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We studied time-averaged square displacement (TASD) in continuous-time random walks. TASD grows linearly with lag time and steps, with relative scatter decreasing as 1/sqrt[N].

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Area of Science:

  • Statistical Mechanics
  • Stochastic Processes
  • Physical Chemistry

Background:

  • Continuous-time random walks (CTRWs) are fundamental models for anomalous diffusion.
  • Understanding the statistical properties of CTRWs is crucial for various fields, including physics and finance.
  • Time-averaged square displacement (TASD) is a key observable for characterizing diffusion dynamics.

Purpose of the Study:

  • To analyze the behavior of TASD in CTRWs as a function of the number of steps (N).
  • To investigate the impact of step accumulation and thermal history on TASD fluctuations.
  • To determine the scaling of relative scatter in TASD and its implications for nonlinear features.

Main Methods:

  • Theoretical analysis of TASD for CTRWs.
  • Asymptotic analysis with respect to the number of steps (N) and lag time (τ).
  • Investigation of fluctuation properties and their dependence on N.

Main Results:

  • TASD grows asymptotically linear in lag time (τ) and number of steps (N).
  • Fluctuations in TASD are dominated by N, rendering thermal history fluctuations irrelevant.
  • Relative scatter in TASD decays as 1/sqrt[N], suppressing nonlinear features.

Conclusions:

  • The study provides a rigorous framework for understanding TASD in CTRWs.
  • The findings clarify the dominant factors influencing TASD fluctuations and their scaling behavior.
  • The results are applicable to CTRWs with correlated steps or waiting times.