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Diffusion of an Active Particle Bound to a Generalized Elastic Model: Fractional Langevin Equation

Alessandro Taloni

Abstract

We investigate the influence of a self-propelling, out-of-equilibrium active particle on generalized elastic systems, including flexible and semiflexible polymers, fluid membranes, and fluctuating interfaces, while accounting for long-ranged hydrodynamic effects. We derive the fractional Langevin equation governing the dynamics of the active particle, as well as that of any other passive particle (or probe) bound to the elastic system. This equation demonstrates analytically how the active particle dynamics is influenced by the interplay of both the non-equilibrium force and of the viscoelastic environment. Our study explores the diffusional behavior emerging for both the active particle and a distant probe.The active particle undergoes three different surprising and counterintuitive regimes identified by the distinct dynamical time-scales: a pseudo-ballistic initial phase, a drastic decrease of the mobility and an asymptotic subdiffusive regime.

Diffusion of an Active Particle Bound to a Generalized Elastic Model: Fractional Langevin Equation

Abstract

We investigate the influence of a self-propelling, out-of-equilibrium active particle on generalized elastic systems, including flexible and semiflexible polymers, fluid membranes, and fluctuating interfaces, while accounting for long-ranged hydrodynamic effects. We derive the fractional Langevin equation governing the dynamics of the active particle, as well as that of any other passive particle (or probe) bound to the elastic system. This equation demonstrates analytically how the active particle dynamics is influenced by the interplay of both the non-equilibrium force and of the viscoelastic environment. Our study explores the diffusional behavior emerging for both the active particle and a distant probe.The active particle undergoes three different surprising and counterintuitive regimes identified by the distinct dynamical time-scales: a pseudo-ballistic initial phase, a drastic decrease of the mobility and an asymptotic subdiffusive regime.
Paper Structure (10 sections, 43 equations, 1 figure)

This paper contains 10 sections, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Generalized elastic model with active Brownian particle
  3. Fractional Langevin equation
  4. $h$-autocorrelation function
  5. Mean Square Displacement
  6. AOUP's MSD
  7. $t\ll\tau_A$
  8. $t\gg\tau_A$
  9. MSD at a generic position $\vec{x}$
  10. Concluding remarks

Figures (1)

  • Figure 1: MSD of the AOUP. The three situations described in the text, $0<\beta<1/2$ (black curve), $\beta=0$ (red curve) and $1/2<\beta<1$ (green curve) are qualitatively shown. Assuming $\tau_{sub}\gg\tau_A$ the three regimes appear well distinct. After a pseudo- ballistic initial phase, the behaviors in \ref{['MSD_OUN_t>tau_A_xstar']} represent a considerable slowing down of the diffusive dynamics, which is followed by the asymptotic subdiffusive GEM usual behaviour \ref{['MSD_FGN']}.