# Eliminating Inertia in a Stochastic Model of a Micro-Swimmer with Constant Speed

## TL;DR

This paper derives a simplified stochastic model for a micro-swimmer with constant speed by eliminating orientation inertia, providing a clearer understanding of its long-term positional dynamics.

## Contribution

It introduces a novel adiabatic elimination method for the orientational variable in a stochastic micro-swimmer model, simplifying the analysis of its long-term behavior.

## Key findings

- Derived a reduced stochastic equation for position
- Established equivalence between Langevin and Fokker-Planck approaches
- Provided insights into long-time dynamics of micro-swimmers

## Abstract

We are concerned with the dynamical description of the motion of a stochastic micro-swimmer with constant speed and fluctuating orientation in the long time limit by adiabatic elimination of the orientational variable. Starting with the corresponding full set of Langevin equations, we eliminate the memory in the stochastic orientation and obtain a stochastic equation for the position alone in the overdamped limit. An equivalent procedure based on the Fokker-Planck equation is presented as well.