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Scaling behavior in a nonlocal and nonlinear diffusion equation

Cecconi1, Banavar, Maritan

  • 1International School for Advanced Studies (SISSA/ISAS), Via Beirut 2-4, I-34014 Trieste, ItalyINFM and "The Abdus Salam" International Centre for Theoretical Physics, strada costiera 11, 34100 Trieste, Italy.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|December 2, 2000
PubMed
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This study reveals a universal scaling behavior in a one-dimensional diffusion equation. This transition occurs dynamically between different solutions and can also emerge spontaneously through symmetry transformations.

Area of Science:

  • Mathematical Physics
  • Nonlinear Dynamics
  • Complex Systems

Background:

  • Nonlocal and nonlinear diffusion equations model diverse nonequilibrium processes.
  • Understanding emergent behaviors in these systems is crucial for fields like aggregation and social dynamics.

Purpose of the Study:

  • To analytically investigate a one-dimensional nonlocal and nonlinear diffusion equation.
  • To identify and characterize a dynamical transition and its associated universal scaling behavior.

Main Methods:

  • Analytical studies of the one-dimensional diffusion equation.
  • Tuning initial conditions to observe dynamical transitions.
  • Application of mirror symmetry transformations to the evolving equation.

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Main Results:

  • A dynamical transition with universal scaling behavior was observed between two asymptotic solutions by adjusting initial conditions.
  • This universal scaling behavior was also achieved in a self-organized manner, independent of initial conditions, via temporal evolution under mirror symmetry.

Conclusions:

  • The study demonstrates a robust universal scaling phenomenon in a complex diffusion model.
  • The findings highlight the interplay between initial conditions, symmetry, and emergent universal behaviors in nonlinear systems.