Beyond the Markovian limit: Exact solutions for active motion in a power-law viscoelastic bath

Active particles from bacteria to synthetic microswimmers often navigate viscoelastic media with complex relaxation dynamics. The classical active Brownian model that assumes instantaneous friction is clearly not applicable to describe such motility, while the non-Markovian processes combined with viscoelasticity are relatively unexplored. Here, we develop an analytical theory for an active particle in a power-law viscoelastic medium by solving coupled non-Markovian generalized Langevin equations for translational and rotational degrees of freedom. The viscoelastic memory results in novel phenomena, such as fractional short-time transport, enhanced long-time persistence, and de-correlation of the instantaneous force and the swimmer orientation. We demonstrate that the memory kernel controls the anomalous scaling exponents, while the activity determines the crossover between sub-diffusive, ballistic, and diffusive regimes. Our work provides a framework for theoretical description of biological and synthetic microswimmers in complex biological and polymeric environments.