The fractal properties of generalised random walks in one dimension

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Abstract
The properties of a simple random walk and random walks with persistence of velocity, in one dimension, are reported. The finite-fractal properties of the walks are obtained, with exact expressions for indefinitely long walks. Remarkably simple expressions are found for the length, L( epsilon ), at scale epsilon . Computer simulations of these walks are reported for finite trajectories which help to explain the properties of atomic trajectories in realistic liquids determined by molecular dynamics simulation. Some new molecular dynamics simulations are reported for a typical liquid, its coexisting vapour and the crystal, all at the same temperature. The general form of the fractal properties of the three states of matter are well represented by the present one-dimensional model. Indeed, even the quantitative nature is quite well represented, which suggests that results for random walks in three dimensions will be substantially the same.

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