Generalized Fokker-Planck equation for the active Brownian motion

TL;DR

The paper addresses the challenge of describing active particles in non-Markovian baths by deriving generalized Fokker-Planck equations (FPEs) for inertial active Ornstein-Uhlenbeck particles in viscoelastic media modeled by a Jeffrey fluid kernel. The method casts the dynamics as a linear Gaussian process with a time-dependent covariance matrix $\Xi(t)$, yielding Gaussian phase-space distributions with explicit, memory-dependent coefficients $Q_i(t)$ that reduce to known Markovian results in the appropriate limit. The authors provide closed-form FPEs and PDFs for four configurations: a free particle, a particle in harmonic confinement, a particle in a magnetic field, and a particle in both confinement and a magnetic field, under general memory kernels. This framework enables systematic analysis of relaxation, transport, and external-field responses of active matter in mucus-like or polymeric environments, with potential applications in biological transport and designed active materials.

Abstract

We investigate the dynamics of an inertial active Ornstein-Uhlenbeck particle suspended in a non-Markovian environment. The particle is additionally subjected to external forces, such as harmonic confinement and a magnetic field. Motivated by the importance of understanding the non-Markovian behavior of complex environments, we examine the impact of a viscoelastic medium by employing the Jeffrey fluid framework for modeling the particle motion, which effectively captures both viscous and elastic contributions of the environment. Within this model, we explicitly derive the corresponding Fokker-Planck equation for each case. Building on this, we extend the analysis to general non-Markovian framework and derive the corresponding generalized Fokker-Planck equation for a free active particle. Furthermore, we obtain the probability distribution function valid for arbitrary memory kernel under various conditions, including both free and confined motion with and without a magnetic field. To the best of our knowledge, this represents the first attempt to establish such a comprehensive formalism for the probabilistic description of an active particle subjected to non-Markovian memory effects. This formulation provides a solid basis for analyzing the dynamics of an active particle in a non-Markovian environment, such as mucus and polymer solutions, and further allows the study of relaxation in confined geometries and responses to external fields.