Distribution and pressure of active Levy swimmers under confinement

Many active matter systems are known to perform Levy 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 active Levy 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 Levy walk model

but now under confinement. We study the density distribution in the channel and

force exerted on the walls by the Levy 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 Levy 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 Levy 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 Levy walks under confinement

with sliding no-flux boundary conditions and provides a foundation for

studies of interacting active Levy swimmers.