Theoretical and numerical study of fractal dimensionality in self-avoiding walks

It is shown that the concept of fractal dimensionality, recently proposed by Mandelbrot, provides a useful characterization of the configurational properties of a single polymer. From numerical studies of self-avoiding walks, computer generated by Monte Carlo methods, we find that a single-chain configuration possesses a statistical self-similarity property and therefore has a well-defined fractal dimensionality. The fluctuations in fractal dimensionality measured on a single-chain configuration vanish as the number of steps increases. It is shown that renormalization-group theory provides a theoretical basis for the concept of fractal dimensionality in polymers, as well as for its relation to the end-to-end exponent $\nu$.