Distribution and pressure of active L\'evy swimmers under confinement

Abstract

Many active matter systems are known to perform L\'{e}vy walks during migration or foraging. Such superdiffusive transport indicates long-range correlated dynamics. These behavior patterns have been observed for microswimmers such as bacteria in microfluidic experiments, where Gaussian noise assumptions are insufficient to explain the data. We introduce \textit{active L\'evy swimmers} to model such behavior. The focus is on ideal swimmers that only interact with the walls but not with each other, which reduces to the classical L\'evy walk model but now under confinement. We study the density distribution in the channel and force exerted on the walls by the L\'evy swimmers, where the boundaries require proper explicit treatment. We analyze stronger confinement via a set of coupled kinetics equations and the swimmers' stochastic trajectories. Previous literature demonstrated that power-law scaling in a multiscale analysis in free space results in a fractional diffusion equation. We show that in a channel, in the weak confinement limit active L\'evy swimmers are governed by a modified Riesz fractional derivative. Leveraging recent results on fractional fluxes, we derive steady state solutions for the bulk density distribution of active L\'evy swimmers in a channel, and demonstrate that these solutions agree well with particle simulations. The profiles are non-uniform over the entire domain, in contrast to constant-in-the-bulk profiles of active Brownian and run-and-tumble particles. Our theory provides a mathematical framework for L\'evy walks under confinement with sliding no-flux boundary conditions and provides a foundation for studies of interacting active L\'evy swimmers.