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Time-dependent random walks and the theory of complex adaptive systems
1Department of Condensed Matter Physics, Weizmann Institute, Rehovot 76100, Israel.
Physical Review Letters
|April 12, 2003
Summary
Complex adaptive systems reveal that time-dependent jumping probabilities in random walks can maximize survival. Oscillating probabilities benefit unbiased walks, while constant probabilities aid drifted walkers.
Area of Science:
- Complex adaptive systems
- Statistical mechanics
- Stochastic processes
Background:
- Random walks are fundamental models in various scientific fields.
- Understanding survival probabilities is crucial for predicting system dynamics.
- Complex adaptive systems exhibit emergent behaviors from simple rules.
Purpose of the Study:
- To analyze the dynamics of random walks with time-dependent jumping probabilities.
- To determine the survival probability in the presence of an absorbing boundary.
- To connect these dynamics to phenomena like self-segregation in evolutionary games.
Main Methods:
- Mathematical analysis of random walk models.
- Calculation of survival probabilities under different boundary conditions.
- Investigation of time-dependent jumping probability functions.
Main Results:
- For unbiased random walks, survival probability is maximized with large temporal oscillations in jumping probabilities.
- For random walks drifted towards an absorbing boundary, constant jumping probabilities yield the best survival.
- Identified the underlying dynamics governing self-segregation and clustering in the evolutionary minority game.
Conclusions:
- Time-dependent jumping probabilities offer a mechanism to control survival in random walks.
- The findings provide insights into the collective behavior observed in evolutionary game theory.
- This research bridges theoretical complex adaptive systems with practical applications in evolutionary dynamics.
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