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One-dimensional annihilating random walk with long-range interaction.

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Summary
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This study examines the annihilating random walk model with biased hopping. The long-time behavior of particle density decay is classified into seven universal categories based on model parameters.

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Area of Science:

  • Statistical Physics
  • Stochastic Processes
  • Mathematical Physics

Background:

  • Annihilating random walks are fundamental models for systems with particle interactions and removal.
  • Long-range interactions and biased movement introduce complex dynamics not fully understood in standard models.

Purpose of the Study:

  • To analyze the long-time asymptotic behavior of an annihilating random walk model with biased, long-range interactions.
  • To classify the different decay patterns of particle density based on model parameters.

Main Methods:

  • Investigated survival probability and mean spreading for two particles.
  • Analyzed density decay for initially fully occupied systems.
  • Derived classifications of asymptotic behaviors based on parameters ε, μ, and σ.

Main Results:

  • Identified seven distinct categories of asymptotic behaviors for particle density decay.
  • Demonstrated universality in these behaviors, independent of μ and sometimes ε.
  • The interaction parameter ε dictates attraction or repulsion between particles.

Conclusions:

  • The long-time dynamics of this biased annihilating random walk are highly sensitive to the interplay of interaction range (σ) and bias strength (ε).
  • The seven identified categories provide a comprehensive classification of the model's universal asymptotic behaviors.