# Anomalous diffusion and run-and-tumble motion of a chemotactic particle in low dimensions

## TL;DR

This paper investigates the anomalous diffusion behaviors of a self-chemotactic particle in low dimensions, revealing superdiffusive, logarithmic, and normal diffusion regimes depending on the interaction type and spatial dimension.

## Contribution

It introduces a non-linear, non-Markovian model for chemotactic particles and maps the 1D case to a run-and-tumble dynamics, highlighting new diffusion regimes.

## Key findings

- Superdiffusive MSD with exponent 4/3 in 1D repulsive chemotaxis
- Logarithmic growth of MSD in attractive chemotaxis
- Diffusion reverts to normal in 3D with a renormalized coefficient

## Abstract

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 non-linear, non-Markovian effective dynamics, which results into anomalous diffusion for spatial dimensions $d \leq 2$. 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 standard diffusive behavior with a renormalized diffusion coefficient.