  Periodic reversals of the direction of motion in systems of self-propelled
rod shaped bacteria enable them to effectively resolve traffic jams formed
during swarming and maximize their swarming rate. In this paper, a connection
is found between a microscopic one dimensional cell-based stochastic model of
reversing non-overlapping bacteria and a macroscopic non-linear diffusion
equation describing dynamics of the cellular density. Boltzmann-Matano analysis
is used to determine the nonlinear diffusion equation corresponding to the
specific reversal frequency. Macroscopically (ensemble-vise) averaged
stochastic dynamics is shown to be in a very good agreement with the numerical
solutions of the nonlinear diffusion equation. Critical density $p_0$ is
obtained such that nonlinear diffusion is dominated by the collisions between
cells for the densities $p>p_0$. An analytical approximation of the pairwise
collision time and semi-analytical fit for the total jam time per reversal
period are also obtained. It is shown that cell populations with high reversal
frequencies are able to spread out effectively at high densities. If the cells
rarely reverse then they are able to spread out at lower densities but are less
efficient at spreading out at at higher densities.
