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Langevin dynamics for a Lévy walk with memory.

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Memory effects in continuous-time random walks are modeled by correlated waiting times. Stronger correlations suppress diffusion, leading to slower movement regardless of friction.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Stochastic Processes

Background:

  • Memory effects are crucial in many physical systems.
  • Continuous-time random walks (CTRWs) are a standard model for anomalous diffusion.
  • Incorporating memory effects into CTRWs requires modeling correlated waiting times.

Purpose of the Study:

  • To derive the probability distribution of correlated waiting times in CTRWs.
  • To present a Langevin description for Lévy walks with memory.
  • To investigate the impact of correlated waiting times on diffusion properties.

Main Methods:

  • Derivation of two-point probability distributions for correlated waiting times and their inverse.
  • Development of a Langevin description for Lévy walks with correlated waiting times.
  • Analysis of aging and nonstationary properties within the Langevin framework.

Main Results:

  • The proposed model exhibits sub-ballistic superdiffusion with friction and superballistic diffusion/hyperdiffusion without friction.
  • Correlated waiting times were found to suppress diffusion, with stronger correlations leading to slower diffusion.
  • The study analyzed correlation functions, ergodicity, and scaling properties of the stochastic process.

Conclusions:

  • Correlated waiting times significantly alter diffusion dynamics in Lévy-walk-type models.
  • The strength of waiting time correlation is a key parameter controlling diffusion speed.
  • The Langevin approach provides a powerful tool for studying complex stochastic processes with memory.