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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Nonlinear Dynamics

Background:

  • Random walks are fundamental models in physics, but standard models often assume memoryless processes.
  • Understanding diffusion with memory is crucial for modeling complex systems exhibiting aging and nonergodiality.

Purpose of the Study:

  • To investigate the ensemble properties and time-averaged observables of a diffusive-superdiffusive transition induced by memory.
  • To analyze the impact of memory on the nonstationary behavior and aging phenomena in random walks.

Main Methods:

  • A random walker model where transitions depend on a weighted combination of previous steps.
  • Analysis of nonstationary diffusion, aging, and the phenomenon of nonergodicity.
  • Calculation of time-averaged mean squared displacement and response to bias.

Main Results:

  • Ensemble behavior can be normal, superdiffusive, or ballistic depending on memory parameters.
  • Time-averaged mean squared displacement matches normal diffusion, indicating nonergodicity.
  • Finite-duration trajectories show time-averaged displacements dependent on measurement and memory properties.

Conclusions:

  • The study demonstrates a memory-induced transition between diffusive and superdiffusive regimes.
  • The process exhibits aging and nonergodicity, differentiating ensemble averages from time averages.
  • The findings offer insights into systems with complex temporal correlations and memory effects.