrium, we demand that , and isolate this equation for , yielding:
This equation is straightforward to integrate, leading to the following expression for , known as the Boltzmann relation, due to its resemblance to the statistical Boltzmann distribution (see canonical ensemble):
Where the linearity factor represents the electron density for . We can do the same for ions instead of electrons, leading to the following ion density :
But due to their large mass, ions respond much slower to fluctuations in the above equilibrium. Consequently, after a perturbation, the ions spend more time in a non-equilibrium state than the electrons, so this formula for is only valid if the perturbation is sufficiently slow, such that the ions can keep up. Usually, electrons do not suffer the same issue, thanks to their small mass and hence fast response.
- P.M. Bellan, Fundamentals of plasma physics, 1st edition, Cambridge.
- M. Salewski, A.H. Nielsen, Plasma physics: lecture notes, 2021, unpublished.