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

Jacopo Romano* and Andrea Gambassi†

  • *Contact author: jromano@sissa.it
  • †Contact author: gambassi@sissa.it
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Phys. Rev. Lett. 136, 107102 – Published 12 March, 2026

DOI: https://doi.org/10.1103/zn4t-gv6y

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 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.

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Phys. Rev. Lett. 136, 107102– Published 12 March, 2026
  • Received 16 September 2025
  • Revised 16 December 2025
  • Accepted 28 January 2026
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