Fractal Geometry and Brownian Motion: A New Parameter to Describe Molecular Motion

Abstract

The lengths of molecular trajectories in an argonlike liquid simulated by molecular dynamics have been found to be fractal over a substantial range of scales with a Richardson coefficient of $\alpha =0.65\mathrm{}$ (Mandelbrot dimensionality of $d=1.65\mathrm{}$). The results obey a simple empirical relation with three parameters. We believe $\alpha$ to be a useful new parameter to describe the form of trajectories of molecules in fluids where none has previously been available, and that it is a state function.