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Area of Science:

  • Physics
  • Statistical Mechanics
  • Physical Chemistry

Background:

  • Brownian diffusion is fundamental to modeling random processes.
  • Stochastic resetting enhances search efficiency by periodically returning particles to a specific location.
  • Previous studies focused on resetting to a fixed position.

Purpose of the Study:

  • Investigate the optimization of random search processes using Brownian diffusion with stochastic resetting to random positions.
  • Analyze how the distribution of resetting points influences the mean first-passage time and optimal resetting strategy.
  • Characterize the transition between different optimal search regimes.

Main Methods:

  • Analytical study of Brownian diffusion in one dimension with stochastic resetting to random positions.
  • Mathematical analysis of the mean first-passage time.
  • Application of Ginzburg-Landau theory to characterize critical phenomena.
  • Calculation of the probability density function of the last resetting position.

Main Results:

  • The optimal resetting rate can vary smoothly or exhibit a discontinuity depending on the resetting position distribution.
  • A critical line and singular point separate these two regimes.
  • The probability density function of the last resetting position reveals distinct optimal search strategies.
  • Discontinuous transitions lead to strategies favoring either distant likely positions or closer unlikely positions.

Conclusions:

  • Resetting to random positions introduces new complexities and optimization possibilities compared to fixed-position resetting.
  • The shape of the resetting position distribution critically determines the search strategy's effectiveness.
  • Understanding these regimes offers insights into optimizing search processes in various scientific fields.