Fractal structure of a three dimensional Brownian motion on an attractive plane

TL;DR

This study explores how a three-dimensional Brownian particle attracted to a plane exhibits fractal structures, with properties varying based on attraction strength, revealing a transition towards planar Brownian motion characteristics.

Contribution

It introduces a detailed analysis of the fractal geometry of a Brownian particle's visited sites influenced by attraction strength, connecting 3D and 2D Brownian motion properties.

Findings

Fractal dimensions depend on attraction strength α.

Cluster size distribution follows a scaling law with exponent τ.

Perimeter of clusters approaches that of planar Brownian motion as α increases.

Abstract

Consider a Brownian particle in three dimensions which is attracted by a plane with a strength proportional to some dimensionless parameter $\alpha$. We investigate the fractal spatial structure of the visited lattice sites in a cubic lattice by the particle around and on the attractive plane. We compute the fractal dimensions of the set of visited sites both in three dimensions and on the attractive plane, as a function of the strength of attraction $\alpha$. We also investigate the scaling properties of the size distribution of the clusters of nearest-neighbor visited sites on the attractive plane, and compute the corresponding scaling exponent $\tau$ as a function of $\alpha$. The fractal dimension of the curves surrounding the clusters is also computed for different values of $\alpha$, which, in the limit $\alpha\rightarrow\infty$, tends to that of the outer perimeter of planar Brownian motion i.e., the self-avoiding random walk (SAW). We find that all measured exponents depend significantly on the strength of attraction.