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Optimal first-arrival times in Lévy flights with resetting
Łukasz Kuśmierz1, Ewa Gudowska-Nowak2
1Marian Smoluchowski Institute of Physics, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland and AGH University of Science and Technology, Department of Automatics and Biomedical Engineering, Al. Mickiewicza 30, 30-059, Kraków, Poland.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 15, 2015
Summary
This study explores particle random walks with Lévy jumps and resetting, revealing superdiffusive motion. We derived a formula for mean first arrival time (MFAT) to optimize search strategies.
Area of Science:
- Statistical Physics
- Stochastic Processes
Background:
- Random walks are fundamental models for particle motion.
- Lévy flights and resetting mechanisms introduce complex behaviors like superdiffusion.
Purpose of the Study:
- To analyze particle motion with Lévy distributed jump lengths and a resetting mechanism.
- To derive and investigate the mean first arrival time (MFAT) for target reaching.
- To determine criteria for optimizing search strategies based on MFAT.
Main Methods:
- Modeling particle motion as a random walk with Lévy jumps and position resetting.
- Deriving a formula for the mean first arrival time (MFAT).
- Analyzing the long-time and infinite-step limits to identify superdiffusive behavior.
Main Results:
- The process converges to superdiffusive motion with replenishment in the long-time limit.
- A formula for MFAT to a target position was derived.
- Conditions for an optimal (shortest) MFAT were analyzed.
Conclusions:
- Particle motion with Lévy jumps and resetting exhibits superdiffusive characteristics.
- The derived MFAT formula provides a tool for analyzing and optimizing search strategies.
- Understanding MFAT is crucial for designing efficient search processes in complex systems.
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