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Area of Science:

  • Physics
  • Applied Mathematics
  • Statistical Mechanics

Background:

  • The continuous-time random walk (CTRW) model is a fundamental framework for describing anomalous transport phenomena.
  • Analytical and numerical solutions for CTRW models are often limited to specific cases, necessitating further investigation.

Purpose of the Study:

  • To analytically and numerically investigate the uncoupled CTRW model.
  • To derive and analyze the probability density function (PDF) and n-moment for various jump and waiting time distributions.

Main Methods:

  • Utilized analytical and numerical methods to solve the uncoupled CTRW model.
  • Employed exponential and Gaussian functions for jump length PDFs.
  • Applied Mittag-Leffler and exponential/power-law functions for waiting time PDFs.

Main Results:

  • Obtained and analyzed the PDF and n-moment for the CTRW model.
  • Demonstrated that exponential and Gaussian jump length PDFs yield the same second moment but differ near the origin.
  • Showcased that a combination of exponential and power-law waiting time PDFs can induce a crossover from anomalous to normal diffusion regimes.

Conclusions:

  • The behavior of the n-moment is largely independent of the jump length PDF parameter across all time scales for both exponential and Gaussian jump length distributions.
  • The choice of waiting time and jump length distributions significantly influences the CTRW model's emergent dynamics, including anomalous diffusion characteristics.