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Ultra slow sub-logarithmic diffusion of a sluggish random walker subject to resetting with memory

Denis Boyer, Satya N. Majumdar

Abstract

We solve a model of sluggish stochastic motion in which a Brownian particle diffuses with a diffusion coefficient that decays algebraically with the distance to the origin, as $|x|^{-α}$. Additionally, the particle resets with a constant rate $r$ to positions previously visited in the past, so that frequently visited regions are more likely to be revisited. An exact expression is obtained at all times for the position distribution in arbitrary spatial dimensions. At late times, the typical displacement of the walker from the origin grows extremely slowly, as $[\ln(rt)]^{1/(α+2)}$, and the position distribution tends to a scaling law. For any $α>0$, the scaling function has a bimodal shape with a minimum at $x=0$ and has non-Gaussian tails. Although the mean square displacement is hard to compute, some generalized moments of this process can be calculated exactly at all times in one dimension, and are shown to be closely related to the moments of the well-studied model with a constant diffusion coefficient.

Ultra slow sub-logarithmic diffusion of a sluggish random walker subject to resetting with memory

Abstract

We solve a model of sluggish stochastic motion in which a Brownian particle diffuses with a diffusion coefficient that decays algebraically with the distance to the origin, as . Additionally, the particle resets with a constant rate to positions previously visited in the past, so that frequently visited regions are more likely to be revisited. An exact expression is obtained at all times for the position distribution in arbitrary spatial dimensions. At late times, the typical displacement of the walker from the origin grows extremely slowly, as , and the position distribution tends to a scaling law. For any , the scaling function has a bimodal shape with a minimum at and has non-Gaussian tails. Although the mean square displacement is hard to compute, some generalized moments of this process can be calculated exactly at all times in one dimension, and are shown to be closely related to the moments of the well-studied model with a constant diffusion coefficient.
Paper Structure (10 sections, 86 equations, 3 figures)

This paper contains 10 sections, 86 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The model and its exact solution in one dimension
  3. Time-dependent part
  4. Space-dependent part
  5. The full solution
  6. Asymptotic behaviour at late times for general $r$ and $\alpha$
  7. Generalized moments of the position distribution
  8. Generalization to higher dimensions
  9. Summary and Conclusion
  10. Numerical Simulations

Figures (3)

  • Figure 1: Scaling functions of the sluggish random walker in one dimension, given by Eq. (\ref{['scaling.1']}), for various parameter values. The walker follows a Brownian motion where the diffusion coefficient decreases with the distance from the origin as $D(x)=|x|^{-\alpha}$.
  • Figure 2: Moment of order $\alpha+2$ rescaled by $K_{\alpha}\equiv (\alpha+1)(\alpha+2)$ as a function of time, for various values of $\alpha$. The thin lines correspond to simulations with $r=1$, a time-step $\Delta t=10^{-5}$ and averages performed over $10^4$ trajectories, see \ref{['app:simul']} for details. The results are independent of $\alpha$ and follow the evolution predicted by Eq. (\ref{['mom_n1.2']}).
  • Figure 3: Moment ratio defined by the lhs of Eq. (\ref{['int_rel.1']}) as a function of $rt$, obtained from simulations with different values of $\alpha$ (a-b-c). The horizontal lines are given by the rhs of Eq. (\ref{['int_rel.1']}) for $n=1,2,3$. The simulations parameters are the same as in Fig. \ref{['fig:msd']}. For larger values of $n$, the fluctuations become increasingly visible due to the finite number of realizations.