Biased active particles in a three-dimensional tube: Theory, simulations, and validity criterion

We investigate the biased transport of active Brownian particles (ABPs) confined in a three-dimensional (3D) corrugated tube under a constant external bias. Within a sufficiently low-activity regime, we derive a generalized Fick-Jacobs (FJ) equation that reduces the 3D Fokker-Planck equation to an effective one-dimensional (1D) equation under the assumption that particles quickly reach a steady state along the transverse directions of the tube. The transport characteristics of ABPs are computed semi-analytically using the FJ approximation and corroborated by numerical simulation results of the 3D tube. We study the influence of activity, external bias, rotational diffusion constant, and geometric confinement on the transport characteristics of ABPs. We find that ABP activity enhances mobility (μ) and induces a non-monotonic behavior of the effective diffusion (Deff), with a maximum at an intermediate bias. Both tube shape and Dr show a strong impact on the behaviors of μ and Deff. In particular, higher strengths of the rotational diffusion constant Dr favor the particles to reach the steady state quickly, thereby supporting the aforementioned assumption. These findings are well described by the FJ equation for a wider range of parameter strengths, demonstrating its robustness. Furthermore, a validity criterion for the FJ approximation is derived and validated through numerical simulations. These findings are helpful for the design of micro- and nanofluidic devices, enabling enhanced control over particle transport and precise manipulation in technological applications.