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Humboldt-Universität zu Berlin - Mathematisch-Naturwissen­schaft­liche Fakultät - International Research Training Group 1740

Humboldt-Universität zu Berlin | Mathematisch-Naturwissen­schaft­liche Fakultät | Institut für Physik | International Research Training Group 1740 | Publications | Adiabatic elimination of inertia of the stochastic microswimmer driven by α-stable noise

J Noetel, I M Sokolov, and L Schimansky-Geier (2017)

Adiabatic elimination of inertia of the stochastic microswimmer driven by α-stable noise

Physical Review E, 96(4):042610.

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 (α-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 τϕ 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≫τϕ, 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.