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

We investigate the influence of a self-propelling, out-of-equilibrium active particle ongeneralized elastic systems, including flexible and semi-flexible polymers, fluid membranes, andfluctuating interfaces, while accounting for long-ranged hydrodynamic effects. We derive the frac-tional Langevin equation governing the dynamics of the active particle, as well as that of any otherpassive particle (or probe) bound to the elastic system. This equation analytically demonstrateshow the active particle dynamics is influenced by the interplay of both the non-equilibrium forceand of the viscoelastic environment. Our study explores the diffusional behavior emerging for boththe active particle and a distant probe. The active particle undergoes three different surprising andcounter-intuitive regimes identified by the distinct dynamical time-scales: a pseudo-ballistic initialphase, a drastic decrease in the mobility, and an asymptotic subdiffusive regime.