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This study explores a modified diffusion process where particles reset upon reaching a threshold. We found conditions to optimize net gain in finite domains by balancing reset costs and proximity rewards.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Stochastic Processes

Background:

  • Classic diffusion models describe particle movement over time.
  • Introducing reset mechanisms alters the stationary properties of diffusion.
  • Understanding these modified processes is crucial for applications in various fields.

Purpose of the Study:

  • To analyze a novel diffusion process with particle resetting at a threshold.
  • To investigate the nonstationary behavior and probability distribution in infinite domains.
  • To establish an optimization framework for net gain in finite domains.

Main Methods:

  • Mathematical modeling of the reset diffusion process.
  • Analysis of probability distributions in infinite and finite domains.
  • Derivation of optimization conditions for net gain.

Main Results:

  • The process exhibits nonstationary behavior and rich probability distribution features in infinite domains.
  • A cost-reward optimization problem is defined for finite domains.
  • The condition for optimizing net gain (reward minus cost) is derived.

Conclusions:

  • Reset diffusion presents complex dynamics not seen in classic diffusion.
  • The derived optimization condition provides a framework for practical applications.
  • This work offers insights into controlling and utilizing stochastic processes.