Adiabatic elimination of inertia of the stochastic microswimmer driven by $\alpha-$stable noise

Abstract

We consider a microswimmer that moves in two dimensions at a constant speed and changes the direction of its motion due to a torque consisting of a constant and a fluctuating component. The latter will be modeled by a symmetric L\'evy-stable ($\alpha$-stable) noise. The purpose is to develop a kinetic approach to eliminate the angular component of the dynamics in order to find a coarse grained description in the coordinate space. By defining the joint probability density function of the position and of the orientation of the particle through the Fokker-Planck equation, we derive transport equations for the position-dependent marginal density, the particle's mean velocity and the velocity's variance. At time scales larger than the relaxation time of the torque $\tau_{\phi}$ the two higher moments follow the marginal density, and can be adiabatically eliminated. As a result, a closed equation for the marginal density follows. This equation which gives a coarse-grained description of the microswimmer's positions at time scales $t\gg \tau_{\phi}$, is a diffusion equation with a constant diffusion coefficient depending on the properties of the noise. Hence, the long time dynamics of a microswimmer can be described as a normal, diffusive, Brownian motion with Gaussian increments.