(500h) Self-Avoiding Random Walks Generated By Iterated Function Systems

Molecular simulations of polymeric systems commonly rely on the generation of an initial self-avoiding polymer chain configuration. Several initial configurations may be sampled to determine the dependence of thermodynamic, mechanical, or transport properties on configuration and evaluate configurational average properties. A representative set of configurations must be used to obtain accurate estimates of average values and the standard deviation of the estimate.

Chain configurations are generated in a variety of ways including growing the chain from one end and monomer/oligomer polymerization from solution. Additionally, simulations may be limited by computational resources to a unit cell that contains only a portion of the entire chain.

A new approach to generating self-avoiding random walks (SAWs) representative of polymer chain configurations is presented. The walks are generated using iterated function systems (IFSs) like those used to generate fractal geometries [1-2]. IFS systems consist of a set of contraction mappings on a complete metric space. The mappings commonly are affine transformations. Application of the IFS from an arbitrary initial subset of the space converges to the same fixed point (subset of the space). The fixed point is self-similar upon dilation (i.e., possesses similar topology when a portion of the fixed point is expanded) and can possess a non-integral Hausdorff dimension.

The proposed method assumes a SAW possesses self-similarity over a sufficiently broad range of length scales that it can be represented by an appropriate IFS. The IFS systems used here are based on identifying all random walks between two points in a finite regular grid. These walks form the set of contraction mappings for the IFS. Application of the IFS generates a self-avoiding random walk of arbitrary length. The fixed point of the IFS and its Hausdorff dimension are readily calculated. Both depend on the probabilities assigned for application of each contraction mapping to generate a SAW.

Results are presented for two-dimensional random walks. Square and triangular regular grids are used from the smallest possible (three by three) to arbitrarily large grids. Additionally, two methods for assigning probabilities to the contraction mappings are used: 1) uniform change in walk length after each mapping and 2) uniform probability of generating all walks of a fixed length.

The results indicate the Hausdorff dimension depends on the size of the grid used but approaches a fixed value as the size of the grid increases. The dimension approaches 1.5 for both square and triangular grids with probabilities that lead to uniform changes in walk length after each mapping. The dimension approaches 1.0 for both square and triangular grids with probabilities that generate all walks of a fixed length with equal probability.

For a broad range of grid sizes, the Hausdorff dimension lies between 1.3 and 1.4 for IFSs that generate walks of a fixed length with uniform probability. Such a value is comparable to the accepted value for two-dimensional random walks of 1.33. Moreover, the smallest IFS systems yield values between 1.2 and 1.35. Such a finding suggests the SAW IFS accurately captures the statistics of two-dimensional walks.

Extensions of the approach to three-dimensions and use of chain conformation free energy (in addition to the entropic contribution) are discussed.

1. Hutchinson, John E. (1981). "Fractals and self similarity" (PDF). Indiana Univ. Math. J. 30 (5): 713–747. doi:10.1512/iumj.1981.30.30055.
2. Michael Barnsley (1988). Fractals Everywhere, p.82. Academic Press, Inc. ISBN 9780120790623.