  The linear Boltzmann equation approach is generalized to describe fractional
superdiffusive transport of the Levy walk type in external force fields. The
time distribution between scattering events is assumed to have a finite mean
value and infinite variance. It is completely characterized by the two
scattering rates, one fractional and a normal one, which defines also the mean
scattering rate. We formulate a general fractional linear Boltzmann equation
approach and exemplify it with a particularly simple case of the Bohm and Gross
scattering integral leading to a fractional generalization of the Bhatnagar,
Gross and Krook kinetic equation. Here, at each scattering event the particle
velocity is completely randomized and takes a value from equilibrium Maxwell
distribution at a given fixed temperature. We show that the retardation effects
are indispensable even in the limit of infinite mean scattering rate and argue
that this novel fractional kinetic equation provides a viable alternative to
the fractional Kramers-Fokker-Planck (KFP) equation by Barkai and Silbey and
its generalization by Friedrich et al. based on the picture of divergent mean
time between scattering events. The case of divergent mean time is also
discussed at length and compared with the earlier results obtained within the
fractional KFP.
