Anomalous diffusion of active Brownian particles cross-linked to a networked polymer: Langevin dynamics simulation and theory

Quantitatively understanding the dynamics of an active Brownian particle (ABP) interacting with a viscoelastic polymer environment is a scientific challenge. It is intimately related to several interdisciplinary topics such as the microrheology of active colloids in a polymer matrix and the athermal dynamics of the in vivo chromosomes or cytoskeletal networks. Based on Langevin dynamics simulation and analytic theory, here we explore such a viscoelastic active system in depth using a star polymer of functionality f with the center cross-linker particle being ABP. We observe that the ABP cross-linker, despite its self-propelled movement, attains an active subdiffusion with the scaling similar to t(alpha) with alpha <= 1/2, through the viscoelastic feedback from the polymer. Counter-intuitively, the apparent anomaly exponent alpha becomes smaller as the ABP is driven by a larger propulsion velocity, but is independent of functionality f or the boundary conditions of the polymer. We set forth an exact theory and show that the motion of the active cross-linker is a Gaussian non-Markovian process characterized by two distinct power-law displacement correlations. At a moderate Peclet number, it seemingly behaves as fractional Brownian motion with a Hurst exponent H = alpha/2, whereas, at a high Peclet number, the self-propelled noise in the polymer environment leads to a logarithmic growth of the mean squared displacement (similar to ln t) and a velocity autocorrelation decaying as -t(-2). We demonstrate that the anomalous diffusion of the active cross-linker is precisely described by a fractional Langevin equation with two distinct random noises.