  This paper is devoted to the approximation of the linear Boltzmann equation
by fractional diffusion equations. Most existing results address this question
when there is no external acceleration field. The goal of this paper is to
investigate the case where a given acceleration field is present. The main
result of this paper shows that for an appropriate scaling of the acceleration
field, the usual fractional diffusion equation is supplemented by an advection
term. Both the critical and supercritical case are considered.
