Anomalous Diffusion and Run-and-Tumble Motion of a Chemotactic Particle in Low Dimensions

We study the stochastic dynamics of a symmetric self-chemotactic particle and determine the long-time behavior of its mean squared displacement (MSD). The attractive or repulsive interaction of the particle with the chemical field that it generates induces a nonlinear, non-Markovian effective dynamics, which results in anomalous diffusion for spatial dimensions d ≤ 2 . In one spatial dimension, we map the case of repulsive chemotaxis onto a run-and-tumble-like dynamics, leading to an MSD which, as a function of the elapsed time t , grows superdiffusively with exponent 4 / 3 . In the presence of attractive chemotaxis, instead, the particle exhibits a slowdown, with the MSD growing logarithmically with time. In d = 2 , we find logarithmic aging of the diffusion coefficient, while in d = 3 the motion reverts to standard diffusive behavior with a renormalized diffusion coefficient.