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Smoluchowski diffusion equation for active Brownian swimmers

Francisco J. Sevilla1,* and Mario Sandoval2,†

  • 1Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000, México D.F., Mexico
  • 2Department of Physics, Universidad Autonoma Metropolitana-Iztapalapa, Distrito Federal 09340, Mexico

  • *fjsevilla@fisica.unam.mx
  • †sem@xanum.uam.mx
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Phys. Rev. E 91, 052150 – Published 29 May, 2015

DOI: https://doi.org/10.1103/PhysRevE.91.052150

Abstract

We study the free diffusion in two dimensions of active Brownian swimmers subject to passive fluctuations on the translational motion and to active fluctuations on the rotational one. The Smoluchowski equation is derived from a Langevin-like model of active swimmers and analytically solved in the long-time regime for arbitrary values of the Péclet number; this allows us to analyze the out-of-equilibrium evolution of the positions distribution of active particles at all time regimes. Explicit expressions for the mean-square displacement and for the kurtosis of the probability distribution function are presented and the effects of persistence discussed. We show through Brownian dynamics simulations that our prescription for the mean-square displacement gives the exact time dependence at all times. The departure of the probability distribution from a Gaussian, measured by the kurtosis, is also analyzed both analytically and computationally. We find that for the inverse of Péclet numbers ≲0.1, the distance from Gaussian increases as ∼t−2 at short times, while it diminishes as ∼t−1 in the asymptotic limit.

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Phys. Rev. E 91, 052150– Published 29 May, 2015
  • Received 29 January 2015

©2015 American Physical Society

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