Asymptotic analysis of a Vlasov-Boltzmann equation with anomalous   scaling
  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. 
Pedro Aceves-Sanchez and Antoine Mellet
math.AP