# Aging two-state process with L\'{e}vy walk and Brownian motion

## TL;DR

This paper develops a theoretical framework for a two-state aging process combining Lévy walk and Brownian motion, analyzing how heavy-tailed sojourn times influence dynamics and mean squared displacements in complex systems.

## Contribution

It introduces a novel theoretical analysis of two-state processes with heavy-tailed sojourn times, revealing how state fractions affect long-term dynamics and MSDs.

## Key findings

- The state fraction determines long-time behavior of MSDs.
- Heavy-tailed sojourn times lead to anomalous diffusion characteristics.
- Velocity correlation functions can be generalized to other multi-state processes.

## Abstract

With the rich dynamics studies of single-state processes, the two-state processes attract more and more interests of people, since they are widely observed in complex system and have effective applications in diverse fields, say, foraging behavior of animals. This report builds the theoretical foundation of the process with two states: L\'{e}vy walk and Brownian motion, having been proved to be an efficient intermittent search process. The sojourn time distributions in two states are both assumed to be heavy-tailed with exponents $\alpha_\pm\in(0,2)$. The dynamical behaviors of this two-state process are obtained through analyzing the ensemble-averaged and time-averaged mean squared displacements (MSDs) in weak and strong aging cases. It is discovered that the magnitude relationship of $\alpha_\pm$ decides the fraction of two states for long times, playing a crucial role in these MSDs. According to the generic expressions of MSDs, some inherent characteristics of the two-state process are detected. The effects of the fraction on these observables are detailedly presented in six different cases. The key of getting these results is to calculate the velocity correlation function of the two-state process, the techniques of which can be generalized to other multi-state processes.