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Continuous time random walks under Markovian resetting.

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Markovian resetting enhances random walks with power-law waiting and jump times, enabling efficient target searching. An optimal reset rate exists but depends on specific power-law tail relationships.

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Area of Science:

  • Statistical Physics
  • Complex Systems

Background:

  • Continuous time random walks (CTRWs) are fundamental models for anomalous transport.
  • Power-law distributions in waiting times and jump lengths lead to complex, non-Markovian dynamics.
  • Markovian resetting introduces periodic restarts to stochastic processes.

Purpose of the Study:

  • To analyze the impact of Markovian resetting on CTRWs with power-law distributed waiting times and jump lengths.
  • To determine conditions for the existence of a nonequilibrium stationary state and finite mean first arrival time.
  • To investigate the search efficiency and identify optimal random walk parameters for minimizing target acquisition time.

Main Methods:

  • Theoretical analysis of CTRWs with power-law waiting and jump distributions.
  • Mathematical derivation of stationary state properties and mean first arrival time.
  • Optimization techniques to find the optimal reset rate and transport exponents for search efficiency.

Main Results:

  • The study proves the existence of a nonequilibrium stationary state and finite mean first arrival time for the considered CTRW model.
  • An optimal reset rate for efficient searching is shown to exist only under a specific relationship between the power-law exponents.
  • The optimal random walk parameters (reset rate, initial distance, transport exponents) for minimizing mean first arrival time are identified.

Conclusions:

  • Markovian resetting can significantly alter the behavior of CTRWs, leading to a stationary state and finite mean first arrival time.
  • The efficiency of target search via resetting is highly sensitive to the interplay between resetting frequency and the underlying power-law dynamics.
  • This research provides a framework for optimizing search strategies in systems exhibiting anomalous diffusion and resetting phenomena.