In this paper we consider a Boltzmann-type kinetic description of
Follow-the-Leader traffic dynamics and we study the resulting asymptotic
distributions, namely the counterpart of the Maxwellian distribution of the
classical kinetic theory. In the Boltzmann-type equation we include a
non-constant collision kernel, in the form of a cutoff, in order to exclude
from the statistical model possibly unphysical interactions. In spite of the
increased analytical difficulty caused by this further non-linearity, we show
that a careful application of the quasi-invariant limit (an asymptotic
procedure reminiscent of the grazing collision limit) successfully leads to a
Fokker-Planck approximation of the original Boltzmann-type equation, whence
stationary distributions can be explicitly computed. Our analytical results
justify, from a genuinely model-based point of view, some empirical results
found in the literature by interpolation of experimental data.Comment: 18 pages, 7 figure
