Aging two-state process with Lévy walk and Brownian motion

Area of Science:

• Complex Systems Dynamics
• Statistical Physics
• Theoretical Biology

Background:

• Single-state process dynamics are well-studied, but two-state processes are increasingly important due to their prevalence in nature.
• These processes, like animal foraging, exhibit complex behaviors with significant real-world applications.
• Previous research highlights Lévy walk and Brownian motion as efficient intermittent search strategies.

Purpose of the Study:

• To establish a theoretical framework for two-state processes involving Lévy walk and Brownian motion.
• To analyze the dynamical behaviors, specifically mean-squared displacements (MSDs), under varying conditions.
• To investigate the influence of sojourn time distributions on the system's long-term characteristics.

Main Methods:

• Modeling two-state processes with heavy-tailed sojourn time distributions (exponents α_{±} ∈ (0,2)).
• Analyzing ensemble-averaged and time-averaged mean-squared displacements (MSDs).
• Calculating the velocity correlation function to understand system dynamics.

Main Results:

• The relationship between the exponents α_{±} dictates the long-term fraction of time spent in each state, significantly impacting MSDs.
• Generic MSD expressions reveal inherent characteristics of the two-state process.
• Detailed analysis of six cases demonstrates the effect of state fractions on observable dynamics.

Conclusions:

• The interplay of Lévy walk and Brownian motion in a two-state system is crucial for understanding complex search dynamics.
• Sojourn time exponent relationships are key determinants of long-term system behavior.
• The developed theoretical techniques for analyzing velocity correlations can be extended to more complex multi-state processes.