Fractal trace of earthworms.

Area of Science:

• Statistical Physics
• Condensed Matter Physics
• Complex Systems

Background:

• Random walks are fundamental in modeling diffusion and transport.
• Lattice-based models with obstacles introduce complex dynamics and emergent properties.
• Understanding equilibrium states is crucial for predicting system behavior.

Purpose of the Study:

• To analyze the equilibrium properties of a random walk on a 2D lattice with holes and obstacles.
• To determine the distribution of obstacles moved during a single step at equilibrium.
• To characterize the fractal nature of the hole distribution at equilibrium.

Main Methods:

• Simulating a point particle random walk on an n x n square lattice with periodic boundary conditions.
• Introducing a fraction 'p' of holes (macroporosity) and obstacles on the lattice.
• Analyzing the system's approach to equilibrium and the distribution of moved obstacles (M).
• Calculating the fractal dimension of the equilibrium hole distribution.

Main Results:

• The system reaches equilibrium in O(n^2 ln n) steps.
• The distribution of obstacles moved, F(M), follows a power law M(γ) with γ ranging from -1.18 to -1.28.
• The equilibrium distribution of holes exhibits fractal behavior with a dimension 'a' between 1.42 and 1.56.
• The random walker's path forms a distribution with an expected fractal dimension of 2.

Conclusions:

• The dynamics of obstacle movement and hole distribution in this model are governed by power laws and fractal geometry.
• The fractal dimension of the hole distribution is dependent on the macroporosity 'p'.
• Random walks on disordered lattices can lead to complex, self-organized structures at equilibrium.