Fractal trace of earthworms (original) (raw)
We investigate a process of random walk of a point particle on the two-dimensional square lattice of size n × n with periodic boundary conditions. A fraction p ≤ 20% of the lattice is occupied by holes (p represents macro-porosity). A site not occupied by a hole is occupied by an obstacle. Upon a random step of the walker, a number of obstacles, M , can be pushed aside. The system approaches equilibrium in (n log n) 2 steps. We determine the distribution of M pushed in a single move at equilibrium. The distribution F (M) is given by M γ where γ = −1.18 for p = 0.1 decreasing to γ = −1.28 for p = 0.01. Irrespective of the initial distribution of holes on the lattice the final, equilibrium distribution of holes forms a fractal with fractal dimension changing from a = 1.56 for p = 0.20 to a = 1.42 for p = 0.001 (for n = 4, 000). The trace of a random walker forms a distribution with expected fractal dimension 2.
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