In general, electron transport is described by the Boltzmann equation, which is a nonlocal continuity equation in phase space (r, u). The Boltzmann equation is an extremely complicated integrodifferential equation and solving it in an efficient manner is not currently possible. The Boltzmann equation can be approximated by fluid equations by multiplying by a weighting function and then integrating over velocity space. This reduces the governing equations to a three-dimensional, time-dependent problem. The fluid equations describe the electron number density, the mean electron momentum and the mean electron energy as a function of configuration space and time. The rate of change of the electron density is described by:

where ne is the electron density, Γe is the electron flux vector and Re is either a source or a sink of electrons. The rate of change of the electron momentum is described by:

where me is the electron mass (SI unit: kg), ue is the drift velocity of the electrons (SI unit: m/s), pe is the electron pressure tensor (SI unit: Pa), q is the electron change (SI unit: s A), E is the electric field (SI unit: V/m) and νm is the momentum transfer frequency (SI unit: 1/s). The rate of change of the electron energy density is described by:

where nε is the electron energy density (V/m3) and Sen is the energy loss or gain due to inelastic collisions (V/m3 s).

The expression for the flux, Γe in Equation 4-2 is derived from the momentum conservation, Equation 4-3. Under the assumptions that the ionization and attachment frequencies and also the angular frequency are much less than the momentum transfer frequency, the first term on the left hand side of Equation 4-3 can be neglected. Under the assumption that the electron drift velocity is smaller than the thermal velocity, the second term on the left side of Equation 4-3 can also be neglected. For a Maxwellian distribution the pressure term, pe can be replaced using the equation of state:

where I is the identity matrix and Te is the electron temperature. With these assumptions an expression for the electron drift velocity can be derived:

where the electron mobility, μe (SI unit: m2/(V s)) is defined as:

and the electron diffusivity, De (SI unit: m2/s) is defined as:

The drift-diffusion approximation is suitable when the gas pressure is above about 10 mtorr (1.33 Pa) and the reduced electric field is not too high (typically less than 500 Townsends). Furthermore, the number density of charged species should be much less than the number density of the background gas. In other words, the discharge must be weakly ionized. The plasma must also be collisional which means that the mean free path between electrons and the background gas must be much less than the characteristic dimension of the system. This means the drift diffusion approximation is not suitable for modeling fusion plasmas.