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Superdiffusion in self-reinforcing run-and-tumble model with rests.

Sergei Fedotov1, Daniel Han2, Alexey O Ivanov3

  • 1Department of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom.

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This study introduces a run-and-tumble model demonstrating how self-reinforcing directionality and rest periods can lead to superdiffusion. Superdiffusion emerges when mean running time is at least 2/3 of mean resting time.

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Area of Science:

  • Statistical Physics
  • Non-equilibrium Systems
  • Stochastic Processes

Background:

  • Random-walk models are crucial for understanding particle and cell movement.
  • The telegrapher's equation describes motion with constant velocity and instantaneous switching.
  • Superdiffusion, a faster-than-diffusive transport, is observed in various complex systems.

Purpose of the Study:

  • To introduce a novel run-and-tumble model incorporating self-reinforcing directionality and rest states.
  • To investigate the criteria for transitioning between normal diffusion and superdiffusion in this model.
  • To extend the applicability of the telegrapher's equation to include self-reinforcement and resting dynamics.

Main Methods:

  • Derivation of a hyperbolic partial differential equation for probability density.
  • Analytical calculation of the second moment in the long-time limit.
  • Validation through Monte Carlo simulations.

Main Results:

  • Identified criteria for the transition from diffusion to superdiffusion.
  • Superdiffusion emergence depends on self-reinforcement strength and the ratio of mean running to resting times.
  • Superdiffusion is possible only if mean running time is at least 2/3 of mean resting time.

Conclusions:

  • The proposed model successfully generates superdiffusion even with the inclusion of rest states.
  • Self-reinforcing directionality is key to achieving superdiffusion in systems with intermittent movement.
  • This work provides a theoretical framework extending the telegrapher's equation for complex transport phenomena.