Anomalous diffusion of self-propelled particles

The transport equation of active motion is generalised to consider time-fractional dynamics for describing the anomalous diffusion of self-propelled particles observed in many different systems. In the present study, we consider an arbitrary active motion pattern modelled by a scattering function that defines the dynamics of the change of the self-propulsion direction. The exact probability density of the particle positions at a given time is obtained. From it, the time dependence of the moments, i.e., the mean square displacement and the kurtosis for an arbitrary scattering function, are derived and analysed. Anomalous diffusion is found with a crossover of the scaling exponent from $2\alpha$ in the short-time regime to $\alpha$ in the long-time one, $0<\alpha<1$ being the order of the fractional derivative considered. It is shown that the exact solution found satisfies a fractional diffusion equation that accounts for the non-local and retarded effects of the Laplacian of the probability density function through a coupled temporal and spatial memory function. Such a memory function holds the complete information of the active motion pattern. In the long-time regime, space and time are decoupled in the memory function, and the time fractional telegrapher's equation is recovered. Our results are widely applicable in systems ranging from biological microorganisms to artificially designed self-propelled micrometer particles.