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Diffusion with partial resetting.

Ofir Tal-Friedman1, Yael Roichman1,2,3, Shlomi Reuveni2,3,4

  • 1School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 6997801, Israel.

Physical Review. E
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Summary
This summary is machine-generated.

This study introduces partial resetting for stochastic processes, generalizing diffusion models. The research reveals a transition from Laplace to Gaussian distributions, offering new insights into particle motion dynamics.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Probability Theory

Background:

  • Stochastic resetting is a key concept in random process modeling, inspired by natural phenomena.
  • Existing models often assume full resetting to the origin, limiting their applicability.
  • Generalizing resetting mechanisms is crucial for a broader understanding of dynamic systems.

Purpose of the Study:

  • To generalize diffusion with resetting models to incorporate partial resetting.
  • To analyze the steady-state distributions and temporal evolution of such processes.
  • To explore the transition between different distributional forms based on resetting strength.

Main Methods:

  • Mathematical modeling of diffusion with partial resetting.
  • Derivation of steady-state distributions as infinite sums of Laplace random variables.
  • Analysis of drift diffusion with partial resetting, including temporal evolution.
  • Probabilistic construction for closed-form solutions in Fourier-Laplace space.

Main Results:

  • The generalized model always achieves a steady-state distribution.
  • Steady-state distributions transition from Laplace (full resetting) to Gaussian (no resetting) forms.
  • Drift diffusion with partial resetting exhibits similar transitional behavior.
  • A closed-form solution for the time-dependent distribution of drift diffusion with partial resetting was obtained.

Conclusions:

  • Partial resetting provides a more flexible framework for modeling stochastic processes.
  • The study elucidates the relationship between resetting strength and emergent distribution shapes.
  • The findings have implications for understanding and predicting particle dynamics in various systems.
  • The developed methods offer tools for analyzing complex temporal evolutions in stochastic systems.