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Non-Poisson renewal events and memory.

Rohisha Tuladhar1, Mauro Bologna2, Paolo Grigolini1

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Summary

This study explores two fluctuation-dissipation processes causing anomalous relaxation. Both exhibit nonexponential Mittag-Leffler relaxation, but differ in crucial event occurrence and trajectory analysis.

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Area of Science:

  • Statistical Mechanics
  • Non-equilibrium Physics
  • Complex Systems Analysis

Background:

  • Anomalous relaxation processes deviate from standard exponential decay towards equilibrium.
  • Fluctuation-dissipation processes are fundamental to understanding system dynamics.
  • Distinguishing between different mechanisms driving anomalous relaxation is crucial for accurate interpretation.

Purpose of the Study:

  • To investigate two distinct fluctuation-dissipation processes leading to anomalous relaxation.
  • To analyze the role of stationary correlations versus crucial events in these processes.
  • To characterize the relaxation dynamics using single trajectory analysis and waiting time probability density functions (PDFs).

Main Methods:

  • Modeling two types of fluctuation-dissipation processes: one with slow stationary correlations, another with non-Poisson renewal events.
  • Analyzing single trajectories by measuring time distances between origin recrossings.
  • Deriving and comparing the waiting time probability density function (ψ(t)) for each process.

Main Results:

  • Both processes demonstrate nonexponential relaxation to equilibrium, characterized by the Mittag-Leffler function.
  • Processes without crucial events show exponential waiting time PDFs.
  • Processes with crucial events exhibit an inverse power-law waiting time PDF with a diverging first moment.

Conclusions:

  • The presence or absence of crucial events fundamentally alters the waiting time statistics of anomalous relaxation processes.
  • Understanding these distinct mechanisms is essential for correctly interpreting experimental observations of anomalous relaxation.
  • The Mittag-Leffler structure unifies these seemingly different relaxation behaviors.