Aging two-state process with Lévy walk and Brownian motion

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.