  The run-and-tumble (RT) dynamics followed by bacterial swimmers gives rise
first to a ballistic motion due to their persistence, and later, through
consecutive tumbles, to a diffusive process. Here we investigate how long it
takes for a dilute swimmer suspension to reach the diffusive regime as well as
what is the amplitude of the deviations from the diffusive dynamics. A linear
time dependence of the mean-squared displacement (MSD) is insufficient to
characterize diffusion and thus we also focus on the excess kurtosis of the
displacement distribution. Four swimming strategies are considered: (i) the
conventional RT model with complete reorientation after tumbling, (ii) the case
of partial reorientation, characterized by a distribution of tumbling angles,
(iii) a run-and-reverse model with rotational diffusion, and (iv) a RT particle
where the tumbling rate depends on the stochastic concentration of an internal
protein. By analyzing the associated kinetic equations for the probability
density function and simulating the models, we find that for models (ii),
(iii), and (iv) the relaxation to diffusion can take much longer than the mean
time between tumble events, evidencing the existence of large tails in the
particle displacements. Moreover, the excess kurtosis can assume large positive
values. In model (ii) it is possible for some distributions of tumbling angles
that the MSD reaches a linear time dependence but, still, the dynamics remains
non-Gaussian for long times. This is also the case in model (iii) for small
rotational diffusivity. For all models, the long-time diffusion coefficients
are also obtained. The theoretical approach, which relies on eigenvalue and
angular Fourier expansions of the van Hove function, is in excellent agreement
with the simulations.
