Boltzmann's equation

The distribution function $f_{\alpha} ({\bf v}, {\bf x}, t)$ for species $\alpha$ satisfies

$$\frac{\partial f_{\alpha}}{\partial t} + {\bf v }\cdot\frac{\... ...rac{\partial f_{\alpha}} {\partial t} \right)_{\rm coll.} \, .$$ (3.11)

The left-hand side is equal to the total time derivative (in six-dimensional phase space) of the distribution function. The term on the right-hand side treats collisional effects due to fields generated by other particles within a Debye radius $\lambda_{D}$. Collisional effects are only important in this small range because of Debye shielding of the long-range electrostatic force in a plasma. The characteristic frequency of collisions is denoted by $\nu_{e}$.

Due to the presence of many oppositely charged particles around a central positive test charge in a plasma, the scalar electrostatic potential falls off as ${\rm exp}(-r/\lambda_{D})/r$, where the extra exponential factor is due to Debye shielding. The Debye length is given by $\lambda_{D \alpha} = V_{\alpha}/\omega_{p}$, where $\omega_{p}$ is the electron plasma frequency, with $\omega_{p}^{2}=n_{e} e^{2}/m_{e} \varepsilon_{0}$.