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This study introduces particle-particle interactions into continuous time random walks (CTRWs), modeling them with a modified Chapman-Kolmogorov equation. Results show CTRWs with interactions converge to standard models over long times, but not under specific boundary conditions.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Nonlinear Dynamics

Background:

  • Continuous time random walks (CTRWs) are widely used for modeling particle diffusion in disordered systems.
  • Existing CTRW models typically neglect particle-particle interactions, limiting their applicability to dense systems.
  • Understanding particle interactions is crucial for accurately describing complex transport phenomena.

Purpose of the Study:

  • To incorporate local particle-particle interactions into the CTRW framework.
  • To derive and analyze a novel time-fractional nonlinear advection-diffusion equation governing interacting particle systems.
  • To investigate the influence of attractive and repulsive interactions on particle dynamics.

Main Methods:

  • Modification of the generalized Chapman-Kolmogorov equation using a perturbation function to model interactions.
  • Derivation of a time-fractional nonlinear advection-diffusion equation from the modified CTRW.
  • Numerical solution of the derived equation under specific initial and free-boundary conditions.
  • Validation of numerical results using Monte Carlo simulations.

Main Results:

  • The derived time-fractional nonlinear advection-diffusion equation successfully models interacting particle systems.
  • Both attractive and repulsive interactions were studied numerically.
  • The interacting CTRW system was found to converge to the conventional CTRW in the long-time limit.
  • Convergence to the conventional model fails when free-boundary conditions and the long-time limit are not simultaneously met.

Conclusions:

  • Particle-particle interactions can be effectively incorporated into CTRW models via modified Chapman-Kolmogorov equations.
  • The derived fractional advection-diffusion equation provides a new tool for studying interacting particle transport.
  • The long-time convergence of interacting CTRWs depends critically on boundary conditions, highlighting the importance of system-specific analysis.