In a plasma, the Boltzmann relation connects the electron density n_e to the plasma potential φ_pl as follows: The word plasma has a Greek root which means to be formed or molded (the word plastic shares this root). ...

n_e = n₀ exp(eφ_pl/k_BT_e)

The reference for the potential potential is taken to be a position where the electron density is n₀. The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ...

It can be dreived in a particle view by equating the density of states to the physical density and applying the Boltzmann factor. In physics, the Boltzmann factor is a weighting factor determining the relative probability of a system in thermodynamic equilibrium at a temperature T being in a state with energy E: (kB is Boltzmanns constant. ...

Alternatively, it can be derived from the fluid equation for the electrons by equating the force density due to the electron pressure gradient assuming isothermal electrons, , to the force density due to the electric field on the electron charge density, .

In many problems of plasma physics, it is not useful to calculate the electric potential on the basis of the Poisson equation because the electron and ion densities are not known a priori, and if they were, because of quasineutrality the net charge density is the small difference of two large quantities, the electron and ion charge densities. If the ion density is known and the assumptions hold sufficiently well, the electric potential can be calculated simply from the Boltzmann relation. This article is about plasma in the sense of an ionized gas. ...

Discrepancies with the Boltzmann relation can occur, for example, when oscillations occur so fast that the electrons cannot find a new equilibrium (see e.g. plasma oscillations) or when the electrons are prevented from moving by a magnetic field (see e.g. lower hybrid oscillations). In physics, plasma oscillations, often referred to as Langmuir waves or plasma waves, are periodic oscillations of charge density in conducting media such as plasmas or metals. ...