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We developed a general framework for resetting processes, applicable to any interval distribution and underlying system. This approach reveals that stationary distributions and reduced mean first-passage times are achievable even for non-stationary systems.

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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Stochastic Processes

Background:

  • Resetting processes are crucial in various scientific fields.
  • Understanding the impact of resetting on system dynamics is key.
  • Existing models often lack generality.

Purpose of the Study:

  • To introduce a universal mathematical framework for resetting problems.
  • To investigate the conditions for stationary distributions under resetting.
  • To analyze the effect of resetting on first-passage times.

Main Methods:

  • Developed a general mathematical formulation for resetting processes.
  • Analyzed the conditions for the existence of stationary distributions.
  • Investigated the mean first-passage time in resetting systems.
  • Applied the formalism to anomalous diffusion with Poissonian resetting.

Main Results:

  • A stationary distribution can emerge even if the original process is not stationary.
  • Resetting significantly decreases the mean first-passage time.
  • Explicit solutions are derived for anomalous diffusion with Poissonian resetting.

Conclusions:

  • The proposed general formulation provides a powerful tool for studying resetting phenomena.
  • Resetting is an effective strategy for controlling system dynamics and enhancing efficiency.
  • The framework is applicable to a wide range of complex systems and diffusion processes.