Geometric Properties of Off-Lattice Self-Avoiding Random Walks

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Abstract
Simple geometric properties of self‐avoiding random walks not restricted to lie on a lattice have been investigated. Both the mean‐square end‐to‐end distance and the mean‐square radius of gyration are found to deviate in their exponential dependence upon the number of steps in the walk from values established for lattice‐restricted walks. As the allowed deviation of each step from its lattice‐defined position increases, the exponent increases markedly from its lattice value and then attains a limiting value as free rotation is approached. For all degrees of deviation the exponent is significantly larger than the accepted value of 6/5. Although normal random walks have the same exponential dependence whether restricted to a lattice or not, self‐avoiding random walks behave differently. Thus any generalization to off‐lattice systems of properties calculated from self‐avoiding walks constrained to a lattice may be difficult. Since all linear macromolecules have some rotational freedom, the lattice‐constrained self‐avoiding random walk may be an insufficient model of such systems, even though it is clearly better than the normal random‐walk model.