Semidivergence of the Legendre expansion of the Boltzmann equation

The Boltzmann equation for a uniform gas in an electric field is immediately soluble if we neglect the scattering in, and these solutions will be termed SOO functions (for scattering out only). They are complementary functions to the conventional $P_1$ approximation, and for isotropic scattering they satisfy all the Legendre component equations except the zero-order equation. With the taking of the Legendre components of the complete Boltzmann equation, the zero-order equation determines the ratio $\frac{f_1}{f_0}$ and no more, and shows this to be an increasing function of the energy $u$ so that $f_1$ crosses $f_0$ at $u_1$. Above $u_1$, the behavior of the Legendre expansion is defined as semidivergent. The first-order equation relates $f_2$, $f_1$, $f_0$, and as $\frac{f_2}{f_1}\to 0$ at low energies the $P_1$ approximation is valid there, whereas when $\frac{f_0}{f_1}\to 0$ at high energies a SOO function is valid there. This is illustrated by an analytical solution of a simple model in the integral-equation form of the Boltzmann equation.