Aging two-state process with Lévy walk and Brownian motion.

Abstract

With the rich dynamics studies of single-state processes, the two-state processes are attracting more interest, since they are widely observed in complex system and have effective applications in diverse fields, such as foraging behavior of animals. This paper builds the theoretical foundation of the process with two states: Lévy 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 α_{±}∈(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 α_{±} 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 presented in detail for 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 multistate processes.