# Generalized Ornstein-Uhlenbeck Model for Active Motion

## TL;DR

This paper introduces a generalized active Ornstein-Uhlenbeck model incorporating memory effects, providing analytical solutions for velocity autocorrelation and mean-squared displacement, revealing damped oscillations and long-time subdiffusion.

## Contribution

It extends the active Ornstein-Uhlenbeck model by including exponential and power-law memory kernels, offering analytical insights into active particle dynamics with persistent self-propulsion.

## Key findings

- Analytical expressions match numerical simulations well.
- Damped oscillations arise from memory and velocity persistence.
- Long-term memory leads to active subdiffusion in power-law models.

## Abstract

We investigate a one-dimensional model of active motion, which takes into account the effects of persistent self-propulsion through a memory function in a dissipative-like term of the generalized Langevin equation for particle swimming velocity. The proposed model is a generalization of the active Ornstein-Uhlenbeck model introduced by G. Szamel [Phys. Rev. E {\bf 90}, 012111 (2014)]. We focus on two different kinds of memory which arise in many natural systems: an exponential decay and a power law, supplemented with additive colored noise. We provide analytical expressions for the velocity autocorrelation function and the mean-squared displacement, which are in excellent agreement with numerical simulations. For both models, damped oscillatory solutions emerge due to the competition between the memory of the system and the persistence of velocity fluctuations. In particular, for a power-law model with fractional Brownian noise, we show that long-time active subdiffusion occurs with increasing long-term memory.