Random walks with feedback on fractal lattices

Area of Science:

• Complex Systems
• Statistical Physics
• Dynamical Processes

Background:

• Fractal lattices inherently promote subdiffusive random walk behavior.
• Self-organized feedback coupling introduces attractive (lambda>0) or repulsive (lambda<0) dynamics.
• The interplay between fractal geometry and feedback dictates particle transport.

Purpose of the Study:

• To numerically investigate random walk dynamics on fractal lattices with self-organized feedback.
• To analyze how feedback strength (lambda) influences subdiffusion, superdiffusion, and localization.
• To determine the effect of combined fractal and feedback processes on the dynamical exponent (z).

Main Methods:

• Numerical simulations of random walks on fractal lattices (Sierpinski gasket, Sierpinski carpet).
• Introduction of a self-organized feedback coupling with variable strength (lambda).
• Analysis of transport properties and calculation of the dynamical exponent (z) using scaling arguments.

Main Results:

• Repulsive feedback coupling on Sierpinski gaskets and carpets leads to superdiffusion (2/z=1.04 and 2/z=1.08, respectively).
• Attractive feedback coupling results in particle localization, similar to random walks on regular lattices.
• The dynamical exponent (z) is significantly modified by the competitive feedback and fractal structure.

Conclusions:

• The competition between fractal subdiffusion and feedback dynamics determines the overall transport behavior.
• Attractive feedback can induce localization, overriding the subdiffusive tendency of fractal lattices.
• Numerical findings are robustly supported by analytical scaling arguments.