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Area of Science:

  • Physics
  • Physical Chemistry
  • Statistical Mechanics

Background:

  • Expanding and contracting media are prevalent in diverse scientific fields, influencing transport properties.
  • The continuous-time random-walk (CTRW) model describes anomalous diffusion, but its behavior in dynamic media is less understood.
  • Superimposing medium expansion on particle motion can drastically alter diffusion dynamics.

Purpose of the Study:

  • To characterize the effects of medium expansion on diffusion processes described by the CTRW model.
  • To derive and analyze a general bifractional diffusion equation for expanding media.
  • To investigate specific cases like Lévy flights and subdiffusive CTRWs in dynamic media.

Main Methods:

  • Derivation of a general bifractional diffusion equation for CTRW in expanding media.
  • Analytical solutions for Green's function (propagator) in Lévy flight scenarios.
  • Analysis of moment hierarchies and recurrence relations for subdiffusive CTRWs.
  • Numerical simulations to validate analytical findings.

Main Results:

  • A general bifractional diffusion equation is derived, reducing to normal diffusion in limits.
  • For Lévy flights, an exact propagator solution is found; fast expansion leads to a stationary profile.
  • In contracting media, a
  • big crunch
  • effect (particle localization) is observed for subdiffusive CTRWs.
  • Medium expansion hinders particle mixing, quantified by the
  • Lévy horizon
  • for Lévy flights.

Conclusions:

  • Medium expansion significantly modifies anomalous diffusion, introducing behaviors not seen in static media.
  • The CTRW model in dynamic media offers insights into particle transport, localization, and mixing.
  • The derived bifractional diffusion equation provides a versatile framework for studying diffusion in expanding/contracting environments.