# Two-dimensional active motion

## TL;DR

This paper derives exact analytical expressions for the diffusion behavior of two-dimensional active particles with arbitrary motion patterns, revealing how direction distribution influences transport properties and connecting to generalized diffusion equations.

## Contribution

It provides a comprehensive analytical framework for understanding two-dimensional active particle diffusion with arbitrary direction distributions, including effects of persistence and circular motion.

## Key findings

- Exact formulas for particle position distribution and mean-square displacement.
- Connection established between direction distribution moments and diffusion properties.
- Analysis of persistence and circular motion effects on diffusion behavior.

## Abstract

The diffusion in two dimensions of non-interacting active particles that follow an arbitrary motility pattern is considered for analysis. Accordingly, the transport equation is generalized to take into account an arbitrary distribution of scattered angles of the swimming direction, which encompasses the pattern of motion of particles that move at constant speed. An exact analytical expression for the marginal probability density of finding a particle on a given position at a given instant, independently of its direction of motion, is provided; and a connection with a generalized diffusion equation is unveiled. Exact analytical expressions for the time dependence of the mean-square displacement and of the kurtosis of the distribution of the particle positions are presented. For this, it is shown that only the first trigonometric moments of the distribution of the scattered direction of motion are needed. The effects of persistence and of circular motion are discussed for different families of distributions of the scattered direction of motion.