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Ergodicity breaking and localization.

Elvis Geneston1, Rohisha Tuladhar2, M T Beig2

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Non-Poisson renewal events combined with fractional Brownian motion or scaled Brownian motion lead to localization. This phenomenon, ergodicity breaking, results in time averages causing particles to become fixed, even with superdiffusion.

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Area of Science:

  • Statistical Physics
  • Stochastic Processes
  • Complex Systems

Background:

  • Continuous-time random walks (CTRW) with non-Poisson renewal events (NPR) exhibit anomalous diffusion.
  • Fractional Brownian motion (FBM) and scaled Brownian motion (SBM) are key models for describing anomalous diffusion with Hurst coefficients (H) not equal to 0.5.
  • Ergodicity breaking is a phenomenon where time averages do not converge to ensemble averages, often leading to localization.

Purpose of the Study:

  • To investigate the combined effects of NPR-driven CTRW and two distinct anomalous diffusion generators (FBM and SBM).
  • To analyze the emergence of ergodicity breaking and its consequences, specifically localization, in these combined systems.
  • To explore the theoretical implications of memory-induced localization in correlated jump processes.

Main Methods:

  • Mathematical modeling of joint stochastic processes involving NPR, CTRW, FBM, and SBM.
  • Analysis of ergodicity breaking conditions for the combined systems.
  • Investigation of time averages and their convergence properties.

Main Results:

  • The joint action of NPR and SBM leads to localization, particularly when SBM alone would exhibit subdiffusion.
  • The joint action of NPR and FBM can induce localization even when FBM alone is a source of superdiffusion.
  • The NPR and FBM combination is equivalent to a CTRW with correlated jumps, suggesting memory effects drive localization.

Conclusions:

  • Combining NPR with SBM or FBM consistently results in ergodicity breaking and particle localization.
  • Localization can occur under conditions (superdiffusion with FBM) that would typically suggest delocalization.
  • Memory-induced localization in correlated jump processes necessitates a re-evaluation of deterministic versus random behavior in theoretical frameworks.