We study the long-time behaviour of light particles, for example
an electron swarm
in which Coulomb interactions are unimportant, subjected to an external
elastic collisions with an inert neutral gas. The time evolution of the
position distribution function is described by a linear Boltzmann equation
The small ratio of electron to neutral masses, ε, makes the energy
them very inefficient. We show that, under suitable scalings, the LBE reduces,
the limit ε→0, to a formally exact equation for the speed (energy)
distribution of the electrons that contains mixed spatial and speed derivatives.
the system is spatially homogeneous, this equation reduces to and thus
for small-enough ε, the commonly used ‘two-term’ approximation.